Basic Theory of Ordinary Differential Equations Universitext

{n) on the interval 20 if r E Zo and

)

(ii) the solution x = qi(t, r, ti, ... , {n) is continuous in (t, r, t1, ... , n) and satisfies the condition t t 'r, 6, W) e(t r O("-1)(t r

(j=1,2,...,n);

II. DEPENDENCE ON DATA

38

(iii) the partial derivative

exists and is continuous in (t, r, Ft,

t8tn

n) on the domain D = {(t, r, 1, ... , {n) : t E To, t1 < T < t2, -')(T)I < p (1 = 1,2,...,n)}; (iv) u = Lo (t, r, dnU

dtn

- rn `J= axff

is the unique solution of the initial-value problem -1 ` tt (t,7,6,... ,Sn)) dtj_1

U(T) = -6, u'(T)

,u(n-2)(7)

u(n_1) =

,Ln).

11-7. Assume that f (t, X1, x2) is a real-valued, continuous, and continuously differentiable function of (t, xl, x2) on an open set A in the (t, x1, x2}space. Assume also that O0(t) is a real-valued solution of the second-order differential equation

x" = f (t, x, x') and (t, Oo(t), 00'(t)) E A on the interval Zo = (t : 0 < t < 1). Set ¢0(0) = a and 0o(0) = b. Denote by t(t,/3) the unique solution of the initial value-problem x" = f (t, x, x'), x(0) = a, x'(0) = ,B, where lb - 81 is sufficiently small. Show that (1,b) > 0 if of 0 for t C013

11-8. Let g(t) be a real-valued and continuous function oft on the interval 0 < t < 1. Also, let A be a real parameter. (1) Show that, if Q(t, A) is a real-valued solution of the boundary-value problem

Ez + (g(t) + A)u = 0,

u(0) = 0, u(1) = 0,

and if 8 &2 (t, A) is continuous on the region A = {(t, A) : 0 < t < 1, a < A < b}, then d(t, A) is identically equal to zero on A, where a and b are real numbers. (2) Does the same conclusion hold if 0(t,,\) is merely continuous on A? 11-9. Let a(x, y) and b(x, y) be two continuously differentiable functions of two variables (x, y) in a domain Do = {(x, y) : IxI < a:, IyI < Q} and let F(x, y, z) be a continuously differentiable function of three variables (x, y, z) in a domain Do = {(x, y, z) : Ixl < a:, IyI < 3, Izl < 7}. Also, let x = f (t, i7), y = 9(t, ij), and z = h(t,17) be the unique solution of the initial-value problem dt

= a(x,y),

x(0) = 0,

L = b(x,y),

= F(x,y,z),

y(0) = n,

z(0) = c(+1). where c(q) is a differentiable function of 11 in the domain 14 = (q: In1 < p). Assume that (t, rl) = (d(x, y), O(x, y)) is the inverse of the relation (x, y) = (f (t,17), g(t, rt)), where we assume that O(z, y) and i,1(x, y) are continuously differentiable with re-

spect to (x, y) in a domain Al = {(x, y)

:

IxI < r, IyI < p}. Set H(x,y) =

h(p(x, y), tp(x, y)). Show that the function H(x, y) satisfies the partial differential equation a(x, y)

+ b(x, y) M = F(x, y, H)

8 y) = c(y). and the initial-condition H(0,

EXERCISES II

39

Hint. Differentiate H(x,y) with respect to (x,y).

11-10. Let F(x, y, z, p, q) be a twice continuously differentiable function (x,y,u,p,q) for (x,y,u,p,q) E 1R5. Also, let x = x(t, s),

z = z(t, s),

y = y(t, s),

p = P(t, s),

of

q = q(t, s)

be the solution of the following system: dx dy

-

OF

_

OF

(x+ y, z, p, q), flq(T, y, z, p, q),

dt dz

OF

OF

dt = pij (x,y,z,P,q) + gaq(x,y,z,P,q),

dp dt

OF

dq dt

OF 8y

OF

8x (T, y, z, P, q) - P 8z (x, y, z, p, q), OF

(x, y, z, P, q) - q az (x, y, z. P, q)

satisfying the initial condition x(O,s) = xo(s),

y(O,s) = yo(s),

p(O,s) = Po(s),

q(O,s) = qo(s),

z(O,s) = zo(s),

where x0 (s), yo(s), zo(s), po(s), and qo(s) are differentiable functions of s on R such

that F(xo(s),yo(s),zo(s),Po(s),go(s))=0,

dzo(s)=Po(s)ds

(s)+go(s)dyo(s)

ds

ds

on R. Show that F(x(t, s), y(t, s), z(t, s), p(t, s), q(t, s)) = 0,

at (t, s) = At, s) 5 (t, s) + q(t, s) flz(t,s) fls

(t, s),

= P(t,s)L(t,s) + q(t,s)ay(t,8) as as

as long as the solution (x, y, z, p, q) exists.

Comment. This is a traditional way of solving the partial differential equation For more details, see (Har2, pp. F(x, y, z, p, q) = 0, where p = 8 and q = 131-143].

H. DEPENDENCE ON DATA

40

II-11. Let H(t, x, y, p, q) be a twice continuously differentiable function of (t, x, y, p, q) in RI. Show that we can solve the partial differential equation Oz

Z 8z\ =0 + H t, x, y,-,

by using the system of ordinary differential equations

8H

dx

dt = dp dt dz

dy 8p'

t=

8H q

,

8H (LL dq _ _ 8H W, 8x ' dt 8H du 8H

= u + p-p + q8q,

dt

OH _ = --6T.

CHAPTER III

NONUNIQUENESS

We consider, in this chapter, an initial-value problem dy

dt = f(t,y1,

(P)

without assuming the uniqueness of solutions. Some examples of nonuniqueness are given in §III-1. Topological properties of a set covered by solution curves of problem (P) are explained in §§III-2 and 111-3. The main result is the Kneser theorem (Theorem III-2-4, cf. [Kn] ). In §1II-4, we explain maximal and minimal solutions

and their continuity with respect to data. In §§III-5 and 111-6, using differential inequalities, we derive a comparison theorem to estimate solutions of (P) and also some sufficient conditions for the uniqueness of solutions of (P). An application of the Kneser theorem to a second-order nonlinear boundary-value problem will be given in Chapter X (cf. §X-1).

111-1. Examples In this section, four examples are given to illustrate the nonuniqueness of solutions of initial-value problems. As already known, problem (P) has the unique solution if f'(t, y) satisfies a Lipschitz condition (cf. Theorem I-1-4). Therefore, in order to create nonuniqueness, f (t, y-) must be chosen so that the Lipschitz condition is not satisfied. Example III-1-1. The initial-value problem dy dt

(III.1.1)

= yt/3

y(to) = 0

has at least three solutions

y(t) = 0

(S.1.1)

(-oo < t < +oc), 3/2

(S.1.2)

[3 t - to)

y(t) _

]I

,

t > to, t < to,

0,

and

[3(t - to), 2

(S.1.3)

y(t) 0,

41

3/2

t > to, t < to

111. NONUNIQUENESS

42

(cf. Figure 1). Actually, the region bounded by two solution curves (S.1.2) and (S.1.3) is covered by solution curves of problem (I11.1.1). Note that, in this case, solution (S.1.1) is the unique solution of problem (P) for t < to. Solutions are not

unique only fort ? to. Example 111-1-2. Consider a curve defined by

y = sin t

(111.1.2)

(-oo < t < +oo)

and translate (111.1.2) along a straight line of slope 1. In other words, consider a family of curves y = sin(t - c) + c,

(111.1.3)

where c is a real parameter. By eliminating c from the relations

dt =

c os(t - c),

y - t = sin(t - c) - (t - c),

we can derive the differential equation for family (111.1.3). In fact, since sin u - u

is strictly decreasing, the relation v = sinu - u can be solved with respect to u to obtain u = G(v) - v, where G(v) is continuous and periodic of period 27r in v, G(2n7r) = 0 for every integer n, and G(v) is differentiable except at v = 2n7r for every integer n. The differential equation for family (111.1.3) is given by dy (II1.1.4)

dt

= 1OS[G(y-t)-(y-t)1.

Since G(2n7r) = 0 and cos(-2n7r) = 1 for every integer n, differential equation (111. 1.4) has singular solutions y = t + 2mr, where n is an arbitrary integer. These lines are envelopes of family (111.1.3) (cf. Figure 2).

FIGURE 1.

FIGURE 2.

Example 111-1-3. The initial-value problem (111.1.5)

dt =

y(to) = 0

has at least two solutions (S.3.1)

y(t) = 0

(-oo < t < +oo),

43

1. EXAMPLES and

4 (t -

(S.3.2)

t,0)2,

t > to

,

y(t) -

t < to

4(t - to)2,

(cf. Figure 3). The region bounded by two solution curves (S.3.1) and (S.3.2) is covered by solution curves of problem (111.1.5).

FIGURE 3.

Consider the following two perturbations of problem dy

dy

=

_

+ E,

y(to) = 0

y2 y2 + C2

y(to) = 0,

lyl

where a is a real positive parameter. Each of these two differential equations satisfies the Lipschitz condition. In particular, the unique solution of problem (111. 1.6) is given by

J4(t-to+2VI -E)2 - E, (S.3.3)

y(t) =

-4(t-to-2f)2

+ E,

t > to t < to

(cf. Figure 4). On the other hand, (S.3.1) is the unique solution of problem (111.1.7). Figure 5 shows shapes of solution curves of differential equation (111.1.7). Note that nontrivial solution of (111.1.7) is an increasing function of t, but it does not reach y = 0 due to the uniqueness.

III. NONUNIQUENESS

44

y

FIGURE 5.

FIGURE 4.

Generally speaking, starting from a differential equation which does not satisfy any uniqueness condition, we can create two drastically different families of curves by utilizing two different smooth perturbations. In other words, a differential equation without uniqueness condition can be regarded as a branch point in the space of differential equations (cf. [KS]). Example 111-1-4. The general solution of the differential equation 2

d) +y2=1

(1II.1.8)

is given by y = sin(t + c),

(111.1.9)

where c is a real arbitrary constant. Also, y = 1 and y = -1 are two singular solutions. Two solution curves (111.1.9) with two different values of c intersect each other. Hence, the uniqueness of solutions is violated (cf. Figure 6). This phenomenon may be explained by observing that (111. 1.8) actually consists of two differential equations: (III.1.10)

dy = dt

1 - yz

and

dy=- l-y2. dt

Each of these two differential equations satisfies the Lipschitz condition for lyI < 1. Figures 6-A and 6-B show solution curves of these two differential equations, respectively. y=I

y=-I FIGURE 6.

FIGURE 6-A.

FIGURE 6-B.

Observe that each of these two pictures gives only a partial information of the complete picture (Figure 6).

2. THE KNESER THEOREM

45

We can regard differential equation (111.1.8) as a differential equation dy

(111.1.11)

dt

=w

on the circle

w2 + y2 = 1.

(111.1.12)

If circle (111.1.12) is parameterized as y = sinu, w = cosu, differential equation (III.1.11) becomes (III.1.13)

d=1

or

cos u = 0 .

Solution curves u = t of (111.1.13) can be regarded as a curve on the cylinder

{(t, y, w) : y =sin u, w = cos u, -oc < u < +oo (mod 2w), -oo < t < +oo} (cf. Figure 7). Figure 6 is the projection of this curve onto (t,y)-plane.

FIGURE 7.

In a case such as this example, a differential equation on a manifold would give a better explanation. To study a differential equation on a manifold, we generally use a covering of the manifold by open sets. We first study the differential equation on each open set (locally). Putting those local informations together, we obtain a global result. Each of Figures 6-A and 6-B is a local picture. If these two pictures are put together, the complete picture (Figure 6) is obtained.

111-2. The Kneser theorem We consider a differential equation

dt = f(t,y-)

(III.2.1)

under the assumption that the R"-valued function f is continuous and bounded on a region (111.2.2)

S2 = {(t, y-)

:

a:5 t < b , Iy1 < +oo }.

Under this assumption, every solution of differential equation (III.2.1) exists on the interval Zfl = {t : a < t < b} if (to, y"(to)) E f2 for some to E Zo (cf. Theorem 1-3-2 and Corollary I-3-4). The main concern in this section is to investigate topological properties of a set which is covered by solution curves of differential equation (111.2.1).

47

2. THE KNESER THEOREM

Theorem 111-2-4 ((Kn]). If A is compact and connected, then SS(A) is also compact and connected for every c E To. Proof.

The compactness of SS(A) was already explained. So, we prove the connectedness only.

Case 1. Suppose that A consists of a point (r, t), where we assume without any loss of generality that r < c. A contradiction will be derived from the assumption that there exist two nonempty compact sets F1 and F2 such that

F1nF2=0.

SS(A)=F1uF2,

(111.2.3)

If t'1 E F1 and 2 E F2, there exist two solutions 1 and 4'2 of (111.2. 1) such that

01(r) _ , 01(c) _ 1, and 42(r) _ ., 02(c) _ 6, (cf. Figure 9). Set

h(µ) _

1(r+11)

{ &T + JJUJ)

for 05µ5c-r, for

-,r)

Let { fk(t, y-) : k = 1, 2,... } be a sequence of R"-valued functions such that (a) the functions fk (k = 1, 2, ...) are continuously differentiable on 0, (b) I fk(t, y1I < M for (t, y) E f2, where M is a positive number independent of (t, yl and k, ra+A = f'uniformly on each compact set in Q (c) k El oo

(cf. Lemma I-2-4). For each k, let &(t,µ) be the unique solution of the initialvalue problem

= fk(t, y-), y(-r+ Iµ4) = h(µ). The solution 15k(t, µ) is continuous

for t E Zo and iµl < c - r. It is easy to show that the family

k=

1, 2,... ; jµj 5 c - r} is bounded and equicontinuous on the interval Io. Note that the functions Z(rk(c, µ) (k = 1, 2, ...) are continuous in µ for 1µI <

c - r and that r'k(c,c - r) = ¢'1(c) E F1 and t k(c,-(c - r)) = &c) E F2. Let d be the distance between two compact sets F1 and F2. Since &(c,µ) is

continuous in p, there exists, for each k, a real number µk such that IµkI < c-r and d. distance(!& (c, µk), F1) = Since the family ilk) : k = 1, 2, ... } is bounded

and equicontinuous on the interval Io, there exists a subsequence j = 1,2.... } such that (i) lim k, _ +oo, (ii) lim µk, = po exists, and (iii) )-.+oo i--+00 urn 1Pk, (t, µk1) = 0(t) exists uniformly on To. It is easy to show that

s-+oo

fi(t) = K(AO) + ft

f (s, ¢(s))ds,

+ {µol

lµo1 < c - r,

distance(¢(c), F1)

=2

Hence, (c) E $.(A) but * (c) 4 F1 U F2. This is a contradiction (cf. Figure 10).

III. NONUNIQUENESS

48

!=c

t=c

FIGURE 10.

FIGURE 9.

Case 2 (general case). Assume (111.2.3) as in Case 1, set AI = AnR((c) x.Fl) and A2 = A n R({c} x F2), where {c} x F; = {(c, yl : y" E Fj } (j = 1, 2). Then, the two sets A, (j = 1, 2) are compact and not empty. Note that A = AI U A2. Since A is connected, we must have AI n A2 0 0. Choose a point (r, {) E AI n A2. Then,

Sc((T,6) = {Ss((T,S)) nF1}U{S,.((T,6) n12} andS,,((r, ))n.Fi 36 0 (j = 1,2). This is a contradiction (cf. Case 1). 0 In order to apply the Kneser theorem (i.e., Theorem III-2-4), it is desirable to remove the boundedness of f from the assumption. To obtain such a refinement of Theorem 111-2-4, consider differential equation (111.2.1) under the following assumptions.

Assumption 1. A set Ao is a compact and connected subset of the region Q such that if ¢(t) is a solution of (111.2.1) satisfying (to, 0(to)) E Ao

(111.2.4)

for some to E To,

then fi(t) exists on To.

As in Definition 111-2-1, denote by Ro the set of all points (t,yr) E Q such that y = fi(t) for some solution 4 of differential equation (111.2.1) satisfying condition (111.2.4). Also, set $S = (9: (c, y-) E Ro) for c E Io.

Assumption 2. The set Ro is bounded. Now, we prove the following theorem.

Theorem 111-2-5. If a compact and connected subset Ao of 12 satisfies Assumptions 1 and 2, then the set Sc is also compact and connected for every c E I. Proof.

Since Ro is bounded, there exists a positive number M such that

Ro c {(t,yj: tETo, Iyl SM) . Set

f(t,yl

9(,t

0=

f ( t,

if j-

y)

tETo, Iyj<2M,

if t E To, ly7 ? 2M.

3. SOLUTION CURVES ON THE BOUNDARY OF R(A)

49

Then, g(t, g) is continuous and bounded on Q. Using the differential equation

dt =

g(t, y), define R(Ao) and SS(Ao) by Definition 111-2-1. Then, we must have

Ro = 1Z(Ao). Otherwise, there would exist a solution ¢o(t) of differential equation (111.2.1) such that (to,&(to)) E Ao for to E Zo and (tt,do(tl)) V Ro for tl E -ToThis is a contradiction. The theorem follows from Theorem 111-2-4. 0

Remark 111-2-6. Any kind of shapes can be made by a wire. Therefore, the set SS(A) needs not to be a convex set even if A is convex. Using Theorem 111-2-4, we can prove the following theorem, which is a refinement of Theorem II-1-4.

Theorem 111-2-7. Assume that the entries of an R"-valued function At' In are continuous in (t, yam) in a domain Do E

Rn+l.

Also, let fi(t) be a solution of system

(111.2.1) on an interval a < t < b. Assume that (t,¢(t)) E Do on the interval a < t < b. Then, for any given positive number e, there exists another solution '(t) of (111.2.1) such that (i) (t, e,1(t)) E Do on the interval a < t < b, (ii) JO(t) - v1'(t)l < e on the interval a < t < b, (iii) a Y(r) # fi(r) for some r on the interval a < t < b. Proof.

Using an idea similar to Step 2 of the proof of Theorem 11-1-2, the local existence

theorem (Theorem 1-2-5), and the connectedness of SS(A), we can complete the proof of Theorem 111-2-7. In fact, subdivide the interval a < t < b (i.e., a = to <

tl <

< tk = b) in such a way that [y(t) - ¢(t)1 < e for t, < t :5 tj+l if g(t)

satisfies (111.2. 1) and Ib"(t,) - i(t,)I < 6. Set A. = {EE E R" : l e - (tj)I < b} and Bl = {y-(t1) : y(to) E Ao}, where g(t) denotes any solution of (111.2.1) with initialvalue y-(to). Since Bl is connected and contains ¢(t1), we must have Bl = if Bl n AI contains only (t1). In such a case, the proof is finished. If Bl n Al contains more than one point, then B1 n A, contains a connected set Sl containing 0(t1) and more than one point. Set B2 = {9(t2) : g(ti) E Si). Since if B2 n A2 contains B2 is connected and contains (t2), we must have B2 = only (t2). In such a case, the proof is finished. In this way, the proof is completed

in a finite number of steps. 0

III-3. Solution curves on the boundary of 1Z(A) We still consider a differential equation (III.3.1)

dt =

under the assumption that the entries of the R°-valued function fare continuous and bounded on a region (111.3.2)

n = {(t, y-) : a < t < b , lyi < +oc}.

M. NONUNIQUENESS

50

As mentioned in §111-2, under this assumption, every solution of differential equation (111.3.1) exists on the interval To = {t : a < t < b} if (to,y-(to)) E ft for some

to E I. Define the sets 1(A) and SS(A) for a subset A of ) by Definition III-2-1. The main concern of this section is to prove the existence of solution curves on the boundary of R(A). We start with the following basic lemma.

Lemma 111-3-1. Suppose that (1) the set A consists of one point (c1, t) (i. e., A = {(c,, f)}), (ii) a < cl < co < c2 < b, and (iii) ij is on the boundary of Sc2(A). Then, SI(B) contains at least a boundary point of S0(A). Let B = Proof A contradiction will be derived from the assumption that SI(B) does not contain

any boundary points of SS (A). Note that SI(B) n 4 (A) 54 0. Set S, = 4 (B) n S,,. (A), S2 = {y : y' E Se. (B) and Ii 0 SS(A)}. Then, S., (B) = 81 U S2 and S, n S2 = 0. It is known that S1 is a nonempty compact set. Also, $2 is closed and bounded, since S,,, (5) is compact and does not contain any boundary points of SS (A). If S2 is not empty, the set Sc.(B) contains a point in Sc,, (A) and another point which does not belong to S,, (A). Then, S,.(8) contains a boundary point of Sc. (A), since S,, (B) is connected. This is a contradiction. Therefore, if it is proved that S2 is not empty, the proof of Lemma 111-3-1 will be completed (cf. Figure 11). To prove that S2 is not empty, observe first that Sam({(a2,()}) nS.,,(A) = 0 if 0 SS,(A) (cf. Figure 12).

FIGURE 11.

FIGURE 12.

Let rj be on the boundary of $, (A). Then, there exists a sequence of points {Sk V Sc, (A) : k = 1, 2, ... ) and a sequence (¢k : k = 1, 2.... ) of solutions of (111.3.1) such that lira (k = '1, O&2) = (k, and mk(co) 0 SS(A). The family k+oo k = 1,2.... } is bounded and equicontinuous on the interval Zo. Therefore, there exists a subsequence { Jk, j = 1, 2, ...) such that lirn k, = +oo and

j+oo

lim bk, (t) = Q(t) exists uniformly on Zo. The limit function ++oo

is a solution of

(111.3.1) such that ¢(c2) = tj. Hence, d(co) E Sc,(B). Since S, (B) does not contain

any boundary points of S,(A), we must have ¢'(co) if $,,(A). This implies that

S2 is not empty. 0 The following theorem is the main result in this section which is due to M. Hukuhara (Hukl( (see also (Huk2J and [HN31).

3. SOLUTION CURVES ON THE BOUNDARY OF R(A)

51

Theorem 111-3-2. Suppose that

(1) a
For a subdivision A : CI = To < Tl < . < Tm-1 < Tm = c2 of the interval cl < t < c2, there exists a solution ¢o of (111.3.1) such that (a) O0(CI) _ ., 4fiG(C2) is on the boundary of R({(cl,O)}), where e = 1,2,... ,m - 1. (p) (Cf. Lemma 111-3-1 and Figure 13.)

FIGURE 13.

Choose a sequence {ok : k = 1, 2, ... } of subdivisions of the interval cl < t < c2 such that

f

Ok : C1 = Tk,O < Tk.1 < ... < Tk,2k-I < Tk,2" = C2, e

1 Tk,t = C1+2k(c2-cl),

e=0,1,...,2k, k=1,2,....

Since Tk+1,21 = Tk,l, we have {Tk,t : e = 0, 1,2,... , 2k} C {Tkrl,e e 2k+1}. Set k =tt jok (k = 1,2,...). Then,

(ak) &(C1) = S, &(C2) = 1,

(Qk) (Tk,t,&(Tk,t)) is on the boundary of where e = 1,2,... , 2k - 1. The family {4k k = 1, 2,... } is bounded and equicontinuous on 10. Hence, there exists a subsequence {&, : j = 1, 2,... } such that lim k, = +oo and that j +00 lira ¢k, exists uniformly on 10. Then, is a solution of (111.3.1) such that i-.+oo

(C2) (Qo) (Tk,t,. k(Tk,t)) is on the boundary of R({(c1i,)}), where f

(ao)

(Cl)

2k - 1 and k = 1,2,.... Since the set {Tk,t : e = 1,2,... 2k - 1; k = 1,2.... } is dense on the interval for cl < t < Cl < t < c2, the curve (t,$(t)) is on the boundary of

C2- 0

III. NONUNIQUENESS

52

Remark 111-3-3. In general, the conclusion of Theorem 111-3-2 does not hold on an interval larger than cl < t < c2.

111-4. Maximal and minimal solutions Consider a scalar differential equation

dt = f (t, y),

(111.4.1)

where t and y are real variables, and assume that f is real-valued, bounded, and continuous on a region H = {(t,y) : a < t < b, -oo < y < +oo}. Choose c in the interval I = (t : a < t < b}. Then, Sr ({ (c, r,) }) is compact and connected for every r E I and every real number r) (cf. Theorem III-2-4). This implies that there exist two numbers 41(r) and ¢2(r) such that ST({(c,17)}) = { y E R : bl(r) < y < &2(r)}. Hence, R({(c,n)}) = {(t,y) E R2 : ¢1(t) < y < 02(t), a < t < b} (cf. Figure 14). The two boundary curves (t,4i(t)) and (t,02(t)) of R({(c,rl)}) are solution curves of differential equation (111.4.1) (cf. Theorem 111-3-2). Every solution ¢(t) of (111.4.1) such that 0(c) = n satisfies the inequalities

01(t) < ¢(t) < ¢2(t) on Z. The solution 02(t) (respectively 01(t)) is called the maximal (respectively minimal) solution of the initial-value problem (111.4.2)

dy

dt = f (t, y),

y(c) = 77

on the interval Z. In this section, we explain the basic properties of the maximal and minimal solutions. Before we define the maximal and minimal solutions more precisely, let us make some observations.

FIGURE 14.

Observation 111-4-1. Assume that f (t, y) is real-valued and continuous on a domain Din the (t,y)-plane. Set lo = {t : a < t < b}. Let 01(t) and ¢2(t) be two solutions of differential equation (111.4.1) such that (t, ¢1(t)) E D and (t, 02(t)) E D

for t E Zo. Note that we do not assume boundedness of f on D. Set 0(t) = max {01(t), 02(t)} for t E Zo. Then, Q(t) is also a solution of (111.4.1) on the interval Zo.

4. MAXIMAL AND MINIMAL SOLUTIONS

53

Proof.

For a fixed to E To, we prove that lim do(t) - O(to)

(111.4.3)

t-.to

= f(to,¢(to)).

t - to

If ¢1(to) > -02(to), then there exists a positive number 6 such that 4(t) = 01(t) for It - tol < J. Therefore, (111.4.3) holds. Similarly, (111.4.3) holds if 02(to) > &(to). Hence, we consider only the case when '(to) _ .01(to) = 02(to). In this case, 0(t) - 0(to) = 0'(t) - 0'(to), where j = 1 or f o r each fixed t E To, we have

t- to

t - to

j = 2, depending on t. Hence, by the Mean Value Theorem on O,(t), there exists r E To such that r --r to as t - to and 0(t) - ¢(to) = f (r, 0,(r)). Also, 03 (r) _

t - to

¢j(to) + f (a,-Oj (a)) (7 - to) for some a E To such that a to as r - to. Since f is bounded on the two curves (t, 01(t)) and (t, 02(t)), (111.4.3) follows immediately. 0

Let f (t, y), D, and To be the same as in Observation 111-4-1. Consider a set F of solutions of differential equation (111.4.1) such that (t, 0(t)) E D on To for every ¢ E F. Assuming that there exists a real number K such that 0(t) < K on lo for every 0 E F, set 00 (t) = sup{Q(t) : ¢ E F} for t E To. Assume also that (t, 00(t)) e D on To. Then, 4'o(t) is a solution of differential

Observation 111-4-2.

equation (111.4.1) on To. Proof.

As in Observation 111-4-1, we prove that (111.4.4)

lim 00(t) - Oo(to) = f (to, bo(to))

t-»to

t - to

for each fixed to E To. Choose three positive numbers po, p, and S so that (i) A = {(t, y) : t E To, Iy - 4o(t)I < po} C D,

(ii) we have (t, 4(t)) E A on the interval t - to I < 6, if d(t) is a solution of (III.4.1) such that 0 < 0o(r) - 4i(r) < p for some r in the interval It - tol < b. There exists a positive number Af such that I f (t, y) I < M on A. Let us fix a point r on the interval It - tol < b. First, we prove the existence of a solution ty(t; r) of (III.4.1) such that I

(111.4.5)

P(r; r) = 0o (r)

and

yJ(t; r) < 4o(t) for

it - tol < b.

To do this, select a sequence {4'k

: k = 1, 2,... } from the family F so that lirao0thk(r) = 40(r). We may assume that (t, hk(t)) E A on It - tol < S for Y+ k (cf. (ii) above). Then, the sequence {4'k : k = 1, 2, ... )is bounded and equicontinuous on It - tot < J. Hence, we may assume that lim 4'k(t) = 0(t; r)

k-»+oo

exists uniformly on the interval It - tol < 6. It is easy to show that V'(t; r) is a solution of (111.4.1) and that (111.4.5) is satisfied. Set ty(t) = max{t/,(t; r), tJ'(t; to)} for Jt - tol < b. Then, v is a solution of (111.4.1)

such that (1) V)(r) = oo(r) and ty(to) = 00(to), (2) iP(t) < 4'o(t) for It - tol < b, and (3) (t, v/ (t)) E A for It - tol < 6 (cf. Figure I5).

III. NONUNIQUENESS

54

Y= #o(') Y s V(1)

FIGURE 15.

O0(T) - ¢040) = t.ti(T) - 0(t0) = f(a,tp(a)) T- to T - to to as r -" to. Since 10(a) - tt'(td)J < for some a such that Ja - tol < 6 and that or Mba - t01, (111.4.4) can be derived immediately. 0

Using this solution tp(t), we obtain

Let us define the maximal (respectively minimal) solution of an initial-value problem {III.4.6

dt = f (t, ii),

)

y(r)

where f is real-valued and continuous on a domain V in the (t, y)-plane and the initial point (T, 1:) is fixed in D.

Definition 111-4-3. A solution tp(t) of problem (111.4.6-() is called the maximal (respectively minimal) solution of problem (111.4.6-() on an interval I = it : T <

t < r' } if (a) tp(t) is defined on I and (t, {1(t)) E D on Z, (b) if 4(t) is a solution of problem (II14.6-{) on a subinterval r < t

r" of Z,

then

0(t) < y(t) (respectively ¢(t) > P(t))

on

r < t < r".

The following two theorems are stated in terms of maximal solutions. Similar results can be stated also in terms of minimal solutions. Such details are left to the reader as an exercise. In the first of the two theorems, we consider another initial-value problem (III.4.6-n)

!LY

dt

= f(t,y),

y(r) = r)

together with problem (111-4.6-{)-

Theorem III-4-4. Suppose that the maximal solution tpt of Problem (IIL4.6-{) exists on an interval Z = {t : r < t < r'}. Then, there exists a positive number So such that the maximal solution 0,1 of problem (III 4.6-r1) exists on I for t; < rl < t + b0. Fjrthermore, tpf(t) < ip,,(t) on I for £ < 17< + 60 and limipR(t) = tb,(t) uniformly on Z.

4. MAXIMAL AND MINIMAL SOLUTIONS

55

Proof

Set 0(p) = {(t, y) : t E Z, yt(t) < y < ot(t) + p}. For a sufficiently small positive number p, we have A(p) C 1), and, hence, if (t, y)I is bounded on A(p) for a sufficiently small positive number p. First, we prove that for a given p > 0, there exists a positive number 6 such that, if f < q < { + b, every solution 0(t) of problem (111.4.6-1?) defined on a subinterval

r < t < r" of I satisfies

0(t) < ,'(t)±p

(III.4.7)

for

r < t <7".

Otherwise, there exist two sequences {qk : k = 1 , 2, ... ) and {rk : k = 1, 2, ... } of real numbers such that (1) qk > {, lim qk = t;, and r < rk < r', (2) 0&-(-r) = nk k -+oo

and Ok(Tk) = v((rk) + p, and (3) Ok(t) < vf(t) + p for r < t < rk. Furthermore, there exists a real number r(p) such that rk > r(p) > r (cf. Figure 16). Y=W4+P

-

1

I

Y

Wt

I

i

I

t=r

t= TA

1=',

FIGURE 16. Set

T
max(0k(t), '' (t)), 11)k (t )

f (t)+p

Tk St ST.

,

It is easy to show that the sequence {0k W : k = 1, 2, ... } is bounded and equicon-

tinuous on the interval Z. Hence, we can assume without any loss of generality that (1)

(2)

tim Tk = rp exists,

k-.+oo

tim 10k(t) = 0(t) exists uniformly on Z.

Note that

(3) d(r) _

and O(ro) _ 'af(ro) + p,

(4) ro?r(p)>r.

It is also easy to show that 0(t) is a solution of (III.4.6-t) on the subinterval r < t < ro of Z. Since 104 is the maximal solution of (III.4.6-1:) on 7, we must have 0(t) < af(t) on the interval r < t < To. This is a contradiction, since 0(ro) = i (ro) + p. Thus, (111.4.7) holds. Now, for a given positive number p, there exists another positive number 6(p) such that (111.4.7) holds for any solution ¢(t) of problem (III.4.6-17) if < ' < t; + 6(p). Hence, using max(4(t), 0((t)), we can extend 6(t) on the interval Z in such a way that (111.4.8)

?Gf(t) 5 y(t) <

p

on

Z.

III. NONUNIQUENESS

56

Let F be the set of all solutions 6(t) of (III.4.6-71) which satisfy condition (111.4.8).

Set t[i,r(t) = sup{¢(t) : 0 E F j for t E I. Then, ty,, is the maximal solution of (III.4.6-r7) on I. Furthermore, 104(t) < 1,1(t) < ot(t) + p if t: < rt < t + 6(p). Letting p - 0, we complete the proof of the theorem. Assuming again that f is continuous on a domain D in the (t, y)-plane, consider initial-value problem (III.4.6-{) together with another initial-value problem

dt =

(III.4.9-E)

y(r)

f (t, y) + E,

where (t, 1;) E V and f is a positive number. We prove the following theorem.

Theorem III-4-5. If the maximal solution tt'(t) of problem (111.4.6-,) on the in-

terval I = {t : r < t < r'} exists, then, for any positive number p, there exists another positive number e(p) such that for 0 < E < E(p), every solution d(t) of (11!.4.9-E) exists on I and tp(t) < d(t) < y',(t) + p on the interval I. In particular, for every sufficiently small positive number f, then exists the maximal solution (111.4.9-E) on I and lim tL, = tV uniformly on I.

tf=, (t) of problem

Proof The maximal solution tI'(t) satisfies the condition (t,>l,(t)) E D on I (cf. Definition 111-4-3). Define a function 9(t, y) by

9(t, y) =

t EI,

f(t,ty+(t)+p),

y>V'(t)+p,

f (t, y), f (t, d,(t)),

do(t) < y < 0(t) + p, t E I, y < rif(t), t E I.

Then, g is continuous and bounded on the domain {t E I, -oo < y < +oo}. Hence, every solution ¢(t, e) of the initial-value problem y(r) _

= 9(t, y) + E,

exists on the interval I. Note that d0(t) = f (t, 0(t)) = 9(t, tl'(t)) < 9(t, tr (t)} + E

(III.4.10)

dt

on I (cf. Figure 17). Hence,

t'(t) < p(t, E)

(111.4.11)

on

t E I.

We prove that for a given positive number p, there exists another positive number e(p) such that if 0 < e <_ E(p), we have 0(t, e) < t1(t)+p on I. Otherwise, there exist a real number r(p) and two sequences {Ek : k = 1, 2.... ) and {rk : k = 1, 2, ... } of real numbers such that (1) ek > O and k

lim Ek = 0, +oo

57

4. MAXIMAL AND MINIMAL SOLUTIONS

(2) T' > rk ? T(p) > T and lim Tk = To exists,

k+00

(3) 0(7k, ek) = IP(rk) + p,

(4) 0(t, (k) < V,(t) + p for r < t < Tk, (5) lim 0(t, ek) = 0(t) exists uniformly on 1.

on the interval r <_ t < ro < T'. Since Then, O(t) is a solution of problem 0(ro) = Ty(ro) + p, this is a contradiction. Thus, it was proved that 0(t) S 0(t, e) < ,y(t) + p on I for 0 < e < e(p). Therefore, Theorem 111-4-5 follows immediately. 0 In the proof of Theorem 111-4-5, (111.4. 11) was derived from (111.4.10) (cf. Figure

17). In a similar manner, we can prove the following result.

Lemma 111-4-6. Assume that f (t, y) is continuous on a region 12 = {(t, y) w_(t) < y < w+(t), t E TO}, where l0 = {t : a < t < b} and (i) w+ and w_ are real-valued, continuous, and differentiable on T0, (ii) w- (t) < w+ (t) on lo. Assume also that dw+(t) dt

J

dw

(t)

dt

and w_(a)

> f(t,w+(t))

on

< Pt"'-(O)

on TO

Zo,

w+(a). Then, every solution 0(t) of the initial-value problem dy dt

(111.4.12)

= f (t, y),

y(a) _

exists on To and w_ (t) < 0(t) < w+(t) on 10 (i.e. (t, y5(t)) E Q). In particular, the maximal solution 01(t) and the minimal solution 02(t) of problem (111.4.12) on lo

exist and w_ (t) < 02 (t) < ¢, (t) < w+ (t) on Zo Proof of this lemma is left to the reader as an exercise (cf. Figure 18). Y = vKW)

v= #(z)

Y=

FIGURE 17.

FIGURE 18.

Using Lemma 111-4-6, we prove the following theorem due to O. Perron (Per2j.

Theorem 111-4-7. Assume that f (t, y) is continuous on a region

fl = {(t, y)

(t) < y < w+(t), t E Zo},

whereto = (t:a
III. NONUNIQUENESS

58

Assume also that dw+(t) dt dw

dt

> f(t ,W+ (t ) )

on

10,

(t) < f(t,w-( t) )

on

4

-

and w_ (a) < 4 < w+(a). Then, there exists a solution Q(t) of initial-value problem (111.4.12) on To such that

w_ (t) < Q(t) < W+(t)

on To,

i.e.,

(t, ¢(t)) E Q

.

Prof. Let us define a function g(t, y) by

g(t, y) =

f(t,W+(t)), f(t, y),

y ? w+(t),

f (t, W_ (t)),

y < W- (t),

W+ (t),

W-

t E To, t E Zo,

t E lo.

Then, g is bounded and continuous on the region {(t, y) : t E 10, -00 < y < +oo). Hence, every solution ¢(t) of the initial-value problem

j = g(t, y),

y(a)

exists on Za.

Set "I+(t,E) = W.}. (t) + E(t - a) and w_(t,e) = w_(t) - e(t - a), where c is a positive number and t E To. Then, +(t, E)

dt dw_(t,E) _ dt

dW+(t) dt

d -(t) dt

+ ( > f(t,w?(t)} + E > g(t,w+(t,e)), - E < f(t,W-(t)) - E < g(t,W_(t,E))-

Hence, w_ (t, c) < ¢(t) < w+ (t, e) on Zo (cf. Lemma 111-4-6). Letting e

0, we

complete the proof of Theorem 111-4-7.

Comment III-4-8. The maximal and minimal solutions given in this section are also defined and explained in [CL, pp. 45-48] and f Har2, p. 251.

111-5. A comparison theorem In this section, we derive an estimate for solutions of a differential equation by means of differential inequalities. Let us introduce the basic assumption.

59

5. A COMPARISON THEOREM

Assumption 1. Let t and u be two real variables and let g(t,u) be a real-valued and continuous function of (t, u) on a domain D in the (t, u)-plane. Also let .y(t) be the maximal solution of the initial-value problem du = g(t,u),

u(a) = uo

on an interval lo = It : a < t < b}, where (a, uo) E D and (t, i,b(t)) E D on To. We first prove the basic theorem given below.

Theorem 111-5-1. Assume that (1) g(t, u) > 0 on D and uo > 0, (2) an R"-valued function ¢'(t) is continuously differentiable on a subinterval I =

it: a< t <,r) of 4, (3)

uo,

(4) (t, 14(t) I) E V on Z, (5)

do(t) dt

< g

on

Z.

Then, P(t)

on

1.

Proof.

Let a be a positive number and let 0(t, e) be any solution of the initial-value problem du

u(a) = u0.

dt = 9(t, u) + e,

If e > 0 is sufficiently small, i/i(t, e) exists on Zo and lim 0(t, e) = P(t) uniformly on To (cf. Theorem III-4-5). Let us make the following observations: (I)

At)I - Im(s)I <-

Jt do

I

ds <

f g(o, I m(o) I )d

and

(II)

0(t, e) - t'(s, e)

t4Y(r,t(a,e)) + e1da,

where t > s. Suppose that l (s)I = r'(s, e) for some s E Z. Then, there exists a positive number b(e) such that Ig(a,+G(o, e)) - 9(o, I$(a)I)I

2

for

Iv - sl < b(e).

This implies that

{

Id(t)I < 0(t, e)

for

s < t < s + b(e),

> 0(s, e)

for

s - b(e) < t
I

111. NONUNIQUENESS

60

(cf. Figure 19).

FIGURE 19.

Note that

- g(a, j¢(o)j))do + e(t - s) t t'(t,e) - WWI > f [g(a,'p(a,e))

for

t>s

a

and

t)! C j [g(o, v(o, )) - g{O, (o )I)do - e(s - t)

for t

t

Thus, it is concluded that I (t) < t'(t,f) on Z. Letting a proof of Theorem III-5-1. 0

0, we complete the

Example 111-5-2. If an R"-valued function 0(t) satisfies the condition

dj(t)

< C + MI¢(t)t

dt

for

a
and

uo,

we can apply Theorem 111-5-1 to fi(t) with g(u) = C + Mu. Note that the initial-

= C + Mu, u(a) = uo has the unique solution

value problem

+G{t) =

uoeM(t-a) + M (eai(t-a)

- i)

.

Hence, W(t)l

<_

M (em('-4) - 1)

for

a < t < b.

We still assume that Assumption 1 is satisfied by the function g(t, u) and O(t) and that uo > 0 and g(t, u) > 0 on D. Consider a differential equation (111.5.1)

dy = f (t, yI dt

under the following assumption: (a) the function f (t, yj is continuous on a domain f2 in the (t, y7)-space, (0) (t, Iyj) E D if (t, y-) E S2,

6. SUFFICIENT CONDITIONS FOR UNIQUENESS

61

(-v) If (t, y-) I 5 g(t, I yl) for (t, yj E 0. Then, the following result is a corollary of Theorem 111-5-1.

Corollary 111-5-3. If fi(t) is a solution of differential equation (111.5.1) on a subinterval a < t < r of the interval Zo such that (t, fi(t)) E 12 for a < t < T and that jd(a)I S uo, then Im(t)I < v' (t) for a < t < T. Proof of this result is left to the reader as an exercise.

111-6. Sufficient conditions for uniqueness In this section, we derive some sufficient conditions for uniqueness by means of differential inequalities. The basic assumption is given below.

Assumption 1. Let t and u be two real variables and let r(t), w(t), and g(t, u) be real-valued functions such that

(1) r(t) and w(t) are continuous on an interval I = {t : a < t < b},

(2) r(t)>0 andw(t)>0 on1. (3) g(t, u) is continuous on the set A = {(t, u) : 0 < u < w(t), t E Z}, (4) g(t, u) > 0 on A and g(t, 0) = 0 on Z, (5) the problem

(111.6.1)

du(t) = g(t,u(t)), dt aim u(t) = 0

(t,u(t)) E A

on Z,

(t, y"(t)) E Q

on Z,

has only the trivial solution u = 0 on Z. Let us consider a problem

(111.6.2)

d9(t) dt lim 19(01 = 0, t-.a r(t)

where S1 = {(t, y) : jyj < w(t), t E Z}. The main result of this section is the following theorem.

Theorem 111-6-1. If the function f (t, y-) is continuous on 1 and if I f(t, y-)I :5 g(t, Iyi)

on

Q,

then problem (111.6.2) has only the trivial solution y" = 6 on Z.

III. NONUNIQUENESS

62

Proof Suppose that problem (III.6.2) has a nontrivial solution j(t) on Z. This means

that (a)

,-a<

for some a E Z. Choose a positive number 6 so that f <

6.

aminbw(t) }.

min

Let us make the following two observations.

Observation 1. Note that the set A(i3) = {(t, u) : a < t < b, 0 < u < B} is a subset ofd and that u = 0 is a solution of the differential equation du

dt =

(111.6.3)

g(t, u)

on the interval a < t < b. Using Theorem 111-2-7, we can construct a nontrivial solution uo(t) of (111.6.3) on the interval a < t < b so that (t, uo(t)) E A(f3) on

a
0 < uo(t) <

(111.6.4)

on a < t < a.

To show this, we first remark that for a given positive number e, any solution t(i(t, c) of the initial-value problem

= g(t, u) + e,

(III.6.5)

satisfies the condition tfi(t, e) < on this subinterval (cf. Figure 21).

u(a) = uo(a)

on any subinterval a < t < a of I if it exists

r=b

r=a I

u= u0(t)

u= 1 ;(r)1

I

I

I

I

1

u ='Kr' E)

-_Ju=P

L U

I

U=0

r=b

TV, El

u

r=a

uo(r)

u=0

FIGURE 21.

FIGURE 20.

Now, define a real-valued function G(t, u) by G(t, u)

(g(t, u),

u > 0, a < t
1

u < 0, a < t < a.

to,

Then, any solution ty(t, e) of problem (III.6.5) can be continued on the interval = G(t, u) + e. Since the set a < t < a as a solution of the differential equation

6. SUFFICIENT CONDITIONS FOR UNIQUENESS

63

{t(i(-, e) : 0 < e < 1) is bounded and equicontinuous on every closed subinterval of a < t < or, we can select a sequence {Ek : k = 1,2.... } of positive numbers such that limEk = 0 and IirW(t,ek) = fi(t) exists uniformly on each closed subinterval k-0 k--0 of a < t < a. It is easy to show that j)(t) is a solution of the initial-value problem du = g(t, u), u(a) = uo(a) which satisfies condition (111.6.4). Thus, we constructed a nontrivial solution u0(t) of differential equation (111.6.3) such that (t,uo(t)) E 0 on I and 0 < uo(t) < ]5(t)l on a < t < a. Since

±L = 0, this contradicts condition (5) of Assumption 1. 0 t-a r(t) Urn

From Theorem III-6-1, we derive the following result concerning uniqueness of solutions.

Theorem 111-6-2. Suppose that Assumption 1 is satisfied and that (i) an 1R"-valued function f (t, y-) is continuous on a domain D in the (t, yl -space, (ii) f satisfies the condition J f (t, 91) - f (f, 92)1 < g(t, Iyi - 92{) whenever t E 1, Iy, - 921:5 w(t), (t, yi) E V, and (t, y2) E D. Let Q(t) be a solution of the differential equation

on I = {t : a < t < b}

dt = f (t, y)

4(t)1 < w(t), t E 1} is contained in D. Then,

such that the set Do = {(t, y-) the problem

dylt) dt

(111.6.6)

= f(t,b(t)),

lun y(t)r

t-a

()

on 1,

(t, y'(t)) E Do 0

has only the solution (t) itself. Proof.

Set l(t) = z(t) + .(t). Then, problem (111-6.6) becomes

di(t) dt

= F(t, i(t)) = A t ' z"(t) + fi(t)) - At' fi(t))

(t 1="(t)!) E o

Since I1 (t,y-Q(t))j < g(t, II1.6.1.

0

on

Z,

and

Hill !AL)!

c-a r(t)

on

Z,

= 0.

on Do, Theorem 111-6-2 follows from Theorem

Let us apply Theorem III-6-2 to some initial-value problems.

Example 1 (The Osgood condition (cf. [Os])). Consider a real-valued function g(t, u) = h(t)p(u) in the case when (1) h(t) is continuous and h(t) > 0 on an interval 1 = {t : a < t < b},

III. NONUNIQUENESS

64

(2) p(u) is continuous on an interval 0 ,.5 u < K, where K is a positive number, (3) p(0) = 0 and p(u) > O for 0 < u < K,

r

(4)

a+

(} (5)

h(t)dt < +oc,

jo+ p(u) = +oo.

Assume that (1) an R"-valued function f (t, y) is continuous on a domain V in the (t, yam)-space, (ii) f satisfies the condition If (t,1%1) - f (t, il2)1 < 9(t, Iii - Y21) = -1721) whenever t E Z, I171 - g2 l < K, (t, y"1) E D, and (t, y2) E D.

Let ¢1(t) and 42(t) be two solutions of an initial-value problem

= f (t, M, y"(a) _

i such that and

(a, T11 E D

(t, 2(t)} E D

(t, p1(t)) E D,

on the interval Zo = {t : a < t < b}. Then, p1(t) = ¢'2(t) on Zo. Proof.

It is sufficient to show that Assumption 1 is satisfied by three functions r(t) = 1,

w(t) = K, and g(t, u) = h(t)p(u), and that lim 1&t)

t-a

1(t))

(t)

= 0. This limit

condition is evidently satisfied, since ¢2(a) = 41(a) and r(t) = 1. Conditions (1), (2), (3), and (4) of Assumption 1 are also evidently satisfied. Therefore, it suffices to prove that condition (5) is also satisfied. Let u(t) be a real-valued function such that

(A) 0
(C) dutt)

= h(t)p(u(t)) on r < t < o.

Then, 1

r(

I

FU (t))

d t)dt =

Jo

h(t) dt.

This contradicts condition (5).

Example 2 (The Lipschitz condition). If we choose h(t) = L and p(u) = u, where L is a positive constant, then the Osgood condition becomes the Lipschitz condition (cf. Assumption 2 of §I-1).

Example 3 (The Nagumo condition). Define r(t), w(t). and g(t, u) by

r(t) = (t -a)',

w(t) = K,

9(t, u) = t L a u,

where A is a non-negative constant and K is a positive constant. Assume also that u > 0 and t > a. Then, the general solution of the differential equation

= g(t, u)

65

6. SUFFICIENT CONDITIONS FOR UNIQUENESS

is given by u(t) = c(t - a)', where c is an arbitrary constant. This implies that Assumption 1 is satisfied by r(t), w(t), and g(t,u). In particular, choosing A = 1, we derive the following result due to M. Nagumo (Nal and Na2J. Assume that (i) f (t, y') is continuous on a domain V in the (t, y-)-space, 191 -921 _ whenever a < t < b, (ii) f satisfies the condition j f(t, 91) - f (t, 92)1 <

a

III, -Y2I
Suppose that 1(t) and y+,2(t) are two solutions of the initial-value problem

dY

dt

=

f (t, y), g(a) = # such that

(a, 1 E V

and

(t, 41(t)) E V.

(t, fi2(t)) E V

on the interval I= {t: a
i and

j _WT

=

(a) = f (a,

then

1*1(t) - 02(t)] = 0.

t-a

Remark 111-6-3. The sufficient conditions of Osgood's [Os] and M. Nagumo's [Nal and Na21 for uniqueness are also explained in [CL. pp. 48-601 and [Hart, pp. 31-351.

Problem 4. Show that the initial-value problem dy dt

_

{tsin(f. 0

if

t 1 0,

if

t = 0,

y(r) = n

has one and only one solution. Answer.

/ \ Note that t sin I tz I - t sin (L2) = yl t

cos (fi) for some y on the interval

between y1 and y2. Therefore, we can use the Lipschitz condition for r - 0, whereas we can use the Nagumo condition at r = 0.

Remark 111-6-4. In order to apply Theorem 111-6-2 to the initial-value problems in Examples 1 and 3, it was assumed that f (t, y-) is continuous at the initial point (a,'qj. However, the reader must notice that problem (111.6.6) is more general than an initial-value problem. Actually, we can apply Theorem 111-6-2 even when f (t, yr) is not continuous at t = a. For example, the Nagumo condition (ii) of

Example 3 is satisfied by the function sin 1 t I at a = 0. Therefore, the problem

{!) = sin (i),

llm -

t-0 t = 0 I has only the trivial solution y(t) = 0.

III. NONUNIQUENESS

66

EXERCISES III

III-1.-Prove the Kneser theorem under the assumption that (i) f'(t, yj is continuous on R = {(t, yj : a < t < b, Myj < +oo), (ii) 111(t, y-)l < K + Llyl on R for some positive numbers K and L.

Hint. Note that if I f (t, yj I < K + LI yj on R for some positive numbers K and L, any solution of the system 9 = f (t, yJ) exists on a < t < b and satisfies the estimate jy(t)l < (jy(a)I + K(t - a))eL(I -a) for a < t < b. 111-2. Find the maximal solution and the minimal solution of the initial-value

problem _, y(0) _ -1 on the interval 0 < t < 10. III-3. Set h(u) _

H(v) = and

1-4t)

- i

for

u > 0,

for

u < 0,

n) hn1J

of Lh \v -H(-v)

for

v > 0,

for

v < 0,

-1
=0

>0

for for

0

for

t = 0

- c(-t)

for

0 < t < 1,

-I+6
,

,

where 0 < d < 1. Show that

(a) v = 0 and v = ± 1 (n = 1,2.... ) are solutions of the differential equation n dv (E)

a

_ c(t) H'(v)

H'(v) = dH

where

dv '

,

(b) solutions of (E) with the initial-values v(-1) = 0 and v(-1) = f 1 are not unique; (c) for any positive constant c,, the differential equation dv

172

ISin

n

(r)

__

I

c(t)

v2 sin (-") +c LH'v) J

vl

satisfies the Lipschitz condition in v for jvi < +oo and Itt < 1. Also, find the maximal solution and the minimal solution of the solutions of (E)

satisfying the initial-condition v(-1) = 0 on the interval -1 < t < 1. Hint. See (KSI.

67

EXERCISES III

111-4. Show that if a continuously differentiable R"-valued function b(t) satisfies an inequality do(t)

then 1¢(t)1 <

I40

<

dt

for

t > 0 and o(0) = 0,

t2 for t > 0.

Hint. Use Theorem 111-5-1.

111-5. Assuming that an R'-valued function j(t) is continuously differentiable for

0 < t < 1, show that if ¢(t) satisfies the condition dd(t)

for

s ia(t)12

dt

0 < t < 1 and lim 1d(t)I = 0, 0+ 9

then ,(t) = 0 for 0 < t < 1. Hint. Use Theorem III-5-1. 111-6. Assuming that an R"-valued function fi(t) is continuously differentiable for

0 < t < +oo, show that if a(t) satisfies the condition d9(t) < 2tlm(t)I dt i

l

for

0 <-

t < +oo,

10(t)i eXpft21 = 0,

then (t) = 0 for 0 < t < +oo. Hint. Use Theorem 111-5-1.

111-7. Show that the problem dy(t) = sin

-

y(t) It1

lim y(t) = 0 (-0

has only the trivial solution y(t) = 0.

Hint. Note that

/

yt sin( It1

for some y between yl and y2.

-sin(/

/

1M l = Y'- Y2cos1

t1)

ltl

!I `t1 j,

M. NONUNIQUENESS

68

111-8. Let f (t, i, y) be an R"-valued function of (t, x, y-) E R x R" x R'". Assume

that (1) the entries of f'(t, i, yl are continuous in the region A

0
a, jxrj < a, jyj < b}, where a, a, and b are fixed positive numbers, (2) there exists a positive number K such that j f (t, x"j, y- )- f (t, x"2, y -)l < Kjxt -x"2 j

if (t,i),y-) E A (j = 1, 2). Let U denote the set of all R"'-valued functions u(t) such that ju(t)j < b for 0 <

t < a and that JiZ(t) - u(r)j < Lit - rj if 0 _< t < a and 0 < r < a, where

L

is a positive constant independent of u E U. Also, let ¢(t; u+) denote the unique u(t)), x(0) = 0', where u" E U. It is solution of the initial-value problem 'jj known that there exists a positive number ao such that for all u E U, the solution

(t, u') exists and

u)j < a for 0 < t < aa. Denote by R the subset of Rn+'

which is the union of solution curves {(t, 0(t, 65)) : 0 < t < ao) for all u E U, i.e.,

R = {(t, ¢(t, u)) : 0 < t < aa, u E U}. Show that R is a closed set in R"+t Hint. See [LM1, Theorem 2, pp. 44-47) and [LM2, Problem 6, pp. 282-283).

111-9. Let f (t, 2, yy be an R"-valued function of (t, a, y) E R x R" x R'". Assume

that (1) the entries of f (t, f, y-) are continuous in the region 0 = {(t, 2, yj : 0 < t < a, Jx"j < a, jy"j < b}, where a, a, and b are fixed positive numbers, (2) there exists a positive number K such that yl- f (t, x2, y-)J < Kjit-i2j

if (t,i1,y)EA (j=1,2). Let U denote the set of all R'"-valued functions g(t) such that I i(t)J < b for 0 <

t < a and that the entries of u are piecewise continuous on the interval 0 <

t < a. Also, let di = f (t, x, u"(t)), dt

(t; u) denote the unique solution of the initial-value problem x(0) = 0, where u" E U. It is known that there exists a positive

number ao such that for all u' E U, the solution (t, u) exists and Jd(t, u)j < a for 0 <- t < ao. Denote by 1Z the subset of R"' which is the union of solution curves {(t, ¢(t, ul)) : 0 < t _< ao) for all u E U, i.e., R = {(t, (t, u)) : 0 < t < ao, u E U). Assume that a point (r, ¢(r, uo)) is on the boundary of R, where 0 < r < 00 and 0 <- t < r} is also on the !!o E U. Show that the solution curve boundary of R. Hint. jLM2, Theorem 3 of Chapter 4 and its remark on pp. 254-257, and Problem 2 on p. 258).

III-10. Let A(t, x) and f (t, x") be respectively an n x n matrix-valued and R"valued functions whose entries are continuous and bounded in (t,x-) E R"+' on a domain 0 = { (t, x) : a < t < b, x" E R"), where a and b are real numbers. Also, assume that (r, {) E A. Show that every solution of the initial-value problem dx

= A(t, x'a + f (t, i), i(r) = t exists on the interval a < t < b.

dt

CHAPTER IV

GENERAL THEORY OF LINEAR SYSTEMS The main topic of this chapter is the structure of solutions of a linear system dt

(LP)

= A(t)f + b(t),

where entries of the n x n matrix A(t) are complex-valued (i.e., C-valued) continuous functions of a real independent variable t, and the Cn-valued function b(t) is continuous in t. The existence and uniqueness of solutions of problem (LP) were given by Theorem 1-3-5. In §IV-1, we explain some basic results concerning n x n matrices whose entries are complex numbers. In particular, we explain the S-N decomposition (or the Jordan-Chevalley decomposition) of a matrix (cf. Definition IV-1-12; also see [Bou, Chapter 7], [HirS, Chapter 6], and [Hum, pp. 17-18]). The S-N decomposition is equivalent to the block-diagonalization which separates distinct eigenvalues. It is simpler than the Jordan canonical form. The basic tools for achieving this decomposition are the Cayley-Hamilton theorem (cf. Theorem IV-1-5) and the partial fraction decomposition of reciprocal of the characteristic polynomial. It is relatively easy to obtain this decomposition with an elementary calculation if all eigenvalues of a given matrix are known (cf. Examples IV-1-18 and IV-1-19). In §IV-2, we explain the general aspect of linear homogeneous systems. Homogeneous systems with constant coefficients are treated in §IV-3. More precisely speaking, we define e'A and discuss its properties. In §IV-4, we explain the structure of solutions of a homogeneous system with periodic coefficients. The main result is the Floquet theorem (cf. Theorem IV-4-1 and [Fl]). The Hamiltonian systems with periodic coefficients are the main subject of §IV-5. The Floquet theorem is extended to this case using canonical linear transformations (cf. [Si4] and [Marl). Also, we go through an elementary part of the theory of symplectic groups. Finally, nonhomogeneous systems and scalar higher-order equations are treated in §1V-6 and §IV-7, respectively. The topics of §§IV-2-IV-4, IV-6, and IV-7 are found also, for example, in [CL, Chapter 3] and [Har2, Chapter IV]. For symplectic groups, see, for example, [Ja, Chapter 6] and [We, Chapters 6 and 8].

IV-1. Some basic results concerning matrices In this section, we explain the basic results concerning constant square matrices. Let Mn(C) denote the set of all n x n matrices whose entries are complex numbers. The set of all invertible matrices with entries in C is denoted by GL(n,C), which stands for the general linear group of order n. We define a topology in Mn (C) by the norm [A] = max [ask[ for A E Mn(C), where a3k is the entry of A on the j-th 1
row and the k-th column; i.e., A =

all

a12

all

a22

and

an2

69

...

aln

a "'

ann

. A matrix A E Mn(C)

IV. GENERAL THEORY OF LINEAR SYSTEMS

70

is said to be upper-triangular if ajk = 0 for j > k. The following lemma is a basic result in the theory of matrices.

Lemma IV-1-1. For each A E Mn(C), there exists a matrix P E GL(n,C) such that P-1 AP is upper- triangular. Proof.

Let A be an eigenvalue of A and pt be an eigenvector of A associated with the eigenvalue A. Then, Apt = A P t and P 1 # 0. Choose n - 1 vectors p", (j = 2, ... , n) so that Q = [#1A p"nJ E GL(n, C), where the p"j are column vectors of the matrix A

Q. Then, the first column vector of Q-1AQ is

0

Hence, ae can complete the

0

proof of this lemma by induction on n.

A matrix A E Mn(C) is said to be diagonal if a,k = 0 for j

k. We denote

by diag(di, d2, ... , d, the diagonal matrix with entries dl, d2, ... , do on the main

diagonal (i.e., d_, = aj.,). A matrix A E Mn(C) is said to be diagonalizable (or semisimple) if there exists a matrix P E GL(n,C) such that P-1AP is diagonal. Denote by Sn the set of all diagonalizable matrices in Mn(C). The following lemma is another basic result in the theory of matrices.

Lemma IV-1-2. A matrix A E Mn(C) is diagonalizable if and only if A has n linearly independent eigenvectors pt, p2,

... , Pn

Proof If A has n linearly independent eigenvectors pt, p'2, ... , p,, set P = [P'tpa ... pnJ

E GL(n,C). Then, P'1AP is diagonal. Conversely, if PAP is diagonal for P = [ 1 6 t h... fl, E GL(n, C), then p1, A, ... , pn 7are n linearly independent eigenvectors of A. In particular, if a matrix A E Mn(C) has n distinct eigenvalues, then n eigenvectors

corresponding to these n eigenvalues, respectively, are linearly independent (cf. [Rab, p. 1861). Therefore, we obtain the following corollary of Lemma IV-1-2.

Corollary IV-1-3. If a matrix A E Mn(C) has n distinct eigenvalues, then A E Sn.

The set Mn(C) is a noncommutative C-algebra. This means that Mn(C) is a vector space over C and a noncommutative ring. The set Sn is not a subalgebra of Mn(C). However, the following lemma shows an important topological property of Sn as a subset of Mn(C).

Lemma IV-1-4. The set Sn is dense in Mn(C). Prioof.

It must be shown that, for each matrix A E Mn(C), there exists a sequence lim Bk = A. To do this, we k +W may assume without any loss of generality that A is an upper-triangular matrix with the eigenvalues At, ... , An on the main diagonal (cf. Lemma IV-1-1). Set {Bk : k = 1,2,... } of matrices in Sn such that

1. SOME BASIC RESULTS CONCERNING MATRICES

71

, Ek,n], where the quantities ek,,, (v = 1, 2, ... , n) are chosen in such a way that n numbers Al + Ek,1, A2 + 4,2, ... , An + Ek,n are distinct and that lim Ek,&, = 0 for v = 1, 2,... , n. Then, by Corollary IV-1-3, we obtain

B k = A + diag[Ek.1, Ek,2,

Bk E Sn and lim Bk = A. k-r+oo

For a matrix A E Mn(C), denote by pA(A) the characteristic polynomial of A with the expansion n

(IV.1.1)

pA(A) = det(AIn - A] = An +

ph(A)An-''

h=1

where In denotes the n x n identity matrix. Note that PA(A)

= An +

Eph(A)An-h,

A° = In.

h=1

Now, let us prove the Cayley-Hamilton theorem (see, for example, [Be13, pp. 200201 and 220], (Cu, p. 220], and (Rab, p. 198]).

Theorem IV-1-5 (A. Cayley-W. R. Hamilton). If A E Mn(C), then its characteristic polynomial satisfies PA(A) = O, where 0 is the zero matrix of appropriate size.

Remark IV-1-6. The coefficients ph(A) of pA(A) are polynomials in entries ap, of the matrix A with integer coefficients. Proof of Theorem IV-1-5.

Since the entries of pA(A) are polynomials of entries a,k of the matrix A, they are continuous in the entries of A. Therefore, if pA(A) = 0 for A E Sn, it is also true for every A E Mn(C), since Sn is dense in Mn(C) (cf. Lemma IV-1-4). Note also that if B = P'1 AP for some P E GL(n, C), then pB(A) = PA(A) and pB(B) = P-'pA(A)P. Therefore, it suffices to prove Theorem IV-1-5 for diagonal matrices. Set A = diag(A1, A2, ... , A.J. Then, pA(A) = (A - A1)(A - A2) ... (A - An) and pA(A) = diag[PA(A1),PA(A2),... ,PA(A.)] = 0. It is an important application of Theorem IV-1-5 that an n x it matrix N satisfies

the condition N" = 0 if its characteristic polynomial pN(A) is equal to A". If N" = O, N is said to be nilpotent. Lemma IV-1-7. A matrix N E Mn(C) is nilpotent if and only if all eigenvalues of N are zero. Proof.

If IV is an eigenvector of N associated with an eigenvalue A of N, then Nkp"= Akp'

for every positive integer k. In particular, N"p = A"p Hence, if N" = 0, then A = 0. On the other hand, if all eigenvalues of N are 0, the characteristic polynomial

pN(A) is equal to A". Hence, N is nilpotent. 0 Applying Lemma IV-1-1 to a nilpotent matrix N, we obtain the following result.

IV. GENERAL THEORY OF LINEAR SYSTEMS

72

Lemma IV-1-8. A matrix N E .Mn(C) is nilpotent if and only if then exists a matrix P E GL(n,C) such that P-I NP is upper-triangular and the entries on the main diagonal of N are all zero. Furthermore, if N is a real matrix, then there exists a real matrix P that satisfies the requirement given above.

To verify the last statement of this lemma, use a method similar to the proof of Lemma IV-1-1 together with the fact that if an eigenvalue of a real matrix is real, then there exists a real eigenvector associated with this real eigenvalue. Details are left to the reader as an exercise. The main concern of this section is to explain the S-N decomposition of a matrix A E Mn(C) (cf. Theorem IV-1-11). Before introducing the S-N decomposition, we need some preparation. Let A. (j = 1, 2,... , k) be the distinct eigenvalues of A and let m., (j = 1,2.... , k) be their respective multiplicities. Then, the characteristic polynomial of the matrix A is given by pA(A) = (A - '\0M'(1\ - A2)m2 ... (A - Ak)m'. Decompose

1

into partial fractions

1

QU(A

=

,

where, for every j, the

(A - A,),n, quantity Q, is a nonzero polynomial in A of degree not greater than m, - 1. Hence, pA(A)

pA(A)

k

I = EQ,(A) J1 (A - Ah)m". Setting )=1

h¢j

P,(A) = Qj(A) 1I (A - A,)me h#1

i=

P2(A}. J=1

Now that this is an identity in A. Therefore, setting (IV.1.4)

Pj(A) = Qj(A)fl(A-AhIn)m"

(y = 1,2,....k),

hv&1

we obtain k

(IV.1.5)

I. _ > Ph(A). h=1

In the following two lemmas, we show that (IV.1.5) is a resolution of the identity in terms of projections Ph (A) onto invariant subspaces of A associated with eigenvalues Ah, respectively.

Lemma IV-1-9. The k matrices P, (A) (j = 1,2,... , k) given by (W-1.4) satisfy the following conditions: (i) A and P, (A) (y = 1, 2, ... , k) commute.

1. SOME BASIC RESULTS CONCERNING MATRICES

73

(ii) (A-),In)'n'Pi(A) =0 (j = 1,2,... ,k), (iii) P,(A)Ph(A) = 0 ifj

h,

k

(iv) >Ph(A) = In, h=1

(v) Pi(A)2=P,(A) (j=1.2,...,k), ( v : ) P, (A)

0 (j = 1, 2, ... , k).

Proof.

Since P,(A) is a polynomial of A, we obtain (i). Using Theorem IV-1-5, we derive (ii) and (iii) from (IV.1.4) and (i). Statement (iv) is the same as (IV.1.5). Multiplying the both sides of (IV.1.5) by P,(A), we obtain k

P,(A) _ >P,(A)Ph(A).

(IV.1.6)

h=1

Then, (v) follows from (IV.1.6) and (iii). To prove (vi), let IT, be an eigenvector of A associated with the eigenvalue Al. Note that (IV.1.2) implies Ph(A)) = 0 if h 0 j. Therefore, we derive P,(A,) = 1 from (IV.1.3). Now, since P. (A)#, = P,(A3)p' # we obtain (vi).

Lemma IV-1-10. Denote by V. the image of the mapping P,(A) : C" -. Cn. Then,

(1) p'E Cn belongs to V. if and only if P,(A)p= p (2) Pj(A)p"=0 for all fl E Vh if j 0 h. (3) Cn = V1 Ei3 V2 e

e Vk (a direct sum).

(4) for each j, V, is an invariant subspace of A. (5) the restriction of A on V, has a coordinates-wise representation: (IV.1.7)

Alv,

:

AjIj + A,,

where I. is the identity matrix and Nj is a nilpotent matrix. (6) dime V, = m, . Proof Each part of this lemma follows from Lemma IV-1-9 as follows. A vector IT E V3 if and only if p" = P, (A)q" for some q' E Cn. If p" = P, (A) q, we

obtain P,(A)p= Pj(A)2q'= Pj(A)q =p""from (v) of Lemma IV-1-9. A vector p" E Vh if and only if ff = Ph (A),y for some q" E Cn. Hence, from (iii) of Lemma IV-1-9 we obtain P, (A)IF = Pj (A)Ph(A)q"= 0 if 0 h. (iv) of Lemma IV-1-9 implies p = P, (A)15 + + Pk- (A)p" for every p3 E C", while (1) implies that P, (A)p E Vj. On the other hand, if p" = )51 + + pk for some g,E V2 (j = 1,2,... , k), then, by (1) and (2), we obtain P,(A)p" = Pi(A)p1+...+PJ(A)pk =pj

Ap"= AP,(A)p = P,(A)Ap E V3 for every 15E Vj. Let n, be the dimension of the space V, over C and let {ff,,t : 1 = 1, 2, ... , n. }

be a basis for V,. Then, there exists an n. X n, matrix N,, such that

IV. GENERAL THEORY OF LINEAR SYSTEMS

74

(A-)'1In)V1,1PJ,2...p'1.n,J = [Pi,1PJ,2...p3i.nsJNj as the coordinates-wise representation relative to this basis. This implies that

(A - A,1.)'Pi(A)(P1,1Pi,2...p),nj) _ for

IP1,191,2...Pj,Nj

I

(t = 1,2,... ).

In particular, from (ii) of Lemma IV-1-9, we derive N, obtain

...P

(IV-1-8)

O . Thus, we

= [l ,1P'r,2 ... p'f n,](A) I, + N,),

where 1, is the n, x n1 identity matrix. This proves (IV.1.7). (6) Let { " l , t : e = 1, 2,... , n, } be a basis for V, (j = 1, 2,... , k). Set (IV.1.9)

Po = (p1,1

.

p'2...,

Then, Po E GL(n, C) and (IV.1.8) implies

Po'AP0 = diagjAllt +N1,A212+N2,...,Aklk+Nk],

([V.1.10)

where the right-hand side of (IV.1.10) is a matrix in a block-diagonal form with entries Al h +N1, A212+N2, ... , Aklk+Nk on the main diagonal blocks. Hence, pA(A) Also, PA(A) _ (A - A1)m'(A A2)'2 X2 )12

(A - Ak)mk. Therefore, dimC V) = n, = m, (j = 1,2, ... , k).

0

The following theorem defines the S-N decomposition of a matrix A E Mn(C).

Theorem IV-1-11. Let A be an n x n matrix whose entries are complex numbers. Then, there exist two n x n matrices S and N such that (a) S is diagonalizable, (b) N is nilpotent,

(c) A = S + N, (d) SN = NS.

The two matrices S and N are uniquely determined by these four conditions. If A is real, then S and N are also real. Furthermore, they are polynomials in A with coefficients in the smallest field Q(a,k, A,1) containing the field Q of rational numbers, the entries ajk of A, and the eigenvalues Al, A2 ... , Ak of A.

Proof We prove this theorem in three steps. Step 1. Existence of S and N. Using the projections P,(A) given by (IV.1.4), define

S and N by

S = A1P1(A)+A2Pz(A)+...+At Pk(A),

N=A - S.

If P0 is given by (IV.1.9), then (IV.1.11)

Po 1SP0 = diag[A111, A212, ... , Aklk)

75

1. SOME BASIC RESULTS CONCERNING MATRICES and

Po 1 NPo = diag[N1 i N2, ... , Nk]

(IV.1.12)

from Lemmas IV-1-9 and IV-I-10 and (IV.1.10). Hence, S is diagonalizable and N

is nilpotent. Furthermore, NS = SN since S and N are polynomials in A. This shows the existence of S and N satisfying (a), (b), (c), and (d). Moreover, from (IV.1.4), it follows that two matrices S and N are polynomials in A with coefficients in the field Q(ajk, Ah).

Step 2. Uniqueness of S and N. Assume that there exists another pair (S, N) of n x n matrices satisfying conditions (a), (b), (c), and (d). Then, (c) and (d) imply that SA = AS and NA = AN. Hence, SS = SS, NS = SN, SN = NS, and NN = NN since S and N are polynomials in A. This implies that S - S is diagonalizable and N - N is nilpotent. Therefore, from S - S = N - N, it follows

that S-S=N-N=O.

Step 3. The case when S and N are real. In case when A is real, let 5 and N be the complex conjugates of S and N, respectively. Then, A = S + N = 3° + N. Hence, the uniqueness of S and N implies that S = 3 and N = N. This completes the proof of Theorem IV-1-11.

Definition IV-1-12. The decomposition A = S + N of Theorem IV-1-11 is called the S-N decomposition of A.

Remark IV-1-13. From (IV.1.11), it follows immediately that S and A have the same eigenvalues, counting their multiplicities. Therefore, S is invertible if and only if A is invertible.

Observation IV-1-14. Let A be an n x n matrix whose distinct eigenvalues are A = S + N be the S-N decomposition of A. It can be shown that n x n matrices P1, P2, ... , Pk are uniquely determined by the following three conditions: (i)

(ii) P,P1 = O if j 36 t, (iii) S = A11P1 + A2P2 + ... + AkPk.

Proof.

Note that

{

In = P1(A) + P2(A) + ... + Pk(A), Pj(A)Ph(A) = O if j h, S = A1P1(A) + A2P2(A) + ... + AkPk(A). k

First, derive that P, S = SP; = \j P,. Then, this implies that .X P1 = >ahPj P1. (A). h=1

Hence, \jPjPA(A) = \h PiPh(A). Thus, PiPh(A) = 0 whenever j it follows that P1 = P1(A) = P2P}(A).

h. Therefore,

IV. GENERAL THEORY OF LINEAR SYSTEMS

76

Observation IV-1-15. Let A = S + N be the S-N decomposition of an n x n matrix A. Let T be an n x n invertible matrix such that if we set A = T-1ST, then A = diag[A1I1, A2I2, ... , AkIk], Where A1, A2, ... , Ak are distinct eigenvalues

of S (and also of A), I, is the m3 x mj identity matrix, and m3 is the multiplicity of the eigenvalue A,. It is easy to show that

(i) if we set M = T-'NT, then M is nilpotent, MA = AM, and M = diag[M1i M2,... , Mk}, where Mj are mj x m j nilpotent matrices,

(ii) if we set Pj = Tdiag[Ej1iE.,2,... ,E,k]T-1, where E,1 = 0 if j Ejj = Ij, we obtain

(I .

PjPh =0

=P1+P2+...+Pk,

1, while h),

(.1

S = A1P1 + A2P2 + ... + AkPk.

Therefore, P. = P, (S) = P? (A) (j = 1, 2, ... , k) (cf. Observation IV-1-14). The following two remarks concern real diagonalizable matrices.

Remark IV-1-16. Let A be a real nxn diagonalizable matrix and let A1, A2 ... , An be the eigenvalues of A. Then, there exists a real n x n invertible matrix P such

that (1) in the case when all eigenvalues A j (j = 1,2,... , n) are real, then P-1 AP is a real diagonal matrix whose entries on the main diagonal are A1, A2, ... , An, (2) in the case when all eigenvalues are not real, then n is an even integer 2m and P-1A fP = diag[D1, D2, ... , DmJ, where A23_1 = a, + ibl, A2J = a) - ibj, and

Dj

abj

a,

(3) in other cases, P1AP = diag[D1, D2i ... , Dhj, where A23_1 = a)+ib,, A2.1 =

a, - ib,, and D. = I b, a j for j = 1, 2, ... , h - 1 and Dh is a real diagonal matrix whose entries on the main diagonal are A, (j = 2h - 1,... , n). Remark IV-1-17. For any given real n x n matrix A, there exists a sequence {Bk : k = 1, 2,. ..} of real n x n diagonalizable matrices such that lim Bk = A. This can be proved in the following way: (i) let A = S + N be S-N decomposition of A,

(ii) using Remark IV-1-16, assume that S = diag[D1, D2, ... , Dhj, as in (3) of Remark IV-1-16,

(iii) find the form of N by SN = NS, (iv) triangularize N without changing S, (v) use a method similar to the proof of Lemma IV-1-4. Details of proofs of Remark IV-1-16 and IV-1-17 are left to the reader as exercises. Now, we give two examples of calculation of the S-N decomposition.

Example IV-1-18. The matrix A

252

498

4134

698

-234

-465

-3885

-656

15

30

252

42

-10

-20

-166

-25

has two

distinct eigenvalues 3 and 4, and PA(A) = (A - 4)2(A - 3)2,

PA(A)

2

2

1

(A

14)2

A

4 + (A 13)2 + A

3

1. SOME BASIC RESULTS CONCERNING MATRICES

77

Set P2(A) _ (A - 4)2 + 2(A - 3)(A - 4)2.

-2(A-4)(A-3)2'

P1(A) = (A - 3)2

Then,

P1(A) =

-1

-2

134

198

2

2

-134

1

-125

-186

-1

-1

125

186

0

2 0

9

12

-12

0

-6

-8

0 0

-8

0

0 0

6

9

P2(A) =

-198

Therefore,

2

-2

1

5

0

0

12

12

10

0

-6

-5

S = 4P1(A) + 3P2(A) =

250

500

4000

500

-235

-470

-3760

-470

15

30

240

30

-10

-20

-160

-20

N = A - S =

Example IV-1-19. The matrix A =

3

4

3

2

7

4 3

-4 8

A1= 11, A2=1,and

(A+9)

_

1

100(A - 11)

PA(A)

I'

has two distinct eigenvalues

_

1

1 ) 21 1 ) ,

PA(A)

198

134

-125 -186

100(A - 1)2*

Hence, 1

-

(A - 1)2

-

(A + 9)(A - 11)

100

100

Set

P1(A)

_

(A - 1)2

- -

P2(A)

100

(A + 9)(A - 11).

100

Then, 1

0

P1(A) =100- 0 0

56 76

28 38

48

24

1

,

P2(A)

100

1100 0 0

-56 -28 24 -48

Therefore,

56 86

28 38

0

48

34

-16 -16

2

20

-40

32

-4

20

N=A - S= 10

0

10

1

S = 11P1(A) + P2(A) = 10

2

1

-38 76

IV. GENERAL THEORY OF LINEAR SYSTEMS

78

In this case, SN = NS = N and N2 = O. Let Vj = Image (PI(A)) (j = 1,2). Then, by virtue of (1) of Lemma IV-1-10, Pj(A)p" = p" for all IT E Vj (j=1,2). 14

Furthermore, V1 is spanned by

Pa =

14 19 12

1

0

0 0

-2

1

19 12

0

1

and V2 is spanned by

0 0

and

1

-2

Set

. Then,

Po 1 SPo =

11

0

0

1

0 0

0

0

1

Pp 1 NPo = 2

[00

1

-1

0

1

-1

.

It is noteworthy that there is only one linearly independent eigenvector x = for the eigenvalue A2 = 1.

It is not difficult to make a program for calculation of S and N with a computer. For more examples of calculation of S and N, see [HKSI.

IV-2. Homogeneous systems of linear differential equations In this section, we explain the basic results concerning the structure of solutions of a homogeneous system of linear differential equations given by (IV.2.1)

dt = A(t)y',

where the entries of the n x n matrix A(t) are continuous on an interval I = It : a < t < b). Let us prove the following basic theorem. Theorem 1V-2-1. The solutions of (1V2. 1) forms an n-dimensional vector space over C.

We break the entire proof into three observations.

Observation IV-2-2. Any linear combination of a finite number of solutions of (IV.2.1) is also a solution of (IV.2.1). We can prove the existence of n linearly independent solutions of (IV.2.1) on the interval Z by using Theorem I-3-5 with n linearly independent initial conditions at t = to. Notice that each column vector of a solution Y of the differential equation (IV.2.2)

dt = A(t)Y

on an n x n unknown matrix Y is a solution of system (IV.2. 1). Therefore, construct-

ing an invertible solution Y of (IV.2.2), we can construct n linearly independent solutions of (IV.2.1) all at once. If an n x n matrix Y(t) is a solution of equation

(IV.2.2)on an interval I={t:a
n columns of a fundamental matrix solution Y(t) of (IV.2.2) are said to form a fundamental set of n linearly independent solutions of (IV.2.1) on the interval T.

2. HOMOGENEOUS SYSTEMS OF LINEAR DIFF. EQUATIONS

79

Observation IV-2-3. Let 4(t) be a solution of (IV.2.2) on Z. Also, let *(t) be a solution of the adjoint equation of (IV.2.2): dZ dt

(IV.2.3)

= -ZA(t)

on the interval Z, where Z is an n x n unknown matrix. Then,

-W(t)A(t)4s(t) + $(t)A(t)4;(t) = 0.

dt

This implies that the matrix %P(t)4i(t) is independent of t. Therefore, W(t)4i(t) = %P(r)t(r) for any fixed point r E Z and for all t E Z. Note that the initial values

44(r) and %P(r) at t = r can be prescribed arbitrarily. In particular, in the case we obtain WY(t)4i(t) = In when 4?(r) E GL(n,C), by choosing %P(r) = for all t E Z. Thus, we proved the following lemma.

Lemma IV-2-4. Let an n x n matrix 4i(t) be a solution of (IV.2.2) on the interval Z. Then, 45(t) is invertible for all t E I (i.e., a fundamental matrix solution of (IV.2.1)) if 4i(r) is invertible for some r E Z. Furthermore, 4 (t)-1 is the unique solution of (IV.2.8) on Z satisfying the initial condition Z(r) = 4i(r)-1.

Observation IV-2-5. Denote by 4?(t; r) the unique solution of the initial-value problem (IV.2.4)

-

dt

= A(t)Y,

Y(r) = In,

where r E Z. Then, 1(t; r) E GL(n, C) for all t E Z. The general structure of solutions of (IV.2.1) and (IV.2.2) are given by the following theorem, which can be easily verified.

Theorem IV-2-6. The C"-valued functwn y(t) = 44(t; r)it is the unique solution of the initial-value problem dt

A(t)y",

y(r) =1,

where it E C. Also, the n x n matrix Y = 4i(t; r)r is the unique solution of the initial-value problem dY = A(t)Y, dt

Y(r) = I

where r E

Theorem IV-2-1 is a corollary of Theorem IV-2-6. 0

Remark IV-2-7. (1) The general form of a fundamental matrix solution of (IV.2.1) is given by Y(t) = 4i(t; r)r, where r E GL(n, C). (2) If a fundamental matrix solution is given by Y(t) = 4'(t; r)r, then Y(r) = F. Hence, (IV.2.5)

0(t; ,r) =

Y(t)Y(r)-1

(t, r E Z)

W. GENERAL THEORY OF LINEAR SYSTEMS

80

for any fundamental matrix solution Y(t). In particular, (IV.2.6)

1(t;r) _

for

t, r, r1 E Z.

(3) In the case when A(t) is a scalar (i.e., n = 1), we obtain easily 1(t; r) = exp I JA(s)ds)] t

(IV.2.7)

t

1

In the general case, we define exp I J A(s)ds] by V,r +M

exp[B(t)] = In

B(t)"',

where

B(t) =

T11.

m=1

Jt A(s)ds.

However, generally speaking, (IV.2.7) holds only in the case when B(t) and

B'(t) = A(t) commute. In particular, 4(t; r) = exp[(t - r )AJ if A = A(t) is independent of t. In §IV-3, we shall explain how to calculate exp((t - r)AJ, using the S-N decomposition of A. Also, (IV.2.7) holds in the case when

A(t) is diagonal on the interval I. A less trivial case is given in Exercise IV-9. It is easy to see that B(t) and A(t) commute if A(t) is a 2 x 2

upper-triangular matrix with an eigenvalue of multiplicity 2. For exam-

ple, the matrix A(t) =

[coStt

I

satisfies the requirement.

case, 4b(t; r) = exp I J t A(s)ds = exp 1

exp(sin t -sin r) 10

t

1 1

(jsin t o sin r

In this

sint - sin r,

)

r J1

t

trA(s)ds) if Y(t) satisfies (IV.2.2), where det A if and trA are the determinant and trace of the matrix A. This formula is

(4) det Y(t) = det Y(r) exp

known as Abel's formula (cf. (CL, p. 28]). Proof.

Regarding detY(t) as a function of n column vectors {y'1(t),... Y(t), set det Y(t) = 7(g, (t), ... , y (t)). Then, (IV.2.8)

d det Y(t) dt

-

[: L P(...

,

of

A(t)ym(t), ... ).

M=1

Then, 9 is multilinear and alternating in y1(t),... ,j,,(t). Furthermore, 9 = trA(t) if Denote the right-hand side of (IV.2.8) by G(y'1(t),...

Y(t) = I,,. Therefore, 9 = tr A(t) det Y(t). Solving the differential equation ddet Y(t) = tr A(t) det Y(t), we obtain Abel's formula. 0 dt

3. HOMOGENEOUS SYSTEMS WITH CONSTANT COEFFICIENTS

81

IV-3. Homogeneous systems with constant coefficients For an n x n matrix A, we define exp[A] by

exp[A] = In + E h Ah.

(IV.3.1)

h=1

It is easy to show that the matrix exp[A] satisfies the condition exp[A + B] = exp[A] exp[B] if A and B commute. This implies that exp[A] is invertible and (exp[A])-1 = exp[-A]. Thus, we obtain a fundamental matrix solution Y = exp[tA]

of the system d = Ay with a constant matrix A by solving the initial-value problem

Y(0) = In.

= AY,

(IV.3.2)

This, in turn, implies that the unique solution of the initial-value problem

d = Ay,

(IV.3.3)

y(T) = P

is given by y = exp[(t - r)A]p, where p E C' is a constant vector. In this section, we explain how to calculate exp[tA] for a given constant matrix A, using the S-N decomposition of A. Assume that an n x n matrix A has k distinct eigenvalues Al, A2, ... , Ak. Let A = S + N be the S-N decomposition of A. Also, let Pj (A) (,j = 1, 2,... , k) be the projections defined in §IV-1 (cf. (IV.1.4)). Then, k

(IV.3.4)

k

In = > PJ(A),

S=

AjPJ(A),

N = A - S,

)=1

J=1

and

(IV.3.5)

PJ(A)Ph(A)

j

h

0 (A)

if

h

= 36

j

(3, h = 1, 2, .... k).

The two matrices S and N commute. Denote by V, the image of the mapping PJ(A) : C" Cn (cf. Lemma IV-I-10). It is known that Sp = \,)5 for pin VJ . Hence, Sl p" = \1)5 and +00

earh=1

1 + E (A2t)n }ic

exp[tS]p' =

n-i h

A Nh since N is nilpotent. Therefore,

On the other hand, exp(tN] = In + h=1

exp[tA)# = exp[t(S + N)]p' = expitN] exp[tS]p" = e'\'` exp[tN]P n- i th

[In

(IV.3.6)

= eA'`

+

p"

n=1

for

fl E Vj.

IV. GENERAL THEORY OF LINEAR SYSTEMS

82

Applying (IV.3.4) and (IV.3.6) to a general p E C", we derive

n-10

C.Njt

exp[tA]p =

(IV.3.7)

In + E hl Nh

j=1

for 9E C".

P, (A)p'

h=I

Thus, we proved the following theorem.

Theorem IV-3-1. The matrix exp[tA] is calculated by formula n-1 th

k

exp[tA] _ > ea+`

(IV.3.8)

=l

In + F, T! Nh

Pj (A).

h=1

Since the general solution of the differential equation (IV.3.9)

is given by (IV.3.7), the following important result is obtained.

Theorem IV-3-2. (i) If R(,\,) < 0 for j = 1, 2, ... , k, then every solution of (IV. 3.9) tends to 06 as t -' +00, (ii) if R(Aj) > 0 for some j, some solutions of (IV.3.9) tend to 00 as t - +00, (iii) every solution of (IV.3.9) is bounded for t > 0 if and only if R(AE) < 0 for j = 1, 2, ... , k and NP, (A) = 0 if R(A)) = 0. Now, we illustrate calculation of exp[tA] in two examples. Note that in the case when A has nonreal eigenvalues, we must use complex numbers in our calculation. Nevertheless, if A is a real matrix, then exp[tA] is also real. Hence, at the end of our calculation, we obtain real-valued solutions of (IV.3.9) if A is real.

-2 Example IV-3-3. Consider the matrix A =

0 3

1

0

2

1

-2 0

. The characteristic

polynomial of A is pA(A) _ (A -1)(A+2)2. By using the partial fraction decomposition of

1

PA(A),

and

P2(A)

we derive 1 =

(A + 2)2 - (A + 5)(A - 1)

+ 5)(A - 1),

9

. Setting P1(A) =

(A + 2)2 9

we obtain

9

P1(A) =

0 0

0 0

1

1

0 0, P2(A) = 1

1

0

0

1

0 0

-1

-1

0

Set

S = PI(A) - 2P2(A) =

-2 0 3

0

0

-2 0 3

1

,

N=A-S=

0

0

1

0

0

0.

0

-1

0

3. HOMOGENEOUS SYSTEMS WITH CONSTANT COEFFICIENTS

83

Note that N2 = 0. Hence,

exp[tA] = e'[ 13 + tN ] PI(A) + e-21 [ 13 + tN ] P2(A) e-2t

to-2t

0

e-2c

0

et - (1 +

et - e-2t

0 t)e-2t

et

The solution of the initial-value problem dt = Ay, y(0) = y is y(t) = exp[tA]i). To find a solution satisfying the condition Jim y(0) = 0, we must choose it so that

t +m

P1(A)ij = 0. Such an iJ is given by it = P2(A)c", where c is an arbitrary constant vector in C3.

Example IV-3-4. Next, consider the matrix A =

0

-1

1

0

-1

1

1

-1 . The charac0

teristic polynomial of A is PA(A) = A(A2 + 3) = A(A - if)(A + i\/3-). Using the 1 partial fraction decomposition of , we obtain PA (A)

(A- if)(A+ if) - A(A+ if)- sA(A- if}.

1= Setting

P1(A) = 3(A2 + 3),

I A(A + if),

P2(A)

P3(A)

A(A - if),

we obtain 1

1

1

1

Pi(A) = -

1

1

1

3

1

1

1

1-if 1+if -2 1-if 6 1-if 1+iv -2 1

,

-2

P2(A) = -- 1+if

and P3(A) is the complex conjugate of P2(A). If we set

S = (if)P2(A) - (if)P3(A), then S = A. This implies that N = 0. Thus, we obtain

exp(tA] = P1(A) + e,.''P2(A) + e-;J1tP3(A) = P1(A) + 23t (e'3tP2(A))

.

Using

(e'' t(1 + if)) = cos(ft) - \/3-sin(ft), 2 (e'-1t(1 - if))

wa(ft) + f sin(ft),

IV. GENERAL THEORY OF LINEAR SYSTEMS

84

we find

a(t)

c(t)

b(t)

exp[tA] = 1 c(t) a(t) b(t) 3

where

a(t)

c(t)

b(t)

a(t) = I + 2cos(ft), b(t) = 1 - {cos(ft) + f sin(ft)} ,

c(t) = 1 - {wa(ft) - f sin(ft)) Remark IV-3-5. Fhnctions of a matnx In this remark, we explain how to define functions of a matrix A.

I. A particular case: Let A0, I,,, and N be a number, the n x n identity matrix, and an n x n nilpotent matrix, respectively. Also, consider a function f (X) in a neighborhood of A0. Assume that f (A) has the Taylor series expansion (i.e., f is analytic at A0) f(A) = f(A0) + 1

f

(h)

h?Ao)(A

-

Ao)h.

h=1

In this case, define f (,\o1 + N) by

= f(AoI. + N) = f(Ao)I. +

f(h)PLO) .A

h=1

h.t

A

=

f(Ao)II +

n-1 f(h)(AO) NA. h=1

h.

n-1 (h)

Since N is nilpotent, the matrix >2f

is also nilpotent. Therefore, the

h=1

characteristic polynomial pf(A0,+N)(A) of f(AoI + N) is

Pf(aol-N)(A) = (A - 1(A0))". II. The general case: Assume that the characteristic polynomial PA(A) of an n x n matrix A is PA(A) = (A - .l1)m1(A - A 2 ) ' - 2 ... (A - Ak)mr.

where A1,... , Ak are distinct eigenvalues of A. Construct P, (A) (j = 1, ... , k), S, and N as above. Then,

A = (A1In + N)P1(A) + (A21n + N)P2(A) + ... + 0k1n + N)Pk(A). Therefore,

A' = (A11n + N)1P1(A) + (A2In + N)'P2(A) 4- ... + (Ak1n + N)'Pk(A) for every integer P.

3. HOMOGENEOUS SYSTEMS WITH CONSTANT COEFFICIENTS

85

Assuming that a function f (A) has the Taylor series expansion

f(A) = f(A3)

00 0f(h)(A,)(A-A,)'' h!

h=1

at A = A., for every j = 1, ... , k, we define f (A) by (IV.3.10)

f(A) = f (A11, + N)Pk(A) + f (A21n + N)P2(A) + ... + f (Akin + N)Pk(A).

Since P2(S) = P,(A) (cf. Observation IV-1-15), this definition applied to S yields

f(S) = f(AIIn)P1(A) + f(A21.)P2(A) + -' + f(AkII)Pk(A) and f (A) - f (S) has a form N x (a polynomial in S and N). Therefore, f (A) - f (S) is nilpotent. Furthermore, f (S) and f (A) commute. This implies that

f(A) = f(S) + (f(A) - f(S)) is the S-N decomposition of f (A). Thus,

Pf(A)(A) = pf(s)(A) = (A - f(A1))m' ... (A -

f(Ak))m

Example IV-3-6. In the case when f (A) = log(A), define log(A) by

log(A) = log(A1In+N)Pk(A) + log(A21n+N)Pk(A) + ... + log(Akln+N)Pk(A), where we must assume that A is invertible so that A, # 0 for all eigenvalues of A. Let us look at log(AoI,, + N) more closely, assuming that A0 0 0. Since

/

log(Ao + u) = 1000) + log t 1 + -o) = logl o) + \\

+O°

m=1

m+1

(-I)m

Io

we obtain

log(.1oIn + N) = log(Ao)ln +

n-1 (-1)m+1 m=1

m

It is not difficult to show that exp[log(A)J = A. In fact, since (log(A))m = (log(A11n + N))mPk(A) + (log(A21n + N))mp2(A) + ... + (log(Akln + N))mPk(A), it is sufficient to show that exp[1og(Aoln + N)J = A01,, + N. This can be proved by using exp[log(Ao +,u)] =A0 +,u.

IV. GENERAL THEORY OF LINEAR SYSTEMS

86

Observation IV-3-7. In the definition of log(A) in Example N-3-6, we used log(A,). The function log(A) is not single-valued.

Therefore, the definition of

log(A) is not unique.

Observation IV-3-8. Let A = S + N be the S-N decomposition of A. If A is invertible, S is also invertible. Therefore, we can write A as A = S(II + M), where

M = S' N = NS-1. Since S and N commute, two matrices S and M commute. Furthermore, M is nilpotent. Using this form, we can define log(A) by

log(A) = log(S) + log(Ih + M), where

log(S) = Iog(A1)P1(A) + log(A2)P2(A) +

. + log(Ak)Pk(A)

and

(1)m+1

log(IR + Al) = E - m

Mm.

M=1

This definition and the previous definition give the same function log(A) if the same definition of log(A,) is used.

Example N-3-9. Let us calculate sin(A) for A =

3

4

2

7

-4 8

3 4

(cf. Example

3

IV-1-19). The matrix A has two eigenvalues 11 and 1. The corresponding projections are 7 -14 -7 0 14

P1(A) =

25

25

0

L9

L9

0

12 25 25

6 50 25

P2(A) =

0 0

25

25

2s

50

-12

19

5

25

Define S = 11P1(A) + P2(A) and N = A - S. Then N2 = 0. Also, (

t

sin(11 + x) = -0.99999 + 0.0044257x + 0.499995x2 + 0(x3 ), sin(1 + x) = 0.841471+0.540302x-0.420735x 2 + 0(x3).

Therefore,

sin(A) _ (-0.9999913 + 0.0044257N)P1(A) + (0.84147113 + 0.540302N)P2(A) 1.92207 1.0806

-1.8957 -1.42252

-0.407549 -0.591695

-2.1612

0.845065

0.1834

It is known that sin x has the series expansion +00

sin x = 1

(2h +)1)I

T2h+1

4. SYSTEMS WITH PERIODIC COEFFICIENTS

87

Therefore, we can also define sin(A) by sin(A)

-

A2h+1 -- + (2h(-1)h + 1)!

However, this approximation is not quite satisfactory if we notice that

_1h sin(11) = -0.99999

and h=O

112h+1

= -117.147.

(2h + 1)!

IV-4. Systems with periodic coefficients In this section, we explain how to construct a fundamental matrix solution of a system dy dt

(IV.4.1)

= A(t)y

in the case when the n x n matrix A(t) satisfies the following conditions: (1) entries of A(t) are continuous on the entire real line R, (2) entries of A(t) are periodic in t of a (positive) period w, i.e., (IV.4.2)

A(t + w) = A(t)

for

t E R.

Look at the unique n x n fundamental matrix solution 4'(t) defined by the initialvalue problem (IV.4.3)

dY = A(t)Y, .it

Y(0) = In

Since 4i '(t + w) = A(t + w)4S(t + w) = A(t)4'(t + w) and +(0 + w) = 4t(w), the matrix 41(t +w) is also a fundamental matrix solution of (IV.4.3). As mentioned in (1) of Remark IV-2-7, there exists a constant matrix r such that 44(t +w) = 4i(t)r and, consequently, r = 4?(w). Thus, (IV.4.4)

41(t + w) = 4(t)t(w)

for t E R.

Setting B = w-' log(4?(w)J (cf. Example IV-3-6), define an n x n matrix P(t) by (IV.4.5)

P(t) = d'(t) exp(-tBJ.

Then, P(t + w) = ((t + w) exp(-(t + w)BI = 4i(t)0(w) exp(-wB) exp(-tBJ = 4'(t) exp(-tBJ = P(t). This shows that P(t) is periodic in t of period w. Thus, we proved the following theorem.

IV. GENERAL THEORY OF LINEAR SYSTEMS

88

Theorem IV-4-1 (G. Floquet [Fl]). Under assumptions (1) and (2), the fundamental matrix solution 4i(t) of (IV.4.1) defined by the initial-value problem (IV.4.3) has the form

4'(t) = P(t) exp[tB],

(IV.4.6)

where P(t) and B are n x n matrices such that (a) P(t) is invertible, continuous, and periodic of period w in t, (Q) B is a constant matrix such that 4'(w) = exp[wB]. Observation IV-4-2. As was explained in Observation IV-3-8, letting 4'(w) _ S + N = S(I + M) be the S-N decomposition of 4(w), we define log($(w)) by log(4?(w)) = log(S) + log(1 + M), where log(S) = log(A1)P1(4'(w)) + log(a2)P2(4'(w)) + - + log(Ak)Pk('6(w)) -

and

n- 1

l)m+l

log(!! + M) = E (- m

Mm.

M=1

In the case when A(t) is a real matrix, the unique solution 4 (t) of problem (IV.4.3) is also a real matrix. Therefore, the entries of 4'(w) are real. Since S and N are real

matrices, the matrix M = S-'N is real. Therefore, log(I + M) is also real. Let us look at log(S) more closely. If \j is a complex eigenvalue of 4'(w), its complex conjugate A3 is also an eigenvalue of 4'(w). In this case, set A,+1 = A3. It is easy to see that the projection Pj+1(4?(w)) is also the complex conjugate of P,(4'(w)). However, if some eigenvalues of 4'(w) are negative, log[S] is not real. To see this more clearly, rewrite log[s] in the form

log[S] _ E log[A,]Pi(4?(w)) + E log1A3]P'(4'(w))other j

A, <0

The matrix

E

log(A,]P,(4'(w))

other j log[aj]Pis

is

a

matrix,

real

while

the matrix

not real. Therefore, log[S] is not real. To rectify this sit-

A, <0

uation, let us look at S2. By virtue of the relations given in Lemma IV-1-9, we obtain S2 = _y2P

(,\J)2P+

other j

A, <0

Notice that

log(S2] _ E log[(Ai)2]Pj(4'(w)) + 2 E log[a3]P,(4(w)) af<0

other 7

is a real matrix. Therefore, we can find a real matrix

log[4'(w)2] = log[S2J + 2log(In + MI. Now, observe that 4'(,)2 =' (2w) and 4'(t + 2w) = 4'(t). (2w). Thus, setting (IV.4.7)

B = _L log{4'(w)2]

we obtain the following theorem.

and

P(t) = 4'(t) exp[-tB],

89

4. SYSTEMS WITH PERIODIC COEFFICIENTS

Theorem IV-4-3. In the case when the matrix A(t) is real, the matrix 4s(t) has the form P(t) exp[tB],

where P(t) and B are n x n real matrices such that (a) P(t) is invertible, continuous, and periodic of period 2w in t, (p) B is a constant matrix such that 4?(w)2 = exp[2wB].

Remark IV-4-4. In the case when 4?(t) = P(t) exp[tB], we have A(t)P(t) - P(t)B. Therefore, (IV.4.1) is changed to dz-

(IV.4.8)

dP(t) dt

= Bz

dt

by the transformation y" = P(t)!. As noted above, P(t) is periodic in t of period w (or, respectively, 2w when A(t) is real), the behavior of solutions of (IV.4.1) is derived from the behavior of solutions of (IV.4.8).

Definition IV-4-5. The eigenvalues p1.... , p of the matrix B are called the characteristic exponents of equation (IV.4.1). The eigenvalues pi = exp[wpjl, ... , of the matrix 4?(w) (or, respectively, when A(t) is real, the eigenp= values pi = exp(2wpi],... , P = exp[2wµn] of 4?(w)2) are called the multipliers of equation (IV.4.1). The following basic result is a corollary of Theorem IV-3-1.

Corollary IV-4-6. All solutions of equation (IV.4.1) tend to 0' as t -. +oo if and only if Ipj I < 1 for all j (or R(pj) < 0 for all j).

Observation IV-4-7. Let be an eigenvector of 45(w) associated with a multiplier p. Then, the solution d(t) = t(t)p" of (IV.4.1) satisfies the condition ¢(t + w) = pi(t). The terminology multiplier came from this fact. (In the case when A(t) is real, choosing an eigenvector p" of 4;(w)2 associated with a multiplier p, we obtain

(t + 2w) = pi(t) if d(t) = 4i(t)p.) Example IV-4-8. Consider the initial-value problem (IV.4.9)

dY = A(t)Y,

Y(0) = 12,

-it-

where cost

A(t) =

0

1

cost]

The solution of (IV.4.9) is given by r

l

I

A(s)dsexp(sin t)

41(t) = exp I

fo

L

t

t]

(cf. (3) of Remark IV-2-7). Therefore, $(27r) = I

0

B=

21 1

1

2-. log[4i(2a)].

IV. GENERAL THEORY OF LINEAR SYSTEMS

90

Since p.(\) = (a - 1) 2, Pt(4i) = 12, we obtain

S = 12i M = This gives B=

[0

20

,

and

log[4s(27r)j = log[I2 + M) = M = I 0

J

1 M= I 0

2,r 0

01. Therefore,

P(t) = i(t) exp [ 0

Ot,

It I = exp(sin t)I2

= 4i(t) l

and fi(t) = P(t) exp[tBj = exp(sin t) exp I 0 t ]. Moreover, the characteristic exponents are 0, 0, and the multipliers of (IV.4.9) are 1, 1. Note that in this case, as both of two multipliers are positive, it is not necessary to take 4i(2a)2 and period 41r to find B.

IV-5. Linear Harniltonian systems with periodic coefficients This section is divided into two parts. In Part 1, we explain the structure of solutions of a linear Hamiltonian system. The basic facts are found in the theory of real symplectic groups and their Lie algebra. If we denote by J the real (2n) x (2n)

matrix

I

0O

1, where In is the n x n identity matrix, the Lie algebra of the

real symplectic group Sp(2n, R) of order 2n is the set of all real 2n x 2n matrices of the form JH, where H is a real (2n) x (2n) symmetric matrix. A linear system

T = A(t)f on y' E R2" is a linear Hamiltonian system, if there exists a real (2n) x (2n) symmetric matrix H(t) such that A(t) = JH(t). Consequently, the fundamental matrix 0(t) of this system belongs to Sp(2n,R) if 0(0) = I. We explain all these elementary facts of Hamiltonian systems in Part 1. We also prove that if G E Sp(2n, R) and G = S+N = S(I2"+M) is the S-N decomposition of G, then S and 12" + M belong to Sp(2n, R). In Part 2, we refine the Floquet theorem (cf. Theorem IV-4-1) for the linear periodic Hamiltonian systems in such a way

that the periodic transformation y = P(t)i given in Remark IV-4-4 is canonical. This means that P(t) belongs to Sp(2n, R). The important consequence is that the linear system

dx

= BE of Remark IV-4-4 is a Hamiltonian system.

Part 1 Consider a linear Hamiltonian system

dp

dt

=

b-KH

(t, y')

dq = dt

-!2c (t, y-), where

8P and j are unknown vectors in R", y = [q1, ?{(t, y7 = 2y"TH(t)y with a real 8H on (2n) x (2n) symmetric matrix H(t), and

ON

8q"

Bpi

8p

an

Ni Here, 8q.

91

5. LINEAR HAMILTONIAN SYSTEMS

yT is the transpose of the column vector Y. Since (t, y-) = [1n, O[ H(t)y,

8 (t, y-) = [0, In[ H(t)y,

the Hamiltonian system can be written in the form

= JH(t)y,

(IV.5.1)

where y is a unknown vector in R2n, H(t) is a real (2n) x (2n) symmetric matrix, and

J=

(IV.5.2)

In

O

[

-In O

The matrix J has the following important properties:

JT = -J,

(IV.5.3)

J-1 = -J,

where JT is the transpose of J. Let fi(t) be the unique real (2n) x (2n) matrix such that do(t)

= JH(t))(t) ,

4(0) = 12n.

Lemma IV-5-1. The matrix 4(t) satisfies the condition 4(t)T J4(t) = J, where t(t)T is the transpose of t(t). Proof Differentiate O(t)TJ4(t) to derive

a

I(t)TH(t)(-J)J4(t) + I(t)T H(t)41(t)

-

O.

Then, 1(t)TJ4(t) = 4(0)T Jo(0) = J. 0 The converse of Lemma IV-5-1 is shown in the following lemma.

Lemma IV-5-2. Assume that a real (2n) x (2n) matrix G(t) satisfies the following conditions:

(i) the entries of G(t) are continuously differentiable in t on an interval 1, (ii) G(t)T JG(t) = J for t E Z. Then, G(t) is invertible and H(t) _ -Jd d(t)G(t)-I is symmetric for t E I. Proof.

Computing the determinant of both sides of condition (ii), we obtain [det G(t)12 = 1. Therefore, G(t) is invertible. To show that H(t) is symmetric, differentiating both sides of G(t)T JG(t) = J, we obtain

[dG(t)]TJG(t) + G(t)T J fi(t) = O.

92

IV. GENERAL THEORY OF LINEAR SYSTEMS

Since [G(t)-']T = [G(t)T]-1, we further obtain

[G(t)-']T [dG(t)]T(- J) = JdG(t )G(t)_1. Observing that the left-hand side of this relation is the transpose of the right-hand

side, we conclude that H(t) is symmetric. 0

Observation 1. If GT JG = J, then (G-' )T JG-' = J and GJGT = J. In fact, GT JG = J implies that (G-' )T JG-' = J. Then, taking the inverse of both sides of the first relation, we obtain the second relation (cf. (IV.5.3)).

Definition IV-5-3. A (2n) x (2n) real invertible matrix G such that GT JG = J is called a real symplectic matrix of order 2n. The set of all real symplectic matrices of order 2n is denoted by Sp(2n, R). It is easy to show that Sp(2n, R) is a subgroup of GL(2n, R) (cf. Observation 1). Hence, Sp(2n, R) is called the real symplectic group of order 2n. Observation 2. If we change Hamiltonian system (IV.5.1) by a linear transformation

(rV.5.1) becomes

dt

-

{P(t)-1JH(t)P(t) - P(t)-1 dt)J i.

Furthermore, if P(t) E Sp(2n, R) for t E Z, this system becomes (IV.5.5)

J [p(t)TH(t)P(t) - JdP( 'P(t) i.

d

Since P-(t)P(t) = 12 and P(t)T E Sp(2n,R), we obtain

I P(t)_1

dP(t)

dP(t) dt

+

di

P(t) =dlzd =

0,

1 P(t)-'J = JP(t)T. Observe that -J

t dP(t)

P(t) is symmetric since P(t) E Sp(2n, R) (cf. Lemma IV-5-2). This implies that (IV.5.5) is a Hamiltonian system. Definition IV-5-4. Transformation (IV.5.4) is called a canonical transformation if P(t) E Sp(2n, R). Observation 3. For a given constant matrix G in Sp(2n, R), let us construct the S-N decomposition G = S + N, where S is diagonalizable, N is nilpotent, and SN = NS. Since G is invertible, S is invertible (cf. Remark IV-1-13). If we set M = S-' N, then M is also nilpotent, and we derive the unique decomposition (IV.5.6)

G = S(12 + M),

SM = MS.

93

5. LINEAR HAMILTONIAN SYSTEMS

The identity GTJG = J implies JGJ-' = (GT)-l. Using (IV.5.6), we can write this last relation in the form

(JSJ-') (J(I2n + M)J-1) =

(ST)-1 (I2, + MT)-'.

Set Si = JSJ-1, M1 = J(I2n + M)J-' - 12,,, S2 = (ST)-', and M2 = = 1, 2) are diagonal(12n + MT) -1 - 12n. It is not difficult to show that S. izable and that Af. (j = 1,2) are nilpotent. Furthermore, S,llf, = MSS, (j = 1, 2). Hence, the uniqueness of S-N decomposition implies that S1 = S2 and 12n + M1 = 12n + M2. Therefore, (IV.5.7)

JSJ-' = (ST)-',

J(I2n + Af)J-' = (12n + AfT)-1.

Thus, we proved the following lemma.

Lemma IV-5-5. If we write a symplectic matrix G of order 2n in form (IV.5.6) by using the unique S-N decomposition of G (where Af = S- IN), the two matrices S and 12n + M are also symplectic matrices of order 2n. Remark IV-5-6. Let J be the 2n x 2n matrix defined by (111.5.2) and let H be a 2n x 2n symmetric matrix. Then, it is important to know that if A is an eigenvalue of JH with multiplicity m, then -A is also an eigenvalue of JH with multiplicity

m. In fact, (JH)T = -HJ = HJ-'. Hence, J-'(JH)TJ = -JH, where

AT

denotes the transpose of A.

Remark IV-5-7. Let G be a 2n x 2n symplectic matrix. Then, it is important to know also that if A is an eigenvalue of G with multiplicity m. then

is also an

eigenvalue of G with multiplicity m. In fact, JGJ-' = (CT)-'. Remark IV-5-8. Assuming that the set of 2n x 2n symplectic matrices with 2n distinct eigenvalues is dense in the symplectic group Sp(2n, R), it can be shown that if I (and/or -1) is an eigenvalue of a symplectic matrix G, its multiplicity is even. Also, det G = 1. In fact, slim, 1. It is also known that det A is equal to the product of all eigenvalues of A.

Remark IV-5-9. Assuming that Sp(2n, R) is a connected set, we can show that det G = 1 for all G E Sp(2n, R). In fact, (det G)2 = 1 and the 2n x 2n identity matrix is in Sp(2n, R).

Part 2 Now, assume that the real symmetric matrix H(t) of system (IV.5.1) is periodic in t of a positive period w, i.e., H(t + w) = H(t). Then, applying Theorem IV-4-3 and Remark IV-4-4 to system (IV.5.1), we conclude that the unique (2n) x (2n) matrix solution 0(t) of the initial-value problem d dtt) = JH(t)4y(t), 4i(O) _

12n can be written in the form 0(t) = P(t) exp[tB], where P(t) and B are real (2n) x (2n) matrices such that P(t) is invertible, P(t+2w) = P(t) for all t E R, and B is a constant matrix such that 4i(w)2 = exp[2wB]. Furthermore, system (IV.5.1) is changed to (IV.5.8)

dzF

dt

= Bi

W. GENERAL THEORY OF LINEAR SYSTEMS

94

by the transformation

y = P(t)f.

(IV.5.9)

The main concern of Part 2 is to show that a matrix P(t) can be chosen so that the transformation (IV.5.9) is canonical. Note that Y = exp[tB] is the unique solution of the initial-value problem = BY, Y(O) = I2,,. Therefore, if JB is symmetric, the matrix exp[tB) is symplectic. Hence, if JB is symmetric, the matrix P(t) = 4i(t) exp[-tB] is symplectic, since 4i(t) is symplectic. This implies that in order to show that the transformation (IV.5.9) is canonical, it suffices to show that we can choose P(t) so that JB is symmetric. Hereafter we use notations and results given in §IV-4. For examples, k

0 (j # t),

I2. = E P,('t(w)),

(1V.5.10)

,=1

Pj(4'(w))2 = P Taking the transpose of (IV.5.10), we obtain k

(IV.5.11)

l2n

0 U 36t),

= yP,(.6(w))T, ,=1 \P(.t(w))T )2

= Pj(.0(W))T.

Note that pAr(A) = PA(A) for any square matrix A. Also,

log[4?(w)2] = log[S2] + 2Iog[I2r, + MI, log[S2]

log[(,\,)2]Pi(I(w)) + 2 E log[Aj]Pj('(w)),

_

other ,

ai <0

1 m+l

1og[I2n + M] =

E (-) m=1

Mm,

and

B=

log[4i(w)2].

In order to prove that JB is symmetric, it suffices to prove that J log[S2] and J log[I2i + M] are symmetric. To do this, let us first prove the following lemma.

Lemma IV-5-10. For the symplectic matrix 4i(w), we have (IV.5.12)

Pj(0(w)T)JPe(0(w)) = 0

if A,Ae 34

where Aj (j = 1, 2,... , k) are distinct eigenvalues of 4i(w).

1,

95

5. LINEAR HAMILTONIAN SYSTEMS Proof.

Upon applying (IV.5.6) to G = 0(w), write 4'(w) in the form -O(w) = S(I2n+M).

Then, S is symplectic, i.e., ST JS = J = 12,,JI2n (cf. Lemma IV-5-5). Using the formulas

t=1=1 k

k

12. = > Pt(4 (w)), (IV.5.13)

12n =

P

k

k

ST = EA,P,(,p(w))T,

S=

J=1

r=1

rewrite the identity ST JS = J = I21 JI2,, in the following form: k

k

Pi(qD(w))TJPt(.D(w))

A31

(IV.5.14)

,.t=1

,,t=1

Fixing a pair of indices j and t and multiplying both sides of (IV.5.14) by Pj (4ti(w))T from the left and by Pt(4i(w)) from the right, we derive A, AtP,(4 (w))T JPt(4 (w)) = P3(4'(w))TJPt(4 (w))

Hence, (IV.5.12) follows. 0

Observation 4. Let us prove that J log[S2] is symmetric. To do this, by using (IV.5.12) and (IV.5.13), we write J log[S2] and Iog[(S2)T](-J) in the following form:

I2nJ log[S2] =

log[(At)2]Pi(.O(w))TJP A,ae=1

log[(S2)T](-J)I2n =

log[(A,)21 A,A =1

Now, choose 1og[(A, )2] for j = 1, 2,... , k in such a way that

log[(At)2] = - log[(A,)2]

if A,A, = 1.

Then, 1og[(S2)Tj(-J) _

log[(A1)2]P,(.t(w))TJPt(+(w))

= J log[S2].

Note that for any square matrix A, we have [log(A)IT = log(AT). Therefore, [J log(S2)]T = - log[(S2)T]J. Hence, J log[S2] is symmetric.

Observation 5. Let us prove that J log[I2 + M) is symmetric. To do this, we write first [12n + M]TJ[I2n + M] = J in the form J[I2n + M] = (12n + MT)-'J.

IV. GENERAL THEORY OF LINEAR SYSTEMS

96

Then, write [12n + MT J -1 in the form [12,, + MT J-1 = 12. + M. Using (IV.5.7), it

is not difficult to show that M is nilpotent and JM = MJ. Hence,

for m= 1,2,....

JMm = (Kf)mJ Therefore,

(IV.5.15)

J

00

=

(-lmm+l

(-lmm+l MM _

km J

m=1

m=1

/

and

J 1og[I2,, + MI = (log(12n + 1GIJ) J

= (log [(I2 +

MT)-1])

]J. J = - (log [12" + Mn)

Thus,

[J log(I2 + M)]T = log(I2i + 111T)(-J) = J log(I2n + M). This proves that J log[I2,, + M] is symmetric. Here, use was made of the fact that if

1 _+X =

1 + y(x), then log I

log[1 + y(x)J = - log(1 + x] 1 + xJ =

as a power series in x. L Thus, we arrived at the following conclusion.

Theorem IV-5-11. In the case when the matrix H(t) of system (IV.5.1) is real, symmetric, and periodic in t of a positive period w, there exist (2n) x (2n) real matrices P(t) and B such that (a) P(t) E Sp(2n, R) for all t E R,

(b) P(t+2w)=P(t) for alit ER, (c) B is a constant matrix such that JB is symmetric, (d) the canonical transformation y" = P(t): changes system (IV.5. 1) to L = BE.

For more information of exponentials in algebraic matrix groups, see (Mar] and (Si4].

IV-6. Nonhomogeneous equations In this section, we explain how to solve an initial-value problem (IV.6.1)

di = A(t)9 +

b'(t),

y(r) = ij,

where the entries of the n x n matrix A(t) and the C"-valued function b(t) are continuous on an interval Z = {t : a < t < b} and r E Z, if E C". As it was

97

6. NONHOMOGENEOUS EQUATIONS

explained in §IV-2, let the n x n matrix 4>(t; r) be the unique fundamental matrix solution of the homogeneous system dy

(IV.6.2)

dt

= A(t)y

determined by the initial-value problem dY = A(t)Y,

(IV.6.3)

Y(r) = In,

dt

where r E I. System (IV.6.2) is called the associated homogeneous equation of (IV.6.1). We treat the solution of (IV.6.1) by using the knowledge of the fundamental matrix solution 4)(t; r) of (IV.6.2) (cf. §§IV-2 and IV-3). Consider the transformation y' = 4)(t; r)z.

(IV.6.4)

Then, the problem (IV.6.1) is changed to

r)z(r) = 4.

dz _

(IV.6.5)

dt

In fact, differentiating both sides of (IV.6.4), we obtain

A(t)4)(t; r)l + b(t) =

r)z""+ 4 (t; r)

di* .

Since the unique solution of problem (IV.6.5) is given by

z= i +

jt(s;rY1(s)ds,

t

r_

y" _ (t;r)Iit + j4(s;T)_1(s)ds] l

(TV.6.6)

(t; r)r1 +

Jr

t

$(t; s)b(s)ds.

Here, use was made of the fact that 0(t; s) = 4'(t; r)O(s; r)-1 (cf. (2) of Remark IV-2-7; in particular (IV.2.6)). In a similar way, using (IV.6.6), we can change a nonlinear initial-value problem

d = A(t)17 + 9(t y),

y(r) = i

to an integral equation (IV.6.7)

y = 4 (t; 7-)17 +

Jr

t

'(t; s)g(s,7(s))ds.

IV. GENERAL THEORY OF LINEAR SYSTEMS

98

The function g(t, y-) is the nonlinear term which satisfies some suitable condition(s). In the case when the matrix A is independent of t, formulas (N.6.6) and (IV.6.7) become

g" = exp[(t - r)A]it + / t exp[(t - s)A]b(s)ds

(IV.6.8)

r

and

exp[(t - r)A]>; +

(IV.6.9)

Jr

exp[(t - s)A]9(s,y1s))ds,

respectively.

Example IV-6-1.

Let us solve the initial-value problem

dt = Ay + 6(t),

(IV.6.10)

where

A=

-2 0 3

1

[2]

0

-2 0 2

y(0)

,

b(t)=

0

#p=

1

1

,

t

1

0

The matrix A was given in Example IV-3-3. As computed in §IV-3, a fundamental matrix solution of the associated homogeneous equation is e-2t

0) = exp(tA) =

to-21

e-2t

0

et - e-2t et - (1 +

t)e-2t

0 0 et

Therefore, the solution of (IV.6.10) is r

t

exp[tA] j i0 + 111

fo

1 + to-2t e-2t

1

exp[-sA]b(s)ds } 111

-9-t+5et-(l+t)e

IV-7. Higher-order scalar equations In this section, we explain how to solve the initial-value problem of an n-th order linear ordinary differential equation (IV.7.1)

ao(t)ucn) + al(t)u(n-1) + ... + an-1(t)u + an(t)u = b(t),

u(r) = 'h, u'(r) = Q2, ...

where the coefficients a°(t), a1(t), ... ,

U(n-1)(,r)

,

=

and the nonhomogeneous term b(t) are

continuous on an interval I = {t : a < t < b}. In order to reduce (IV.7.1) to

99

7. HIGHER-ORDER SCALAR EQUATIONS

a system, setting yj = u and 112 = u', ...I yn = u(n-1), we introduce a vector yl

I.

Then, problem (IV.7.1) is equivalent to

tin d11

(IV.7.2)

dt

= A(t)f + b(t),

where 0 0

1

0

0

1

... ...

0 0

0 0

1

A(t) = an0

an(t) ao(t)

ao(t)

ao(t)

do(t)

ao(t)

al(t)

_a 2(t)

-3(t)

an-1(t)

0 0

b(t) =

0t) b(t)

ao(

Assuming ao(t) 54 0 on Z, let 4?(t) = [ 1(t) 2(t) ... hn(t)J be a fundamental matrix solution of the associated homogeneous equation d

at = A(t)y

(IV.7.3)

of (IV.7.2). Using 4i(t), we can solve problem (IV.7.2) (cf. §IV-6). The first component of the solution of (IV.7.2) is the solution of problem (IV.7.1). Also,

the first components of n column vectors of the matrix $(t) give the n linearly independent solutions of the associated homogeneous equation (IV.7.4)

ao(t)u(n) + al

(t)u(n-1)

+... +

an(t)u = 0

of (IV.7.1).

Example IV-7-1. Let us solve the initial-value problem (IV.7.5)

u"' - 2u" - 5u' + 6u = 3t,

u(0) = 1, u'(0) = 2, u"(0) = 0.

To start with, set

111=u, 112=u',

y2=u",

and y=

111 1122

IY31

IV. GENERAL THEORY OF LINEAR SYSTEMS

100

Then, problem (IV.7.5) is equivalent to dy"

(IV-7.6)

= Ay" + b'(t),

dt

y(0) =

#I,

where

10

A=

(IV.7.7)

0

-6

0

0

1

0 1,

b(t) =

[2].

and

0

3t

2

5

1

0 I,

Three eigenvalues of the matrix A are 1, -2, and 3. The corresponding projections are

-6 -1 P1(A) _ -6 -6 -1

1

-1

-6

I

P2(A)

,

1-2 =

10

-6

15

1

P3 (A)

3

1

1

1

1

-6 3 -18 9

3 9

12

-4

1

8

-2

-16

4

3t

-2

1

1

-18

9

9

and N = 0. Therefore, 4)(t; 0) = exp(tAJ = etP1(A) + e-2tP2(A) + e3tP3(A)

-6 -1

et

-6

_2t

1

-1

3

e -6

1

=-6 -6 -1

15

1

12

-4

1

8

-16

-2 + 10e -6 3 3. 4

Thus, t

4D(s;0)-1 6(s) ds 1

1 - (t + 1)e-t

1 + (2t - 1)e2t

1

1

1-(t+1)e-` +-20-2(1+(2t-1)e2tJ +1 - (t +

1)e-t

4[1 + (2t - 1)e2t]

1 - (3t + 1)e '3t 3(1-(3t+1)e-39

911 - (3t +

1)e-3t1

and, consequently, 1

-

0

1

+

1 - (3t + 3[1 - (3t + 9(1 - (3t +

1

[1_(t+1)e_tl

1

1 + (2t - 1)e2t

1 - (t + 1)e' + - -2(1 + (2t - 1)e2CJ 1 - (t + 1)e-3t

1)e-3t]

1)e-3t]

=

1)e-t

20

4[1 + (2t - 1)e2L]

30t + 25 - 17e'2t + 50et + 2e3L 17e-2t 2(15 + + 25et + 3e3t) 60 2(-34e-2t + 25et + 9e3t) 1

The first component of g(t) gives the solution of (IV.7.5).

,

101

7. HIGHER-ORDER SCALAR EQUATIONS

Remark IV-7-2. In the case of a second-order linear differential equation ao(t)u" + al (t)u' + a2(t)u = b(t),

(IV.7.8)

0' (t) ¢2(t) , .01(t) 02(t) where 01 (t) and ¢2 (t) are two linearly independent solutions of the associated ho-

a fundamental matrix solution of (IV.7.3) has the form 4P(t) = 1

mogeneous equation (IV.7.4). Since 4i(t)-1 =

.02(t)

-02(t) 01(t)

W(t) -011(t) W(t) = det(4i(t)), the first component of the formula

y"(t) _ (t}it + -t(t)

J

t

where

4>(s)-16(s)ds

gives the general solution of (IV.7.8) in the form

u(t) = r)1451(t) + 77202(t) (IV.7.9)

rt

42(s)

- 41(t) It ao(s)W(s)b(s)ds + 02(t)

01(s)

ao(s)W(s)b{s)ds,

where ill and q2 are two arbitrary constants. This is known as the formula of variation of parameters (see, for example, [Rab, pp. 241-2461). Moreover,

W(t) =W(r)exp

(IV.7.10)

f

ds do (s)

J

as in (4) of Remark IV-2-7.

Remark N-7-3. Write (IV.7.10) in the form 01(t)02(t) - dl (t)t2(t) = c2 exp [

(IV.7.11)

-

L

t s)ds

ft ao(s)

I

,

where C2 is a constant. Regarding (IV.7.11) as a differential equation for 02(t), w obtain 1 al(s) 02(t) = C101(t) +C201(t)

where cris anoth1er constant. x exp [ -

J t 01(T )2

Therefore,

[

-J

ao(s)ds }dry

t

1(t)

and ¢1(t)

f 1(r)2

J t _(s)ds] d7- form a fundamental set of solutions of ao(t)u"+al(t)u'+ s

a2(t)u = 0.

Remark IV-7-4. Let 01 (t) and 02(t) be linearly independent solutions of a thirdorder linear homogeneous differential equation (IV.7.12)

ao(t)u"' + a1(t)u" + a2(t)u' + a3(t)u = 0.

IV. GENERAL THEORY OF LINEAR. SYSTEMS

102

Also, let 0(t) be any solution of (IV.7.12). Then, (4) of Remark IV-2-7 implies that 0

[ft

02

01

10'

(IV.7.13)

401

=c3exp

452

'P 4i

ao(s)ds]

oz

where c3 is a constant. Write (IV.7.13) in the form Au(t)o" + A1(t)O' + A2(t)4 = c3exp

(IV.7.14)

[-Ic al(s)ds] ao(s)

,

where 461

02

A1(t) = 101

¢2

A2(t)

4

01

02

- 1,i

'02"

'

As(t)

COI

40

0'2

oz

A fundamental set of solutions of the homogeneous part of (IV.7.14) is given by {01,02}. Therefore, using (IV.7.9), we obtain 0(t) = C101 (t) + c202(t) + C3

I-01(t)

I

02

Ao

(T3 eXp I -

- J ao(s)ds]dr} + 02 (t) I ` Ao(r)2 e

J

T ao(s)

I dT

L

where cl and c2 are constants. This implies that 01(t), 02(t), and 02(r)

Aa(T)2

i)

exp [f'

ao(s)dsj

1(r) dr - -02(t) I Ao(r)2 exp [ -

f

)

ao(s)

ds] dr

form a fundamental set of solutions of (IV.7.12). EXERCISES IV

IV-1. Let A bean n x n matrix, and let A1, A2, ... , Ak be all the distinct eigenvalues of A. On the complex A-plane, consider k small closed disks Of = {A : IA - A2 I <

r)} (j = 1,... , k}. Assume that Aj n At = 0 for j # 1. Set

f P]

2'rf d

ail=rj(AI,, -

and

A)-1dA

(j = 1,... ,k),

N=A-S,

where the integrals are taken in the counterclockwise orientation. Show that

(i) Pl + ... + Pk = In,

(ii) Pi Pt = O if U'41), (iii) A = S + N is the S-N decomposition of A.

EXERCISES IV

103

Hint. Note that 1

r-o

{(u1n - A) -1 - (7-In - A) -1 } = (oln - A) -1 (rln - A) -1 .

Also, if p(A) is a polynomial in A with constant coefficients, then k

p(A) =

1j

p(A)(Aln - A)'1dA.

J=12rri

For general information, see [Ka, pp. 38-43). IV-2. Under the same assumptions as in Exercise IV-1, show that if f (A) is analytic in a neighborhood of each eigenvalue AJ, and an n x n matrix B is in a sufficiently small neighborhood of A, then f (B) is well defined and Biim f (B) = f (A).

Hint. If f (A) is analytic in a neighborhood of each eigenvalue A,, then

f (A) _

1 J=12rri

IV-3. Let A = S + N be the S-N decomposition of an n x n matrix A. Show that, as functions of A, the matrices S and N are not continuous. Hint. Use Lemma IV-1-4. IV-4. Let A be an n x n matrix with n eigenvalues Al, ... , An. Fix a real number r satisfying the condition r > R[AJ] (j = 1,... , n). Show that j+00

(F)

exp[tA] =

2T

io)In - A)-'do 00

for t E R.

Remark. This result means that exp[tA] is the inverse Laplace transform of (sln

A) 1. For example, if A = [

U1

], then (s12 - A) -1 =

s2

taking the inverse Laplace transform, we obtain exp(tA) =

+

1

[_i

cos t

I-sint

-

1 ]. Hence, sin t

For-

cost mula (F) is useful in the study of exp[tA] of an unbounded operator A defined on I.

a Banach space (cf. [HUP, Chapter XII, pp. 356-386]).

IV-5. Show that

J(A2 +T 21" )-'dr

= A-1 arctan(tA-1) if every eigenvalue of an n x n matrix A is a nonzero real number. m

IV-6. For an n x n matrix A, show that

In

lim

+00

(In + A ) = eA. M

IV. GENERAL THEORY OF LINEAR SYSTEMS

104

IV-7. Find the solution of the following initial-value problem

2 - 2A

+ A21r = etAe

b7(0) = n,

V(0) = S,

where A is a constant n x n matrix, {6, ,j,S are three constant vectors in C", and y E C' is the unknown quantity. Hint. If we set y = e`AU, the given problem is changed to

Al =E, d

u(0)

u"(0) =r1',

t2

IV-8. Find the solution of the following initial-value problem

2 +A21 =D,

9(0) = 40 ,

!Ly (0)

= 41

where E C" is an unknown quantity, A is a constant n x n matrix such that det A = 0, and {rjo, >7j } are two constant vectors in C".

Hint. Note that cos(xA) and sin(xA) are not linearly independent when det A = 0. If we define F(u) =

sin u

is

u

then the general solution of the given differential equation

j(x) = cos(xA)c"1 + xF(xA)c2. IV-9. Find explicitly a fundamental matrix solution of the system dy = A(t)y if there exists a constant matrix P E GL(n,C) such that P-"A(t)P is in Jordan canonical form, i.e.,

P-1A(t)P = diag[A, (t)I1 + N1iA2(t)12 + N2,... ,.1k(t)Ik +Nkj, where for each j, Ap(t) is a C-valued continuous function on the interval a S t S 6, I1 is the n, x n j identity matrix, and N, is an nl x n, matrix whose entries are 1 on the superdiagonal and zero everywhere else.

Hint. See [GH]. /r

IV-10. Find log (I

11

]).cos([

\` `

arctan

3

0

1

2

0

1

-1

-1 j).anii 1

698 1).

252

498

4134

-234

-465

-3885

-656

15

30

252

42

-10 -20 -166 -25 IV-11. In the case when two invertible matrices A and B commute, find

log(AB) - log(A) - log B if these three logarithms are defined as in Example IV-3-6.

105

EXERCISES IV

IV-12. Given that A =

0 0

0 0

2

3

1

0

0 0

1

0

-2 0

4

yi

I/2

'

1/3

0

1/4

find a nonzero constant vector ii E C4 in such a way that the solution l(t) of the initial-value problem d9

A9,

9(0)

satisfies the condition lim y"(t) = 0. t -+oo

IV-13. Assume that A(t), B(t), and F(t) are n x n, m x m, and n x m matrices, respectively, whose entries are continuous on the interval a < t < b. Let 4i(t) be

an n x n fundamental matrix of - = A(t)4 and W(t) be an in x in fundamental matrix of Z = B(t)u. Show that the general solution of the differential equation on an ii x m matrix Y dY (E)

= A(t)Y - YB(t)

F(t),

_WT

is given by

Y(t) = 4;(t) C 4,(t)-1

r`

where C is an arbitrary constant n x m matrix.

IV-14. Given the n x n linear system dY dt

= A(t)Y - YB(t),

where A(t) and B(t) are n x n matrices continuous in the interval a < t < b, (1) show that, if Y(to) -I exists at some point to in the interval a < t < b, then Y(t)-i exists for all points of the interval a < t < b, (2) show that Z = Y-1 satisfies the differential equation dZ = B(t)Z - ZA(t). dt

IV-15. Find the multipliers of the periodic system dt = A(t)f, where A(t) _ [cost

simt

-sint

cost

IV. GENERAL THEORY OF LINEAR SYSTEMS

106

Hint. A(t) = exp It 101 0]] IV-16. Let A(t) and B(t) be n x n and n x m matrices whose entries are realvalued and continuous on the interval 0 < t < +oo. Denote by U the set of all R'-valued measurable functions u(t) such that ,u(t)j < 1 for 0 < t < +oo. Fix a

l`

vector { E R". Denote by ¢(t, u) the unique R"-valued function which satisfies the initial-value problem

at

= A(t)i + B(t)u(t), i(0)

where u E U. Also, set

R = {(t, ¢(t, u)) : 0:5 t < +oo, u E R}. Show that R is closed in

Rn+i.

Hint. See ILM2, Theorem 1 of Chapter 2 on pp. 69-72 and Lemma 3A of Appendix of Chapter 2 on pp. 161-163j.

IV-17. Let u(t) be a real-valued, continuous, and periodic of period w > 0 in t on R. Also, for every real r, let ¢(t, r) and t1'(t, r) be two solutions of the differential equation d2y dt2

(Eq)

- u(t) y = 0

such that

o(r, r) = 1,

0'(r, r) = 0,

and

0(7-,T) = 0,

t;J'(T,T) = 1,

where the prime denotes d Set 0(0, r) C(r) = 10'(0.'r)

V(0, r) 1 i,I (0,r) J

,

0(r+w,r) Vl(r+w,r) 0(r+w,r) tG'(r+w,r)

M(r) _ (I) Show that

M(r) = C(r)'1M(O)C(r). (11) Let A+ and A_ be two eigenvalues of the matrix M(r). Let

K _(r)

be eigenvectors of M(r) corresponding to A+ and

I

A_`,

K+(r)J and respectively.

how that (i) K±(r) are two periodic solutions of period w of the differential equation

dK dT

± K2 + u(r) = 0,

(ii) if we set

Qf{t,r) = exp [J t Kt(() d(l r

these two functions satisfy the differential equation (Eq) and the conditions

13f(t+w,r) = A*O*(t,r).

107

EXERCISES IV

Hint for (I). Set 4'(t,r) _ matrix solution of the system

(t

(t'

)

Then, ((t, r) is a fundamental

I JJJ

-

and y = 4'(t, r)c is the solution satisfying the initial condition y(r) = c'. Note that C(r) = 4'(0, r) and M(r) = 4'(r + w, r). Therefore, 4'(t, r) = 4'(t, 0)C(r) and 4'(t + r) = 4'(t, r)M(r). Now, it is not difficult to see 4'(w, r) = 4'(w, 0)C(r) _ 4'(0, r)M(r) = C(r)M(r) and 4'(w, 0) = M(O). Hint for (II). Since eigenvalues of M(r) and M(0) are the same, the eigenvalues of M(r) is independent of r. Solutions 77+ (t) of (Eq) satisfying the condition Y7+ (t +

w) = A+n+(t) are linearly dependent on each other. Hence,

W+(t)

= K+(t) is

independent of any particular choice of such solutions q+(t). In particular K+(t + W) = 17'+ (t + w) = K+(t). Problem (II) claims that the quantity K+(r) can be 17+(t + w)

found by calculating eigenvectors of M(r) corresponding to A+. Furthermore, A+ = exp I

o o

K+(r)drJJ I I.

The same remark applies to A_. The solutions 0±(x, r)

of (Eq) are called the Block solutions.

IV-18. Show that (a) A real 2 x 2 matrix A is symplectic (i.e., A E Sp(2, R)) if and only if det A = 1;

(b) the matrix

01

l

0J

N is symmetric for any 2 x 2 real nilpotent matrix N.

Hint for (b). Note that det(etNJ = I for any real 2 x 2 nilpotent matrix N. IV-19. Let G, H, and J are real (21n ) x (2n) matrices such that G is symplectic,

H is symmetric, and J = I 0n

. Show that G-1JHGJ is symmetric.

IV-20. Suppose that the (2n) x (2n) matrix 4'(t) is the unique solution of the initial-value problem L'P = JH(t)4', 4'(0) = 12n, where J = I 0n n and H(t) l T J is symmetric. Set L(t) = 4'(t)-1JH(t)4'(t)J. Show that d4'dt) = JL(t)4'(t)T, where 4'(t)T is the transpose of 4'(t).

CHAPTER V

SINGULARITIES OF THE FIRST KIND In this chapter, we consider a system of differential equations

xfy

(E)

= AX, Y-),

assuming that the entries of the C"-valued function f are convergent power series in complex variables (x, y-) E Cn+I with coefficients in C, where x is a complex inde-

pendent variable and y E C" is an unknown quantity. The main tool is calculation with power series in x. In §I-4, using successive approximations, we constructed power series solutions. However, generally speaking, in order to construct a power 00

series solution y"(x) = E xmd n, this expression is inserted into the given differM=1

ential equation to find relationships among the coefficients d, , and the coefficients d,,, are calculated by using these relationships. In this stage of the calculation, we do not pay any attention to the convergence of the series. This process leads us to the concept of formal power series solutions (cf. §V-1). Having found a formal power series solution, we estimate Ia',,, i to test its convergence. As the function x-I f (x, y) is not analytic at x = 0, Theorem 1-4-1 does not apply to system (E). Furthermore, the existence of formal power series solutions of (E) is not always guaranteed. Nevertheless, it is known that if a formal power series solution of (E) exists, then the series is always convergent. This basic result is explained in §V-2 (cf. [CL, Theorem 3.1, pp. 117-119) and [Wasl, Theorem 5.3, pp. 22-251 for the case of linear differential equations]). In §V-3, we define the S-N decomposition for a lower block-triangular matrix of infinite order. Using such a matrix, we can represent a linear differential operator (LDO)

G[Yj = x

+ f2(x)lj,

where f2(x) is an n x n matrix whose entries are formal power series in x with coefficients in Cn. In this way, we derive the S-N decomposition of L in §V-4 and a normal form of L in §V-5 (cf. [HKS]). The S-N decomposition of L was originally defined in [GerL]. In §V-6, we calculate the normal form of a given operator C by using a method due to M. Hukuhara (cf. [Sill, §3.9, pp. 85-891). We explain the classification of singularities of homogeneous linear differential equations in §V-7. Some basic results concerning linear differential equations given in this chapter are also found in [CL, Chapter 4].

108

1. FORMAL SOLUTIONS OF AN ALGEBRAIC DE

109

V-1. Formal solutions of an algebraic differential equation We denote by C[[x]] the set of all formal power series in x with coefficients in w

00

b,,,x', we define

C. For two formal power series f = E amxm and g = m=0

m=0

f = g by the condition an = b", for all m > 0. Also, the summf + g and the 00

00

(am +b,,,)xm and f g =

product f g are defined by f +g = m=0

(F am-nbn )x"', m=0 n=0

00

respectively. Furthermore, for c E C and f = >2 a,,,xm E C[[x]], we define cf m=0

00

by cf = >camx'. With these three operations, C[[x]] is a eommutattve algebra m=0

over C with the identity element given by E bmxm, where bo = 1 and b,,, = 0 m=0

if m > 1. (For commutative algebra, see, for example, [AM].) Also, we define the

of f with respect to r by dx _ E(m+ 1)a,,,+,x' and the integral

derivative

m=0 rX

(

airi x"' for f = E a,,,xm E C[[xj]. Then, C[[x]]

f (x)dx by f f (x)dx = o

0

00

"0

r

m=0

m=1

is a commutative differential algebra over C with the identity element. There are some subalgebras of C[[x]j that are useful in applications. For example, denote by C{x) the set of all power series in C[]x]] that have nonzero radii of convergence. Also, denote by C[x] the set of all polynomials in x with coefficients in C . Then, C{x} is a subalgebra of C[[x]] and C]x] is a subalgebra of C{x}. Consequently, C[x] is also a subalgebra of C[[x]]. Let F(x, yo, y, , ... , yn) be a polynomial in yo, y,..... yn with coefficients in C[[x]]. Then, a differential equation dy ,...dxn

y,,d"y

F

(V.1.1)

f

=0

is called an algebrutc differential equation, where y is the unknown quantity and x

is the independent variable. If F ` x, f, ... ,

dxn

= 0 for some f in C[[x]], then

such an f is called a formal solution of equation (V.1.1). In this definition, it is not necessary to assume that the coefficients of F are in C{x}. Example V-1-1. To find a formal solution of xLY

(V.1.2)

+y-x=0,

set y = Famx'. Then, it follows from (V.1.2) that ao = 0, 2a, = 1, and m>0

(m + 1)am = 0 ( m > 2 ). Hence, y = 2x is a formal solution of equation (V.1.2).

V. SINGULARITIES OF THE FIRST KIND

110

In general, if f E C{x} is a formal solution of (V.1.1), then the sum of f as a convergent series is an actual solution of (V.1.1) if all coefficients of the polynomial

FareinC{x}. Denote by x°C([x]j the set of formal series x°f (x), where f (x) E C([x]], a is a complex number, and x° = exp[a logx]. For 0 = x°f E x°C[[x]], 0 =

x°g E x°Cj[x]j, and h E C(jx]], define 0 + iP = x(f + g), hO = x°hf, and xL = ox°f + x°x Then, x°C([xfl is a commutative differential module over .

the algebra C((x]]. Similarly, let x°C{x} denote the set of convergent series x°f (x),

where f (x) E C{x}. The set x°C{x} is a commutative differential module over the algebra C{x}. Furthermore, if a is a non-negative integer, then x°C([x]] C C[[x]]. If F(x, yo,... , y,) is a formal power series in (x, yo,... , yn) and if fo(x) E xC[[x]j, ... , fn(x) E xC([xj], then F(x, fo(x),... , fn(x)) E C[[xJj. Also, if F(x, yo,. .. , yn) is a convergent power series in (x, yo,... , yn) and if fa(x) E xC{x}, ... , fn(x) E xC{x}, then F(x, fo(x), ... , f,,(x)) E C{x}. Therefore, using the notation x°C[[x]]n to denote the set of all vectors with n entries in x°C[(x]], we can define a formal solution fi(x) E xC[[x]]n of system (E) by the condition x = Ax, i) in C[[x]j . (Similarly, we define x°C{x}n.) Also, in the case of a homogeneous sys-

tem of linear differential equations to be a formal solution if x theorem.

xfy

A(x)g, a series d(x) E x°C[[xj]n is said

= A(x)e in x°C[[x]jn. Now, let us prove the following

Theorem V-1-2. Suppose that A(x) is an n x n matrix whose entries are formal power series in x and that A is an eigenvalue of A(O). Assume also that A + k are not eigenvalues of A(O) for all positive integers k. Then, the differential equation (V.1.3)

x!Ly

= A(x)g

has a nontrivial formal solution fi(x) = xa f (x) E x"C[[x]]n Proof. 00

Insert g = xa E x'nam into (V.1.3). Setting A(x) _ y2 xmAm, where An E m=0

m=0

Mn(C) and Ao = A(O), we obtain 00

xa

Co

(A + m)xm= xa

LtAm_i.

Therefore, in order to construct a formal solution, the coefficients am must be determined by the equations m-h ,1ao = Aoa"o

and

(A + m)am = Aodd,n + E Am-hah h=0

(m > 1 ).

1. FORMAL SOLUTIONS OF AN ALGEBRAIC DE

111

Hence, ado must be an eigenvector of Ao associated with the eigenvalue A, whereas m-h

Am-hah form? 1. 0

ii',,, = ((A + m)1. - A0)-1 h

For an eigenvalue Ao of A(O), let h be the maximum integer such that Ao + h is also an eigenvalue of A(O). Then, Theorem V-1-2 applies to A = A0 + h. The convergence of the formal solution ¢(x) of Theorem V-1-2 will be proved in §V-2, assuming that the entries of the matrix A(x) are in C(x) (cf. Remark V-2-9). n

Let P = Eah (x)&h (b = x d h=0

/

be a differential operator with the coefficients

ah(x) in C([x]]. Assume that n > 1 and an(x) 54 0. Set ab

P[x'] = x'

fm(s) xm, m=np

where the coefficients f,,, (s) are polynomials in s and no is a non-negative integer 0. Then, we can prove the following theorem. such that

Theorem V-1-3. (i) The degree of f,,,, in s is not greater than n. (ii) If zeros of fno do not differ by integers, then, for each zero r of f,,, there exists a formal series x' (x) E xr+IC[[x]] such that P[xr(1 +O(x))] = 0.

Proof (i) Since 6h(x'] = shx', it is evident that the degree of f,,,, in s is not greater than n.

+o0

(ii) For a formal power series d(x) = E ckxk, we have k=1

+oo

P[x'(I + b(x))] = P(x'] +

L. Ck P[x'+k

J

k=1 +oo

= x'

+oo

{mnof"

(s)xm + > Ckxk k =1

+oo

+ k)xm/ (frn(s m=no }

((,fm(s)

= x' fno(s)x +

+oo

m -no

Ckfm-k(s + k))

+ k=1

xa+no AK( + M=1

x"`m J1

+ E Ckfno+m-k(s + k)) x"'

.

J

k=1

In order that y = x''(1 +b(x)) be a formal solution of P[y] = 0, it is necessary and sufficient that the coefficients cm and r satisfy the equations m

fno (r) = 0,

fno+m (r) + E Ckfno+m-k(r + k) = 0 k=1

(m > 1).

V. SINGULARITIES OF THE FIRST KIND

112

It is assumed that f,,,, (r + m) 54 0 for nonzero integer m if r is a zero of Therefore, r and c,,, are determined by m-1

(r) = 0 and cm = -7,(

1

r + m)

ckfn+m-k(r + k)

(m > 1).

Convergence of the formal solution x'(1+¢(x)) of Theorem V-1-3 will be proved in s is n and the coefficients at the end of §V-7, assuming that the degree of ah(x) of the operator P belong to C{x}. The polynomial f.,, (s) is called the indicial polynomial of the operator P.

Remark V-1-4. Formal solutions of algebraic differential equation (V.1.1) as defined above are not necessarily convergent, even if all coefficients of the polynomial 00

F are in C{x}. For example, y = Y (-1)m(m!)xm+i is a formal solution of the 2y

m=0

differential equation 2d + y - x = 0. Also, the formal solution x' (I + fi(x)) of 2 Theorem V- 1-3 is not necessarily convergent if the degree of f,,. (s) in s is less than

n. In order that f = >a,,,xm be convergent, it is necessary and sufficient that m=0

la,,, I < KA' for all non-negative integers m, where K and A are non-negative num00

bers. Also, it is known that some power series such as

(m!)mxm do not satisfy

any algebraic differential equation. The following result'n--R gives a reasonable necessary condition that a power series be a formal solution of an algebraic differential equation.

Theorem V-1-5 (E. Maillet [Mail). Let F(x, yo, yi, ... , yn) be a nonzero polynomial in yo, yi, ... , y, with coefficients in C{x}, and let f = E amxm E CI[x]] be m=0

a formal solution of the differential equation

F(x, y, Ly, ... , dny dxn = 0.

/

Then,

there exist non-negative numbers K, p, and A such that (V.1.4)

I am I

< K(m!)PA' (m > 0).

We shall return to this result later in §XIII-8. Remark V-1-6. In various applications, including some problems in analytic number theory, sharp estimates of lower and upper bounds of coefficients am of a formal

solution f = F,00axm of an algebraic differential equations are very important. m =9

For those results, details are found, for example, in [Mah], [Pop], [SS1], (SS2], and [SS3]. The book [GerT] contains many informations concerning upper estimates of coefficients lam I.

2. CONVERGENCE OF FORMAL SOLUTIONS

113

V-2. Convergence of formal solutions of a system of the first kind In this section, we prove convergence of formal solutions of a system of differential equations X

(V.2.1)

fy

= AX, Y-),

where y E C' is an unknown quantity and the entries of the C"-valued function f are convergent power series in (x, y7) with coefficients in C. A formal power series 00

E x' , ,,,

(V.2.2)

(c,,, E C" )

E xC([xfl"

rn-t is a formal solution of system (V.2.1) if (V.2.3)

X

dx

= f(x,0)

as a formal power series. To achieve our main goal, we need some preparations.

Observation V-2-1. In order that a formal power series (V.2.2) satisfy condition (V.2.3), it is necessary that f (O, 0) = 0. Therefore, write f in the form

f (x, y-) = o(2-) + A(x)g + F, y~'fv(x), Iel>2

where

(1) p = (pl,... ,p") and the p, are non-negative integers, (2) jpj = pi +

+ pn and yam' = yi'

yn^, where yi,...are the entries

of g,

(3) fo E xC{x}" and f, E C{x}", (4) A(x) is an n x n matrix with the entries in C{x}. Note that and

fo(x) = Ax, 6)

A(x) _ Lf(x,0).

Setting

A(x) = F, x'A"s

Ao =

M=0

LZ

0,0) I

where the coefficients A", are in M"(C), write condition (V.2.3) in the form xd'o dx

= Ao +

f (x, ¢)

- A0

.

Then, (V.2.4)

"t,,

= A0E

+ rym

for m = 1, 2, ... ,

V. SINGULARITIES OF THE FIRST KIND

114

where (J(x))p

f (x, 0) - AoO = o(x) + IA(x) - Ao1 fi(x) +

fp(x) IpI>2

00

1: xm7m M=1

and Im E C n. Note that ry'm is determined when c1 i ... , c,,,_ 1 are determined and

that the matrices mI" - A0 are invertible if positive integers m are sufficiently large. This implies that there exists a positive integers mo such that if 61, ... , C,,,a are determined, then 4. is uniquely determined for all integers m greater than mo. Therefore, the system of a finite number of equations (V.2.5)

mEm

+

= A0

(m = 1,2,... ,mo)

decides whether a formal solution ¢(x) exists. If system (V.2.5) has a solution {c1

i ... , c,,,a }, those mo constants vectors determine a formal solution (x) uniquely.

Observation V-2-2. Supposing that formal power series (V.2.2) is a formal soluN

x'6,. Since

tion of (V.2.1), set ¢N (x) _ m=1

Ax, (x))

- f (x, N(x)) = A(x)(d(x) - N(x)) + F, [d(x)p - N(x)D] ff(x), Ipl>2

it follows that AX, $(x)) - f (X, dN(x)) E xN+C[Ix]]". Also, xd¢N(x)

= xdd(x) dx

dx

Hence, AX, 4V W) (V.2.6)

-

xdoN(x) E xN+1C((x((n. dx

xdON(x) E xN+1C{x}". Set dx

zddN(x) dx

9N,0(x) =

Now, by means of the transformation y" = z1+ y5N(x), change system (V.2.1) to the system

dz

xdx = JN(x,z

(V.2.7)

on i E C", where

9N(x, l = f (x, z + dN(x)) -

xdjN(x)

dx

= 9N,O(x) + f(x,Z+dN(x))f r- f(x,dN(x))

= 9N,0(x) + A(x)z" + L. [(+ $N(x))" - N(x)'] f,,(x) InI?2

115

2. CONVERGENCE OF FORMAL SOLUTIONS

As in Observation V-2-1, write gN in the form 9N(x,

= 9N,O(x) + BN(x)z + L xr 9N,p(x), Ip1?2

where (1) 9N,o E xN+1C{x}" and g""N,p E C{x}",

(2) BN(x) is an n x n matrix with the entries in C{x}, (3) the entries of the matrix BN(X) - Ao are contained in xC{x}. Observation V-2-3. System (V.2.7) has a formal solution 00

(V.2.8)

ON(x) = O(x) - ON(x) _

x'"cn, E xN+1C1jx11 n.

m=N+1

The coefficients cm are determined recursively by

me,, = Aocm + '7m

for m = N + 1, N + 2, ... ,

where

9N(x,ILN(x)) - AO1GN(x) = f(x,$(x)) - Ao1 N(x) -

-

= f (x, fi(x))

AOcN(x) -

dx

xdVx)

= 9N,O(x) + [BN(x) - Ao1 i'N(x) + E (ZGN(x))p 9N,p(x) Ip1>2 ou

=E

xm-'1'm

m=N+1

Note that the matrices min - Ao are invertible for m = N + 1, N + 2, ... , if N is sufficiently large.

Observation V-2-4. Suppose that system (V.2.1) has an actual solution q(x) such that the entries of i(x) are analytic at x = 0 and that ij(0) = 0. Then, the dim (0) of q(x) at x = 0 is a formal solution of Taylor expansion ¢(x) _

j

(V.2.1). Furthermore, is convergent and E xC{x}". Keeping these observations in mind, let us prove the following theorem.

Theorem V-2-5. Suppose that fo(x) = f (x, 0) E xN+1C{x}n and that the matrices mIn - A0 (m > N + 1) are invertible, where Ao = (V.2.1) has a unique formal solution

69

00

(V.2.9)

t(x) =

E X'n4n E x^'+1C((x}]n. m=N+1

Furthermore, ¢ E xN+1C{x}n.

(0, 0). Then, system

V. SINGULARITIES OF THE FIRST KIND

116

Remark V-2-6. Under the assumptions of Theorem V-2-5, system (V.2.1) possibly has many formal solutions in xC[[x)J". However, Theorem V-2-5 states that there is only one formal solution in xN+1C[[xJJn

Proof of Theorem V-2-5.

We prove this theorem in six steps.

Step 1. Using the argument of Observation V-2-1, we can prove the existence and uniqueness of formal solution (V.2.9). In fact,

f (x, 4) - Ao¢ = o(x) + JA(x) - Ao}fi(x) +

E (_)P_)

IpI?2 00

xmrym.

m=N+1

(m > N + 1) are

This implies that rym = 0 for m = 1, 2, ... , N. Hence, uniquely determined by

for m=N+1,N+2,....

mc,,, = Ao6,,, + rym

Step 2. Suppose that system (V.2.1) has an actual solution q(x) satisfying the following conditions:

(i) the entries of rl(x) are analytic at x = 0, (ii) there exist two positive numbers K and 6 such that

jy(x)I < KIxjN+1 00

xm

Then, the Taylor expansion E m=N+1

ml

a.,.7m'1

IxI < 6.

for

(0) of #(x) at x = 0 is a formal solution

of (V.2.1) (cf. Observation V-2-4). Since such a formal solution is unique, it follows

that $(x) = E

xm d1ij -

(0). Because the Taylor expansion of if(x) at x = 0 is

m=N+1

convergent, the formal solution ¢ is convergent and

E

xN+1C{x}".

Step 3. Hereafter, we shall construct an actual solution f(x) of (V.2.1) that satisfies conditions (i) and (ii) of Step 2. To do this, first notice that there exist three positive numbers H, b, and p such that (I)

If(x,o)I 5

HIxIN+1

for

IxI < 5

and

(II)

If(x,91) - f(x,y2)I 5 (IAoI + 1) 1111 - y2I for

Ix) < b and

Iy, I


(j = 1, 2).

Hence,

(III)

If(x, y-)i <_ HIxl "+1 + (IAoI + 1)Iyl

for

Ixl < b and

Iyl

<_

p.

2. CONVERGENCE OF FORMAL SOLUTIONS

117

Using the transformation of Observation V-2-2, N can be made as large as we want without changing the matrix A0. Hence, assume without loss of any generality

that IAoi + 1

(V.2.10)

1

N+1 < 2

Also, fix two positive numbers K and b so that H + NAol

K>

(V.2.11)

i

1]K

and K6N+1 < p

Step 4. Change system (V.2.1) to an integral equation

: (V.2.12)

J

f

n+(f ))

Define successive approximations

)b(x) = 0 and 1?k+1(x) = IZ

for

k=0,1,2,....

Now, we shall show that

lnk(x)I < KIxIN+1

for

jxl
Since this is true for k = 0, we show this recursively with respect to k as follows. First if this is true for k, then

I i k(x)1 < Kjxl N+l < K&N+I < p

jxj < b.

for

Hence,

I'.F(c nk (o) < {H + IIAol + 1jK}j£VN

for

6.

Therefore, II7k+1(x)I <

H + IAoI

1]K

IxIN+I < K 1xIN+1

for

jxj < S.

1

Step 5. Set I1'Rk+1

--

nkll = maxiInk+iIZI

++1k(x)I

InI < b}.

Then, since

I ik+1(x) - ik(x)I = fo we obtain IInk+1 - llkII

j

N ++1

171k - '1k-11I

2lI'7k - 1k-l II

This implies that lim ilk(x) _

k-.+00 xN+1

exists uniformly for Ixl < J.

#e+1(x) - #((X) e=0

xN+1

V. SINGULARITIES OF THE FIRST KIND

118

Step 6. Setting #(x) = xN+1 ("liM xJ`'+1) = ku> nik(x), it is easy to show that n'(x) satisfies integral equation (V.2.12). It is also evident that i7(x) is analytic for IxI < 6. Thus, the proof of Theorem V-2-5 is completed. 0 Now, finally, by using the argument given in Observations V-2-2 and V-2-3, we obtain the following theorem.

Theorem V-2-7. Every formal solution

E xC[[zjJ" of system (V.2.1) is conver-

gent, i.e., E xC{x}". Remark V-2-8. In general, (V.2.1) may not have any formal solutions. However, Theorem V-2-7 states that if (V.2.1) has formal solutions, then every formal solution is convergent.

Remark V-2-9. If yi(x) = xA f (x) E XAC[[xfj" is a formal solution of a linear system

X!LY

= A(x)y, the formal power series f is a formal solution of

xd

_

[A(x) - A!, }i. Therefore, f E C{x}" by virtue of Theorem V-2-7. This proves convergence of the formal solution constructed in Theorem V-1-2.

V-3. The S-N decomposition of a matrix of infinite order In §V-4, we shall define the S-N decomposition of a linear differential operator. As a linear differential operator will be represented by a lower block-triangular matrix of infinite order, we derive, in this section, the S-N decomposition of such a matrix All 0 0 ... ... ... A21

A22

...

0

Amt Am2 ...

Am,n

... ...

0

.

.

where for each (j, k), the quantity AJk is an n, x nk constant matrix. Set

Al = All, All

0

A21

A22

O O

Am l

Am2

...

.. ..

0 0 (m > 2).

Am =

Form > 2, we write Am - Am-! [

B.

... Am,

0 Amm J

Set Nm =

m

l-1

nt. Then, A. is an

N", x Nm matrix, while Bm is an nm x N,"_ 1 matrix. Also, let Amm = Smm +Nmm

3. THE S-N DECOMPOSITION OF A MATRIX OF INFINITE ORDER 119

and Am = Sm + Aim be the S-N decompositions of Amm and Am, respectively, where Smm is an nm x nm diagonalizable matrix, Sm is an Nm x Nm diagonalizable matrix, Nmm is an nm x nm nilpotent matrix, Aim is an N. x Nn nilpotent matrix, SmmNmm = NmmSmm, and SmAm = NmSm. The following lemma shows how the two matrices Sm and Aim look. Lemma V-3-1. The matrices Sm and Aim have the following forms: S"'

S1 = s11,

J N, = Ni1,

0l

rI Sm_1

=

L Cm

Aim =

(m > 2),

Smm

[Aim_i

fm

(m > 2), O 1 Nmm

where Cm and fm are am x N,_1 matrices. Proof Consider the case m > 2. Since the matrices Sm and Aim are polynomials in Am with constant coefficients, it follows that

Sm = [B m1

and

Aim =

I

fm1 01,

ILM0

where 13m-1 and Dm-1 are Nm-1 x Nm_1 matrices, Cm and fm are nm x Nm_1 matrices, and µm and vm are nm x nm matrices. Furthermore, Dm_1 and Vm are nilpotent. Also, Bm-1Dm-1 = Dm-1Bm-i and tlmvm = 1.m µm. Hence, it suffices to show that Bm-1 and µm are diagonalizable. Note that since Sm is diagonalizable, Sm has Nm linearly independent eigen vectors. An eigenvector of Sm has one of two forms k

PJ

and [?.] , where p is an

eigenvector of Bm_1i whereas q is an eigenvector of µm. Therefore, if we count those independent eigenvectors, it can be shown that Bm-1 has N,, -I linearly independent eigenvectors, while Pm has nm linearly independent eigenvectors. Note that Nm = Nm_ 1 + nm. This completes the proof of Lemma V-3-1. 0 Lemma V-3-1 implies that the matrices Sm and Nn have the following forms:

S1 = SI1, S11

0

C21

S22

0 ... 0 ...

Cm2

...

...

Nil

0

D

...

Y21

JN22

0

Fm 1

Fm2

... ...

Sm = Cmi and

(m>2)

N1 = N11, Aim =

0 0 Smm

0 0 Nmm

(m>2),

V. SINGULARITIES OF THE FIRST KIND

120

where C,k and Fjk are n3 x nk matrices. Set S11

0

C21

S22

0 0

Cml

Cm2

"'

S =

0 N22

0 0

"'

Fm2

...

.Wmm

N11

f f21

1m 1

Smm 0

N= 0

Then,

A=S+N

(V.3.1)

and

SN=NS.

We call (V.3.1) the S-N decomposition of the matrix A.

V-4. The S-N decomposition of a differential operator Consider a differential operator

C(yl = x

(V.4.1)

+ f2(x)Y,

w

where f2(x) _ > xt1 and 1

Let us identify a formal power series

E

t=o ao

xt dt with the vector p =

a1

a2

E C. Then, the operator C is represented

t>o

by the matrix flo

0

f2l

In + f2o

0 0

f12

l1

21, + f2o

11m

f4.-l

fZn-2

0 ... 0 ... 0 ...

A =

...

f21

mI, + fZa

0

.

5. A NORMAL FORM OF A DIFFERENTIAL OPERATOR

121

where I is the n x n identity matrix. Let A = S + N be the S-N decomposition

j)

of A. Since A =

11

R2

A1, where (i =

and I,> is the oo x oo identity

L

matrix, the matrices S and N have the forms al

(V.4.2)

O 0 0

0 0

0

So a2

I. + So al

21n + So

am

am-l

am-2

...

...

...

...

...

...

at

min + SO

0

No 0

...

S=

and

O No

O

V1

V2

V1

No

Vm

Vm-1

Vm-2

No

(V.4.3)

0

N=

...

VI

respectively, where the a. and v, are n x n matrices and S2o = So +No is the S-N decomposition of the matrix S2o. 00

00

Set a(x) = So + E xtat and v(x) = No +

trot.

Then, S represents a

tvl

t_1

+ a(x)y, while N represents multiplication by differential operator Lo[y1 = x v(x) (i.e., the operator: y" -+ v(x)y-). Since A = S + N and SN = NS, it follows that

(V.4.4)

L[gf = Lo[Yj + v(x)y"

and

Lo[v(x)y1 = v(x)Lo[yl-

We call (V.4.4) the S-N decomposition of operator (V.4.1). We shall show in th next section that Lo[yj is diagonalizable and v(x)n = O.

V-5. A normal form of a differential operator Let us again consider the differential operator

L[yj = X

(V.5.1)

f

+ Q(x)U,

00

where f2(x) = Ex1fle and Sgt E WC). In §V-4, we derived the S-N decomposition (V.5.2)

t=o

L[y-] = Lo[yl + v(x)y"

and

Co[v(x)yl = v(x)Lo[yj.

V. SINGULARITIES OF THE FIRST KIND

122

where (V.5.3)

+ a(x)y

Co[YI = x

and

Co

a(x) = So +

00 xtat

and

v(x) = No +

xtvt t_1

t=1

(cf. (V.4.4)). Notice that N = So + No is the S-N decomposition of S2o and that the operator Go[ul and multiplication by v(x) are represented respectively by matrices (V.4.2) and (V.4.3). Set

0 I. +,%

0 0

0 0

al

21 + So

0

am_ 1

am-2

Sm =

Since So is diagonalizable, there exist an invertible n x n matrix P0 and a diagonal matrix Ao such that

SoPo = PoAo,

NO = diag[al, A2, ... , . \n)-

Note that A, (j = 1, 2,... , n) are eigenvalues of So. Hence, A. (j = 1, 2,... , n) are also eigenvalues of flo. For every positive integer m, Sm is diagonalizable. Hence, Po

there exists an (m + 1)n x n matrix P.. =

Pi

such that SmPm = PmA0. If

Pm

m is sufficiently large, we can further determine matrices Pt for t > m + 1 by the equations t (V.5.4)

(tl + S0)Pt + E ahP1_h = PtAo. h=1

Equation (V.5.4) can be solved with respect to Pt, since the linear operator Pt -+ (tl + So)P1 - PtAo is invertible if m is sufficiently large. In this way, we can find

A P

1

an oo x n matrix P =

I

such that SP = PAa. Set P(x) = >x'P,. Then,

Pm

t=D

entries of P(x)-1 are also formal power series in x and (V.5.5)

Co[P(x)j = P(x)Ao.

5. A NORMAL FORM OF A DIFFERENTIAL OPERATOR

123

Define two differential operators and an n x n matrix by X [U1

{

= P(x)-iC(P(x)uZ,

/Co(u1 = P(x)-'Co(P(x)Uj,

vo(x) = P(x)-'v(x)P(x),

respectively. Then 1C(u'j = X 0(u'] + vo(x)u'. Observe that

.- (V.5.6)

_.. _, f _.

_.

=x du

,

l du' \

_

. _.

1

+Aou".

This shows that the operator Co(y1 is diagonalizable. Furthermore, dvo

(V.5.7)

x

dxx) + Aovo(x) =

P(x)-'Lo(P(x)vo(x)1

= P(x)`Co[v(x)P(x)1

= P(x)-'v(x)Lo(P(x)1 = vo(x)P(x)-'Co(P(x)1 = vo(x)Ao

This shows that the entry i,k(x) on the j-th row and k-th column of the matrix vo(x) must have the form v,k(x) = 7jkxAk-a', where ryjk is a constant. Since Lt(x) is a formal power series in x, it follows that (V.5.8)

vjk(x) = 0

if

Ak - )j is not a non-negative integer.

Observe further that the matrix &. (x) can be written in the form

vo(x) = x-^°rx^°,

where

'711

'Yin

'7ni

Inn

r=

Hence, for any non-negative integer p, we have vo(x)P = x-A0rPxAo, where

z^O = exp ((log x)Ao] = diag(za' , Z112'... , za^ I. On the other hand, since No is nilpotent, vo(x)P can be written in a form i (x)P =

X'pQP(x), where mp is a non-negative integer such that lim mP = +oo and P+oo the entries of the matrix Qp(x) are power series in x with constant coefficients. Therefore, L0(x)P = 0 if p is sufficiently large. This implies that the matrix r is nilpotent. Since v(x) = P(x)vo(x)P(x)-1, we obtain v(x)^ = O. Thus, we arrive at the following conclusion.

Theorem V-5-1. For a given differential operator (V.5.1), let 0o = So +No be the S-N decomposition of the matrix N. Then, there exists an n x n matrix P(x) such that (1) the entries of P(x) are formal power series in x with constant coefficients, (2) P(O) is invertible and SoP(0) = P(0)Ao, where A0 is a diagonal matrix whose diagonal entries are eigenvalues of Q0r

V. SINGULARITIES OF THE FIRST KIND

124

(5) the transformation

P(x)u

(V.5.9)

changes the differential operator (V.5.1) to another differential operator (V.5.10)

P(x)-1G(P(x)ui = KOK + vo(x)u,

where

Ko(61 = x

dil

+ Aou,

vo(x) = x-^Orx^0,

and

r= Ynl

Inn

with constants "ljk such that (V.5.11)

Vj k = 0

if

Ak - Aj is not a non-negative integer.

Furthermore, the matrix r as nilpotent.

Remark V-5-2. It is easily verified that the matrix P(x) is a formal solution of the system

xd)

= P(x)(Ao + vo(x)) - Sl(x)P(x).

Since the entries of vo(x) are polynomials in x, the power series P(x) is convergent if f2(x) is convergent (cf. Theorem V-2-7). Therefore, in such a case, v(x) is convergent and, hence, o(x) is convergent.

Observation V-5-3. Choose integers l1, ... , to so that A.,+f j = Ak +fk if Aj -Ak is an integer. Then, vo(x) = x4rx-L, where L = diag(e1, f2, ... , en]. If we set 7t{V1 = x'! 1C(xf and ho{v") = x-LICo(x' i , it follows that dv' ?{o(vl = x'L fXLTf + zLLv + Aoz' 7} = xdx + (Ao + L)v

Hence,

(V.5.12)

R{v = x

+ (Ao + L + r)v.

Note that Ao + L = diag(A1 + P1, A2 + e2, ... , An + enj

and (V.5.13)

(Ao + L)r = r(A0 + L).

It was already shower in §V-4 that r is nilpotent. Thus, the following theorem is proved.

5. A NORMAL FORM OF A DIFFERENTIAL OPERATOR

125

Theorem V-5-4. The transformation y = P(x)xLiT changes the system

G[yi = xdy + fl(x)yl = 0

(V.5.14) to

VV! = x

(V.5.15)

d1U

+ (Ao + L + r)u = 0.

Observation V-5-5. The matrix n-1

o{x) = x-Ao-L-r = x-Ao-L I

+ h-1

h

(-1)h[logx[ rh h!

is a fundamental matrix solution of system (V.5.15). Note that if (A3+tj)-(Ak+tk) is an integer, then A, + ej = Ak + tk in Ao + L.

Remark V-5-6. A fundamental matrix solution 4D(x) of system (V.5.14) is given by 4>(x) = P(x)x4o(x), which can be written in the form n-1

45(x) = P(x)xLx-M-L In +

F {-1)h (lo x h rh h!

h=1

IL

Since L - A0 - L = -A0, the matrix 44(x) can be also written in the form n-1

4i(x) =

P(x)x-ne

In +

h

(-1)h (lohx) rh

h=1

I However, x-Ao and I

n-I

and

In + E(-1)h h=1

n-i

nr >(-1)h [lohx[

h

rh do not commute, whereas

x-A°-L

h=1

[log x[ h h!

rh commute.

I

Remark V-5-7. The methods used in §§V-3, V-4, and V-5 are based on the

original idea given in [GerL[.

Example V-5-8. In order to illustrate the results of this section, consider the differential equation of the Bessel functions (V.5.16)

za (zT)

+ (z2 - a2)y = 0,

where a is a non-negative integer. If we change the independent variable z by x = z2, (V.5.16) becomes x

d

lxdx

2y=0.

V. SINGULARITIES OF THE FIRST KIND

126

This equation is equivalent to the system

-1

0

where 11(x) =

y=

+ Q(x)y = 0,

x

(V.5.17)

0

=10+x11i c o =

x - a2 0

4

a2 4

-1

0

and 0i = 0

0

1

4

0

To begin with, let us remark that the S-N decomposition of the matrix 11 is given by S2o = So + No, where 80 =

O

if

{Q

if

Also, set Ao =

a = 0, a > 0,

a

0

2

a

N

and

if

N0 - to

if

a = 0,

a>0.

Note that two eigenvalues of Q are

0

2

Now, we calculate in the following steps:

Step 1. Fix a non-negative integer m so that m > 2 -

(_a)

= a.

Step 2. Find three 2(m + 1) x 2(m + 1) matrices A,,,, Sm, and Nm (cf. §§V-4 and V-5). Po P1

Step 3. Find a 2(m + 1) x 2 matrix Pm =

, where the Pr are 2 x 2 matrices

Pm

such that SmPm = Pn&. in

m

Step 4. Find two 2 x 2 matrices Mm(x) = No + >2xtvt and Qm(x) _ >xtPt t-o

t=1

(cf. §§V-4 and V-5).

Step 5. We must obtain Qm(x)-1Mm(x)Qm(x) = vo(x) +O(xm+i), where Po InoP0 vo(x)

r0 0

a(a)xn 0

if

a = 0,

if

a>0,

J

where a(a) is a real constant depending on a (cf. Theorem V-5-1). After these calculations have been completed, we come to the following conclusion.

5. A NORMAL FORM OF A DIFFERENTIAL OPERATOR

127

+00

Conclusion V-5-9. There exists a unique 2 x 2 matrix P(x) = >xtPe such that 1=o

(1) (2) (8)

the matrices Pl for t = 0,1, ... , m are given in Step 8, the power series P(x) converges for every x, the transformation y = P(x)u" changes (V.5.17) to

dii xaj + (Ao + vo(x)) i = 0'.

(V.5.18)

We illustrate the scheme given above in the case when a = 0. First we fix m = 2. Then, 0

-1

0

0

0

0

0

0

0

0

00

0

0

0

0

0

0

0

0

0

0

00

0

0

1

-1

0

0

1

0

0

0

0

1

0

0

2

0

0

2

A2 =

S2 = 1

4 0

0

0

0

0

0 0

4

1

0 0

0

0

-1

2

0

-

2

1

1

4

2

1

1

4

4

3

3

1

1

32

32

4

2

1

3

1

1

16

32

4

4

and

N2=

0

-1

0

0

0

0

0

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

-1

0

0

1

1

4

2

0

1

3

3

1

1

32

32

4

2

16

T2-

0

9

64

-320 -128

-16 -32

16 48

1192

Calculating eigenvectors of S2, we find a 6 x 2 matrix P2 =

that S2P2 = 0. Note that, in this case, Ao = 0.

0

1

1

0

such

V. SINGULARITIES OF THE FIRST KIND

128

Set 1

1

3

3

4

2

32

32

0

1

1

4

16

3

32-

M2 (x) = 1 o + xvl + x2v2 and

-320

192

(V.5.19)

-16

P1

-1 8]

PO = [ 64

16 ]

= [ -32 48

'

P2 ,

[0

1

=

01 '

Q2(r) = P° + xPl + x2P2. ]

Then, Qz-' 11f2(x)Q2(x)

-1

+ O(x3). Note that

5

1

1,°

2

12

634

64

[0

[192 64

01]

-128] =

4

[--21

2].

64

Thus, we arrived at the following conclusion. 00

Conclusion V-5-10. There exists a unique 2 x 2 matrix P(x)

Zx'Pr such e=o

that (1) (2)

(3)

the matrices Pt for i = 0, 1, 2 are given by (V.5.19). the power series P(x) conueryes for every x, the transformation y" = P(x)u" changes the system dy

xdx +

(V.5.20)

0

-1

X

0

Y = 0'

4 to

d6

x3i + [_1 2, u" = 0.

(V.5.21)

-

Furthermore, the transformation y" = P(x)Po iv changes (V.5.20) to (V.5.22)

x

+ 10

0

" = 0.

5. A NORMAL FORM OF A DIFFERENTIAL OPERATOR

In the case when a=1, wefixm=2. Then 1 0 -1 0 0 0 0

0

-1

0

0

0

0

0

0

0

0

0

1

-1

0

0

1

0

0

2

-1

1

0

0

0

0

0

0

1

-1

0

0

0

-4

1

0

0

0

0

0

2

-1

129

4

0

A2 = 0 0

1

132=

0

0

-4

2

4

8 3

1

1

16

8

4

1

1

16

16

3

1

3

1

1

64

16

16

8

4

J

1

1

4

2

4

and

Ar2

=

0

0

0

0

0

01

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

8

4

1

1

16

8

16 -3

1

1

1

16

8

4

1

1

1

64

16

16

8

1

00 0

0 384 192

Calculating eigenvectors of S2, we find a 6 x 2 matrix P2 =

_72

32

-16

-124 such

2

0

5

1

1

that S2P2 = P2Ao. Note that, in this case, A0 =

0 1

2

L0 vl =

. Set

1 1

2

1

1

1

1

8

4

16

16

1

1

v2 =

1

3 16

8

64

16

M2(x) = xl/1 + x2v2, and

(V.5.23)

Po

84

62

[19 3 J' Pl -

Q2(x) = Po + xP1 + T2 p2.

-48 -12 [-72 -14]

,

1'2 = [2 1j

,

V. SINGULARITIES OF THE FIRST KIND

130

x 0

Then, Q2-'M2(X)Q2(X) = 0

48 0

+ Q(x3). Thus, we arrived at the following

conclusion. +oo

Conclusion V-5-11. There exists a unique 2 x 2 matrix P(x) = >xtPt such that (1) (2) (3)

e=o

the matrices Pt for P = 0, 1, 2 are given by (V.5.23), the power series P(x) convet es for every x, the transformation y = P(x)u changes the system -1

xdx +

(V.5.24)

V = 0 0

to

x-

(V.5.25)

+

2 0

48I u = U. 1

2

For a further discussion, see IHKS[. A computer might help the reader to calculate S2i N2, and P2 in the cases when a = 0 and a = 1. Such a calculation is not difficult in these cases since eigenvalues of A2 are found easily (cf. §IV-1).

V-6. Calculation of the normal form of a differential operator In this section, we present another proof of Theorem V-5-1. The main idea is to construct a power series P(x) as a formal solution of the system dP(x)

= P(x)(Ao + vo(x)) - Q(x)P(x)

(cf. Remark V-5-2). Another proof of Theorem V-5-1.

To simplify the presentation, we assume that So = A0 = diag[,ulI,, µ2I2, ... , Aklk[, where pi, p2, ... , µk are distinct eigenvalues of no with multiplicities m1, respectively, and the matrix I, is m, x m j identity matrix. Since AOA(o = NoAo, the matrix No must have the form No = diag(N01,No2, ... ,Nok[, Where Nor is an 00

mi x m1 nilpotent matrix. Let us determine two matrices P(x) = I,a + E xmP,,, M=1

6. CALCULATION OF THE NORMAL FORM

131

00

and B(x) = Sto + E xmBm by the equation m=1

(in+mm) (Oo

xad

+

*n=1 xmBm)

(In + m 'nPm)

Cep + m=1

This equation is equivalent to m-1

mPm = PmS2o - S20Pm + Bm - 1m +

(m > 1).

( PhBm-h - fl.-hPh) [=1

Therefore, it suffices to solve the equation

mX + (A0 +N4)X - X(Ao+No) - Y = H,

(V.6.1)

where X and Y are n x n unknown matrices, whereas the matrix H is given. If we write X, Y, and H in the block-form X11

Xkl

...

Xlk

[Y11

...

Xlk

H11

...

Xlk

Xkk

Ykl

...

Ykk

Hkl

...

Hkk

where XXh, YYh, and H,h are m, x mh matrices, equation (V.6.1) becomes

(m + P) - µh)X)h + NO,Xjh - X,hNOh - Yjh = Hjh,

where j, h = 1,... , k. We can determine XJh and Y,h by setting Y,;h = 0 if

m + u, - Ah 4 0, and X,,h = 0 if m + p, - µh = 0. More precisely speaking, if m + p - Ph 96 0, we determine X3h uniquely by solving

(m + p, - µh)Xlh + NojXjh - X3hNoh = H)h. If M + Aj - Ah = 0, we set Y,h = H,h. In this way, we can determine P(z) and B(x). In particular, go + B(x) has the form x ^°I'xA0, where r is a constant d n x n nilpotent matrix. Furthermore, the operator x +A0 and the multiplication ds operator by No + B(x) commute. D The idea of this proof is due to M. Hukuhara (cf. [Si17, §3.9, pp. 85-891).

Remark V-6-1. In the case of a second-order linear homogeneous differential equation at a regular singular point x = a, there exists a solution of the form +00

01(z) = (x - a)aE c, (x - a)n, where the coefficients c,, are constants, co 0 0, n=O

V. SINGULARITIES OF THE FIRST KIND

132

and the power series is convergent. If there is no other linearly independent solution of this form, a second solution can be constructed by using the idea explained in Remark IV-7-3. This second solution contains a logarithmic term. Similarly, a third-order linear homogeneous differential equation has a solution of the form +00

c, (x - a)" at a regular singular point x = a. Using this solu-

01(x) _ (x - a)° n=0

tion, the given equation can be reduced to a second-order equation. In particular, if there exists another solution ¢(x) of this kind such that ¢1 and ¢2 are linearly independent, then the idea given in Remark IV-7-4 can be used to find a fundamental set of solutions. In general, a fundamental matrix solution of system (V.5.14) can be constructed if the transformation y" = P(x)xLV of Theorem V-5-4 is found. In fact, if the P(x)xL2-no-L-r is a fundamental definition of fo(x) of Observation V-5-5 is used, matrix solution of (V.5.14). The matrix P(x) can be calculated by using the method of Hukuhara, which was explained earlier.

V-7. Classification of singularities of homogeneous linear systems In this chapter, so far we have studied a system (V.7.1)

x dt = A(x)y,

y E C",

where the entries of the n x n matrix A(r) are convergent power series in x with complex coefficients. In this case, the singularity at x = 0 is said to be of the first kind. If a system has the form (V.7.2)

xk+1

= A(x)y",

y" E C",

where k is a positive integer and the entries of the n x n matrix A(x) are convergent power series in x with complex coefficients, then the singularity at x = 0 is said to be of the second kind In §V-5, we proved the following theorem (cf. Theorem V-5-4).

Theorem V-7-1. For system (V.7.1), there exist a constant n x n matrix A0 and an n x n invertible matrix P(x) such that (i) the entries of P(x) and P(x)-1 are analytic and single-valued in a domain 0 < jxi < r and have, at worst, a pole at x = 0, where r is a positive number, (ii) the transformation (V.7.3)

y' = P(x)i7

changes (V.7.1) to a system (V.7.4)

du = Aou".

Theorem V-7-1 can be generalized to system (V.7.2) as follows.

7. CLASSIFICATION OF SINGULARITIES OF LINEAR SYSTEMS

133

Theorem V-7-2. For system (V.7.2), there exist a constant n x n matrix A0 and an n x n invertible matrix P(x) such that (i) the entries of P(x) and P(x)-1 are analytic and single-valued in a domain V = {x : 0 < lxi < r}, where r is a positive number, (ii) the transformation

y = P(x)i

(V.7.5) changes (V.7.2) to (V.7.4).

Proof.

Let 4i(x) be a fundamental matrix of (V.7.2) in D. Since A(x) is analytic and single-valued in D, fi(x) _ I (xe2"t) is also a fundamental matrix of (V.7.2). Therefore, there exists an invertible constant matrix C such that 46(x) = 4i(x)C (cf. (1) of Remark IV-2-7). Choose a constant matrix Ao so that C. = exp[2rriAo] (cf. Ex-

ample IV-3-6) and let P(x) = 4(x)exp[-(logx)Ao]. Then, P(x) and P(x)-1 are analytic and single-valued in D. Furthermore,

dP(x) = dam) exp(_(logx)Ao] - P(x)(x-'Ao)

= x-(k+1)A(x)P(x) - P(x)(x-'Ao). This completes the proof of the theorem. 0 An important difference between Theorems V-7-1 and V-7-2 is the fact that the matrix P(x) in Theorem V-7-2 possibly has an essential singularity at x = 0. The proof of Theorem V-7-2 immediately suggests that Theorem V-7-2 can be extended to a system (V.7.6)

dg = F(x)y, dx

where every entry of the n x n matrix F(x) is analytic and single-valued on the domain V even if such an entry of F(x) possibly has an essential singularity at x = 0. More precisely speaking, for system (V.7.6), there exist a constant n x n matrix Ao and an n x n invertible matrix P(x) satisfying conditions (i) and (ii) of Theorem V-7-2 such that transformation (V.7.5) changes (V.7.6) to (V.7.4). Now, we state a definition of regular singularity of (V.7.6) at x = 0 as follows.

Definition V-7-3. Let P(x) be a matrix satisfying conditions (i) and (ii) such that transformation (V.7.5) changes (V.7.6) to (V.7.4). Then, the singularity of (V.7.6) at x = 0 is said to be regular if every entry of P(x) has, at worst, a pole

atx=0.

Remark V-7-4. Theorem V-7-1 implies that a singularity of the first kind is a regular singularity. The converse is not true. However, it can be proved easily that a regular singularity is, at worst, a singularity of the second kind. Furthermore, if (V.7.2) has a regular singularity at x = 0, then the matrix A(0) is nilpotent. This is a consequence of the following theorem.

V. SINGULARITIES OF THE FIRST KIND

134

Theorem V-7-5. Let A(x) and B(x) be two n x n matrices whose entries are formal power series in x with constant coefficients. Also, let r and s be two positive integers. Suppose that there exists an n x n matrix P(x) such that (a) the entries of P(x) are formal power series in x with constant coefficients, (b) det(P(x)] ,E 0 as a formal power series in x, (c) the transformation y = P(x)ii changes the system x'

16 -1

X

= A(x)y to x' 2i _

B(x)il. Suppose also that s > r. Then, the matrix B(O) must be nilpotent.

Proof

Step 1. Applying to the matrix P(x) suitable elementary row and column operations successively, we can prove the following lemma.

Lemma V-7-6. There exist two n x n matrices +oo

+oo

T(x) _

xmTm

and

S(x) _

x"'Sm, m=o

m=0

and n integers 1\1,,\2, ... , an such that (i) the entries of n x n matrices T,,, and Sm are constants. (ii) det To 0 0 and det So 0 0, (iii)

T(x)P(x)S(x) = A(z) = diag(x''',zA2,...

(iv)A1
.

The proof of this lemma is left to the reader as an exercise.

Step 2. Change the two systems

d

x'd

= A(x)y and x'

d6

= B(x)u, respectively,

d

= D(x)ii by the transformations z' = T(x)y and to xr = C(x)a and x' ii = S(x)v. Then, F = A(x)v. Furthermore, if the matrix D(0) is nilpotent, the matrix B(0) is also nilpotent.

Step 3. Look at D(x) = x'-'A(x)-1C(x)A(x) - x'A(x)-ld

). This shows

clearly that if s > r, the matrix D(0) is nilpotent. The following corollary of Theorem V-7-5 is important.

Corollary V-7-7. Assume that conditions (a), (b), and (c) of Theorem V-7-5 are satisfied. Also, assume that A(O) and B(O) are not nilpotent. Then, r = s. Definition V-7-8. If a singularity of the second kind is not a regular singularity, this singularity is said to be irregular. In particular, if the matrix A(O) of (V.7.2) is not nilpotent, the singularity of (V.7.2) at x = 0 is said to be irregular of order k. Also, a regular singularity is said to be of order zero.

In order to define the order of singularity at r = 0 for all systems (V.7.2), the independent variable x must be replaced by x'1P with a suitable positive integer p.

7. CLASSIFICATION OF SINGULARITIES OF LINEAR SYSTEMS

135

For example, for a differential equation (V.7.7)

ah(x)bhl/ = 0, h=0

assume that the coefficients ah are convergent power series in x and that an (x) # 0. Set y; = 6'-1y (j = 0,... , n - 1). Then, (V.7.7) becomes the system f

0

1

0

0

0

0

6y =

(V.7.8)

0

0

0

...

0 .

yl

0 91

1

a0

an-1

an

as

If we change (V.7.8) by the transformation y" = diag[l, x-O, then

/i!

= yn

'.T-(n-1)0

Iii,

(V.7.9)

x°6u 0

1

0

0

0

0

0

0

0

...

0

0

+ x°diag[l, a, 2a,... , (n - 1)a) ...

-x' an-1 an

-xna ao

a.

Set bh(x) =

u'.

1

ah(x)x(n-h)O

. Choose a non-negative rational number a so that bh(X) an(x) (h = 0,... , n - 1) are bounded in a neighborhood of x = 0. Also, if a must be positive, choose a so that bh(0) 0 0 for some h. Then, the matrix on the right-hand side of (V.7.9) is not nilpotent at x = 0 if or > 0. Hence, we define a as the order of the singularity of (V.7.7) at x = 0. System (V.7.2) can be reduced to an equation (V.7.7) (cf. §XIII-5). The following theorem due to J. Moser [Mo) concerns the order of a given singularity-

Theorem V-7-9. Let A(x) = x-a E x'A be an n x n matriz, where p is an =o

integer, A are n x n constant matrices, A0 y6 0, and the power series is convergent. Set

Im(A) = max (P_ 1 + n,0) , µ(A) =

minlm(T-1AT-T-1dx)J

136

V. SINGULARITIES OF THE FIRST KIND

Here, r = rank(Ao) and min is taken over all n x n invertible matrices T of the

form T(x) = x-9 E x°T,,, where q is an integer, T are constant n x n matrices, and the power series is convergent. Note that det T(x) 0 0, but det To may be 0. Assume that m(A) > 1. Then, we have m(A) > p(A) if and only if the polynomial

P(A) = vanishes identically in A, where In is the n x n identity matrix. The main idea is to find out p(A) in a finite number of steps. Using the quantity

p(A), we can calculate the order of the singularity at x = 0. The criterion for m(A) > p(A) is given in terms of a finite number of conditions on the coefficients of A(x). There is a computer program to work with those conditions. In particular, in this way, we can decide, in a finite number of steps, whether a given singularity at x = 0 is regular. Notice that IP(x)I can be estimated at z = 0 for the matrix P(x) of Theorem V-72, using similar estimates for fundamental matrix solutions of (V.7.2) and (V.7.4).

Therefore, an analytic criterion that a singularity of second kind at x = 0 be a regular singularity is that in any sectorial domain V with the vertex at x = 0, every solution fi(x) satisfies an estimate

c K[xj'

(x E V)

for some positive number K and a real number m. These two numbers may depend on 0 and V. In fact, Theorem V-7-2 and its remark imply that a fundamental matrix

solution of (V.7.6) is P(x)xA0. The matrix P(x) has, at worst, a pole at x = 0 if and only if !P(x)I < Kjxjr in a neighborhood of x = 0 for some positive number K and a real number p (cf. [CL, §2 of Chapter 4, pp. 111-1411). Another criterion which depends only on a finite number of coefficients of power series expansion of the matrix A(x) of (V.7.2) was given in a very concrete form by W. Jurkat and D. A. Lutz [JL] (see also 15117, Chapter V, pp. 115-1411). Now, let us look into the problem of convergence of the formal solution which was constructed in Theorem V-1-3. Let n

P = Eah(x)Jh h=0

be a differential operator with coefficients ah(x) in C{x). Assume that n > I and of the operator P as in §V-1, a,(x) 0 0. Defining the indicial polynomial we prove the following theorem.

Theorem V-7-10. (i) In the case when the degree of f b in s is equal to n, if we change the equation

P[y1 = 0 to a system by setting yt = y and yj = 61-1 [y) (j = 2, ... , n), the system has a singularity of the first kind at x = 0.

137

EXERCISES V

(ii) In the case when the degree of fn ins is less than n, if we change the equation P[y] = 0 by setting y1 = y and y, = b'-1 [y] (j = 2,... , n), the system has an irregular singularity at x = 0.

Proof n

00

Ltahi

Look at P(xd] _ >ah(x)bh[x°] = xe= x"

fm(S)xm and set Tn=r+o

ah(x) = xnobh(x). Then, the functions bh(x) are analytic at x = 0. Furthermore, is n, we must have bn(0) # 0. Therefore, in this case we can if the degree of n-1

write the equation P(y) = 0 in the form bo[y] = -bn(Ebh(x)bh(yJ. Claim (i) h=0

follows immediately from this form of the equation.

If the degree of fn0 is less than n, we must have bn(0) = 0 and b,(0) - 0 for some j such that 0 < j < n. To show (ii), change Yh further by zh = x(h-1)ayh with a suitable positive rational number a so that the system for (z1, ... , zn) has a singularity of the second kind of a positive order (cf. the arguments given right after Definition V-7-8, and also §XIII-7).

From Theorem V-7-10, we conclude that the formal solution x''(1 + O(x)) of Theorem V-1-3 is convergent if the degree of f,, (s) in s is n. The following corollary of Theorem V-7-10 is a basic result due to L. Fuchs [Fu].

Corollary V-7-11. The differential equation Ply] = 0 has, at worst, a regular singularity at x = 0 if and only if the functions ah(x) (h = 0, ... , n - 1) are x

analytic at x = 0. Some of the results of this section are also found in [CL, Chapter 4, pp. 108-137].

EXERCISES V

V-1. Show that if A is a nonzero constant, H is a constant n x m matrix, and N1 and N2 are n x n and m x in nilpotent matrices, respectively, then there exists one

and only one n x m matrix X satisfying the equation AX + NIX - XN2 = H. V-2. Show that the convergent power series

y = F(a,x) =

1+a

(a+1)...(a+m-1Q(Q+1)

m=1

satisfies the differential equation

x(1-x)!f 2 + where a, /3, and -y are complex constants.

+1)J

Y

- c3y = 0,

Q+m-1) x,,

V. SINGULARITIES OF THE FIRST KIND

138

Comment. The series F(a, Q, ry, x) is called the hypergeometric series (see, for example, [CL; p. 1351, 101; p. 159], and [IKSYJ).

V-3. For each of the following differential equations, find all formal solutions of c

the form x'' I 1 + > cmxm] . Examine also if they are convergent. L

m=1

(i)

xb2y + aby + /3y = 0,

(ii)

b2y + aby + $y = rxmy,

where b = xa, the quantities a, 3, and 7 are nonzero complex constants and m is a positive integer.

V-4. Given the system (E)

dt = (N + R(t))il,

N = [0

O]

,

and R(t) = t-3 I 0 0J

,

show that (i) (E) has two linearly independent solutions ,

where

(t)

o m!(m+1)!'

and

)1 J2(t) = 1612 (t (t)

where

,

02(t) = t +

+00

bmt-'n - ¢1(t)logt,

m-1

with the constants bm determined by b_1 = 1, bo = 0, and 2m + 1 (m = 1,2,... ), m(m + 1)bm - b,,,-, = m!(m+ 1)! IN) = O for the fundamental matrix solution Y(t) = [if1(t)12(t)J, (ii) 1 lim t-1(Y(t)-e

(iii) the limit of e-'NY(t) as t -+ +oo does not exists.

V-5. Let y be a column vector with n entries and let Ax, y-) be a vector with n entries which are formal power series in n + 1 variables (x, yj with coefficients in C. Also, let u' be a vector with n entries and let P(x, u) be a vector with n entries which are formal power series in n + 1 variables (x, u") with coefficients in C. Find u + xP(x, u7 changes the most general P(x, u") such that the transformation the differential equation

dy"

ds

=

f (X, y-) to

du"

= 0.

Hint. Expand xP(x, u) and f (x, u+xP(x, u)) as power series in V. Identify coefficients of

d[xP(x, V-)]

with those of Ax, u'+zP(x, iX)) to derive differential equations

which are satisfied by coefficients of xP(x, u).

139

EXERCISES V

V-6. Suppose that three n x n matrices A, B, and P(x) satisfy the following conditions:

(a) the entries of A and B are constants, (b) the entries of P(x) are analytic and single-valued in 0 < [xj < r for some positive number r,

dii (c) the transformation y' = P(x)u changes the system xfy = Ay" to xjj = Bu. Show that there exists an integer p such that the entries of xPP(x) are polynomials in x.

Hint. The three matrices P(x), A, and B satisfy the equation

xd) = AP(x) -

+"0

P(x)B. Setting P(x) = >2 xmPm, we must have mPm = APm - PmB for all m=-oo

integers m. Hence, there exists a large positive integer p such that Pm = 0 for ImI ? P. and let A(ye) be an n x n V-7. Let ff be a column vector with n entries {y,... , matrix whose entries are convergent power series in {yj, ... , yn} with coefficients in C. Assume that A(0) has an eigenvalue A such that mA is not an eigenvalue of A(0 for any positive integer m. Show that there exists a nontrivial vector O(x)

with n entries in C{x} such that y" = b(exp[At]) satisfies the system L = A(y-)y.

Hint. Calculate the derivative of (exp(At]) to derive the system Axe = for

.

N

V-8. Consider a nonzero differential operator P = >2 ak(x)Dk with coefficients k=O

ak(x) E C([x]], where D = dx. Regarding C([x]] as a vector space over C, define a homomorphism P : C[[x]] -. C[[x]j. Show that C((xjj/P(C[[x]j] is a finitedimensional vector space over C.

Hint. Show that the equation P(y) = x"`'4(x) has a solution in C([xjj for any O(x) E C[[x]] if a positive integer N is sufficiently large. To do this, use the indicial polynomial of P. N

V-9. Consider a nonzero differential operator P = >2 ak(x)dk with coefficients k=0

ak(x) E CI[x]j, where b = x2j, n > 1, and

54 0. Define the indicial poly-

nomial as in Theorem V-1-3. Show that if the degree of is n and if n zeros {A1, ... , A, } of do not differ by integers, we can factor P in the following form: P = a+,(x)(b - &1(x))(b -

42(x))...(6

- 0n(x)),

V. SINGULARITIES OF THE FIRST KIND

140

where all the functions ¢, (x) (j = 1, 2, ... , n) are convergent power series in x and 4f(O) = Aj.

Hint. Without any loss of generality, we can assume that an(x) = 1. Then, An + n-1

Eak(O)Ak = (A - A1)... (A - An). Define constants {yo... , In-2} by A' ' + k=0 n-2 ,YkAk

= (A - A2) ... (A - An). For v] (X) E xC[[xI] and ch(x) E xC([x]l (h =

k=O

0,. . . , n - 2), solve the equation (

(C)

n-2

P = (6 - Al -

2(7h+Ch(x))6h

h=0

If we eliminate O(x) by Al + V(x) = 1`n-2 + ct-2(x) - an-1(x), condition (C) becomes the differential equations 6(CO(x)1 = a0(x) + (-Yn-2 + Cn-2(x) -a. -1(x))(1'0 +40(x)), 1 b[ch(x)] = ah(x) + (1'n-2 + Cn-2(x) - an-1(x))(7h +Ch(x)) - (1h-1 + Ch-1(x)),

where h=I,...,n-2. n

V-10. Consider a linear differential operator P = E atbt, where ao,... , an are t=o

complex numbers and 6 = x. The differential equation (P)

P[y1= 0

is called the Cauchy-Euler differential equation. Find a fundamental set of solution of equation (P).

Hint. If we set t = log x, then b = dt V-11. Find a fundamental set of solutions of the differential equation

(6-06-8)(6-a-8)[yl =xn'Y' where 6 = xjj and m is a positive integer, whereas a and /3 are complex numbers such that they are not integers and R[al < 0 < 8t[01. V-12. Find the fundamental set of solutions of the differential equation (LGE)

f(1-x2)LJ +a(a+1)y=0

at x = 0, where a is a complex parameter.

141

EXERCISES V

Remark. Differential equation (LGE) is called the Legendre equation (ef. [AS, pp. 331-338] or [01, pp. 161-189]). 1 V-13. Show that for a non-negative integer n, the polynomial Pn(x) = 2nn!

x do [(x2 - 1)'] satisfies the Legendre equation d

]

[(1-x2)

+n(n+1)Pn=0.

Show also that these polynomials satisfy the following conditions:

(1) Pn(-x) = (-1)nPn(x),

(2) Pn(1) = 1, (3) JPn(t)j ? 1 for {xj > 1,

+00

(4)

1 - 2xt + t =

E Pn(x)tn, and (5) lPn(x)I < 1 for Jxj < 1. n=o

Hint. Set g(x) = (1 - x2)n. Then, (1 - x2)g'(x) + 2nxg(x) = 0. Differentiate this relation (n + 1) times with respect to x to obtain

(1,-

x2)g(n+2)(x)

- 2(n + 1)xg(n+1)(x) - n(n + 1)g(n)(x)

+ 2nxg(n+1) (x) + 2n(n + 1)g(")(x) = 0 or

(1 -X 2)g(n+2)(x) - 2ng(n+1)(x) + n(n + 1)g(n)(x) = 0.

Statements (1), (2), (3), and (4) can be proved with straight forward calculations. Statement (5) also can be proved similarly by using (4) (cf. [WhW, Chapter XV, Example 2 on p. 303]). However, the following proof is shorter. To begin with, set F(x) = Pn(x)2

x2)P''(x))2. n(n + 1){1 -x2)((1 2

Then, F(±1) = Pn(±1)2 = 1, p(X) _

,and F(x) = Pn(x)2 if (x) (P"(x)) n(n + 1)

0. Therefore, for 0 < x < 1, local maximal values of Pn(x)2 are less than F(1) = 1, whereas, for -1 < x < 0, local maximal values of Pn(x)2 are less than F(-1) = 1. Hence, Pn(x)I < 1 for JxI < 1 (cf. [Sz, §§7.2-7.3, pp. 161-172; in particular, Theorems 7.2 and 7.3, pp. 161-162]. See also (NU, pp. 45-46].) The polynomials Pn(x) are called the Legendne polynomials.

V-14. Find a fundamental set of solutions of (LGE) at x = oo. In particular, show what happends when a is a non-negative integer.

V-15. Show that the differential equation b"y = xy has a fundamental set of solutions consisting of n solutions of the following form: (logx)k-1-A

A-o

(k - 1 - h)!

OA(x)

(k = 1, ... , n),

where the functions Oh(x) (h = 0,... , n -1) are entire in x, 00(0) = 1, and Oh(O) _ 0 (h = 1, ... , n -1). Also, find O0 (x).

142

V. SINGULARITIES OF THE FIRST KIND

V-16. Find the order of singularity at x = 0 of each of the following three equations. (i) x5{66y - 362y + 4y} = y, (ii) x5{56y - 3b2y + 4y} + x7{b3y - 55y} = y, (iii) x5y"' + 5x2y" + dy + 20y = 0.

V-17. Find a fundamental matrix solution of the system x2 x2

dbi = xyi + y2, dx dy2

dx

= 2xy2 + 2y3,

x2 dx3 = x3y! + 3y3.

V-18. Let A(x) be an n x n matrix whose entries are holomorphic at x = 0. Also, let A be an n x n diagonal matrix whose entries are non-negative integers. Show that for every sufficiently large positive integer N and every C"-valued function fi(x) whose entries are polynomials in x such that those of xA¢(x) are of degree N, there exists a C"-valued function f (x; A, ¢, N) such that (a) the entries off are polynomials in x of degree N-1 with coefficients depending on A, N, and ¢, (b) f is linear and homogeneous in tb, (c) the linear system xAdd9 = A(x)3l + z has a solution #(x) whose entries are holomorphic at x = 0 and il(x) is linear and homogeneous in ', fi(x)) = O(xN+i) as x 0. (d) Hint. See [HSW].

V-19. Let A(x) and A be the same as in Exercise V-18. Assuming that n > trace(A), show that the system xA dz = A(x)y has at least n - trace(A) linearly independent solutions holomorphic at x = 0. Hint. This result is due to F. Lettenmeyer [Let]. To solve Exercise V-19, calculate f (x; A, ¢, N) of Exercise V-18 and solve f(x; A, 4, N) = 0 to determine a suitable function 0. 00

V-20. Suppose that for a formal power series ¢(x) _ > cx' E C[]x]], there exm=0 /dl d ist two nonzero differential operators P = E ak (x) ` and Q = > bk (x) (d) k I

k=0

k

rk=10

= 0. Show

with coefficients ak(x) and bk(x) in C{x} such that P[0] = 0 and Q I

that 0 is convergent.

l

J

EXERCISES V

143

Hint. The main ideas are (a) derive two algebraic (nonlinear) ordinary differential equations (E)

F(x,v,v',... ,v(")) = 0

for v =

and

G(x,v,v',... ,v(' ) = 0

from the given equations P[¢) = 0 and Q ['01 = 0.

(b) eliminate all derivatives of v from (E) to derive a nontrivial purely algebraic equation H(x, v) = 0 on v. See [HaS1] and [HaS2[ for details.

CHAPTER VI

BOUNDARY-VALUE PROBLEMS OF LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND-ORDER

In this chapter, we explain (1) oscillation of solutions of a homogeneous secondorder linear differential equation (§VI-1), (2) the Sturm-Liouville problems (§§VI2-VI-4, topics including Green's functions, self-adjointness, distribution of eigenvalues, and eigenfunction expansion), (3) scattering problems (§§VI-5-VI-9, mostly focusing on reflectionless potentials), and (4) periodic potentials (§VI-10). The materials concerning these topics are also found in [CL, Chapters 7, 8, and 11], [Hart Chapter XI], [Copl, Chapter 1], [Be12, Chapter 61, and [TD]. Singular selfadjoint boundary-value problems (in particular continuous spectrum, limit-point and limit-circle cases) are not explained in this book. For these topics, see [CL, Chapter 9].

VI-1. Zeros of solutions It is known that real-valued solutions of the differential equation L2 + y = 0 are linear combinations of sin x and cos x. These solutions have infinitely many zeros on the real line P. It is also known that real-valued solutions of the differential equation

2 - y = 0 are linear combination of e= and a-x. Therefore, nontrivial solutions has at most one zero on the real line R. Furthermore, solutions of the differential

equation2 + 2y = 0 have more zeros than solutions of L2 + y = 0.

In this

section, keeping these examples in mind, we explain the basic results concerning zeros of solutions of the second-order homogeneous linear differential equations. In §§VI-1-VI-4, every quantity is supposed to take real values only. We start with the most well-known comparison theorem concerning a homogeneous second-order linear differential equation (VI.1.1)

dx2

+ 9(x)y = 0.

Theorem VI-1-1. Suppose that (i) g, (x) and g2(x) are continuous and g2(x) > 91(x) on an interval a < x < b, (ii)

d 2 01 (x)

+ 91(x)o1(x) = 0 and

d2622x)

+ g2(x)02(x) = 0 on a < x < b,

(iii) S1 and 6 are successive zeros of 01(x) on a < x < b. Then. q2(x) must vanish at some point £3 between t:l and C2. 144

145

1. ZEROS OF SOLUTIONS Proof.

Assume without any loss of generality that Sl < 1;2 and 01(x) > 0 on 1;1 < x < £2.

(1:2) < 0. A contradiction will be derived from the assumption that 02(x) > 0 on S1 < z < 1;2. In fact, assumption (ii) implies 1 (l:l) > 0, 01(S2) = 0, and

Notice that 41(1::1) = 0,

02(x)

VI .1. 2

_ 01 ( x ) d202 (x) =

d 20, (7)

[ 92

(x) - 91 (x)10 (x)0 2 (x) 1

and, hence, (VI - 1 - 3 )

0202)

1L(

6)

-

02 V

1

)!!LV ) =

rE2

(x) - 91 (x)j¢ (x)02 (x)dx . 1

1

E1

The left-hand side of (VI.1.3) is nonpositive, but the right-hand side of (VI.1.3) is

positive. This is a contradiction. 0 A similar argument yields the following theorem.

Theorem VI-1-2. Suppose that (i) g(x) is continuous on an interval a < x < b, (ii) 0i(x) and 02(x) are two linearly independent solutions of (VI.1.1), (iii) 1;1 and 6 are successive zeros of 01 (x) on a < x < b. Then, 02(x) must vanish at some point 3 between t;1 and 1;2. Proof.

Assume without any loss of generality that Sl < £2 and .01 (x) > 0 on S1 < x < 1;2.

Notice that 01(11) = 0,

1 (S1) > 0, 01(12) = 0, and X1(1;2) < 0. Note also that

if 02(1) = 0 or 02(6) = 0, then 01 and 02 are linearly dependent. Now, a assumption that 02(x) > 0 on fi < x < 2 contradiction will be derived from the assumption a- (x) - p1(z) d2 02 (x) = 0 and, hence, In fact, assumption (u) implies 02(x) dx2 dS2 (VI.1.4)

02(2)

1(6) - 4'201)

1(W =0.

Since the left-hand side of (VI.1.4) is positive, this is a contradiction. 0 The following result is a simple consequence of Theorem VI.1.2.

Corollary VI-1-3. Let g(x) be a real-valued and continuous function on the interval Zo = {x : 0 < x < +oo}. Then, (a) if a nontrivial solution of the differential equation (VI.1.1) has infinitely many zeros on I , then every solution of (VI.1.1) has an infinitely many zeros on Zo,

(b) if a solution of (VI.1.1) has m zeros on an open subinterval Z = {x : a < x < 3} of Zo, then every nontrivial solution of (VI.1.1) has at most m + 1 zeros on Z.

Denote by W(x) = W(x;01,02) =

I

the Wronskian of the set of

02(x) functions (01(x), 02(x)}. For further discussion, we need the following lemma.

VI. BOUNDARY-VALUE PROBLEMS

146

Lemma VI-1-4. Let g(x) be a real-valued and continuous function on the interval To = {x : 0 < x < +oo}, and let 771(x) and M(x) be two solutions of differential equation (VI 1.1). Then, (a) W (x; 772 , ri2 ), the Wronskian of {rli (x), 712(x)}, is independent of x, 771(x),

(b) if we set t;(x) =

then

772(x)

4(x) = dx

c

, where c is a constant.

772(x)2

Also, if 77(x) is a nontrivial solution of (VI.1.1) and if we set w(x) =

then

we obtain

dw(x)

(VI.1.5)

+ w(x)2 + g(x) = 0 .

Proof.

(a) It can be easily shown that d

771(x)

772(x)

dx I77i(x)

772(x)

(b) Note that

=-

dl;(x}

771(x)

=

dx

77i'(x)

772(x) 772(x)

I rl' (x)

712(x) !

712(x)2 771(x)

772(x)1

1

-g(x) !

712(x) 712(x)

771(x)

and that

dw(x) dx

=0

if'(x) :l(x)

1 = -g(x) - w(x)2. 12 ((xx))

The following theorem shows the structure of solutions in the case when every nontrivial solution of differential equation (VI.1.1) has only a finite number of zeros

on To= {x:0
(a)

lim

771(x)

= 0,

x-+oo 712(x)

(b) +00

(VI.1.6) 1.0

dx x)2 = +oo

and

771(

1

+00

Jxo

dx 7r (x)

2

< +00,

where xo is a sufficiently large positive number, (c) the solution 712(x) is unbounded on To. Proof.

(a) Let (1(x) and (2(x) be two linearly independent solutions of (VI.1.1) such that (, (x) > 0 (j = 1, 2) for x > xo > 0. Using (b) of Lemma VI-1-4, we can derive the

147

1. ZEROS OF SOLUTIONS

(1(x) following three possibilities (1) =limo(2(x) = 0, (ii) =U Urn

( 1( x )>0

x-+oo (2 (x)

.

Set

(i(x) 711(x) _

+oo, and (iii)

(2(x)

in case (i), in case (ii),

(i(x) - 7'(2(x)

in case (iii),

(2(x)

(i(x) 1(2(x)

772(x) =

in case (i), in cage (ii), in case (iii).

Then, (a) follows immediately. (b) Conclusion (b) of Lemma VI-1-4 implies that

_Id(Y12(x)

1

ql (x)2 ^ c dx . q1(x))

1

and

_-1d

>)1(x)

c dx \ 2(x)

712(x)2

,

where c is the Wronskian of q1 and rte. Hence, (VI.1.6) follows. +oc

(c) If q2 is bounded, we must have j()2 -= +00. 0 The following theorem gives a simple sufficient condition that solutions of (V1.1.1) have infinitely many zeros on the interval Zo.

Theorem VI-1-6. Let g(x) be a real-valued and continuous function on the in-

terval lo. If f

g(x)dx = +oo, then every solution of the differential equation

0

(VI.1.1) has infinitely many zeros on Zo. Proof.

Suppose that a solution 17(x) satisfies the condition that 77(x) > 0 for x > (x) Then, w(x) satisfies differential equation (VI.1.5). xo > 0. Set w(x) = !L. 77(x) Hence, lien w(x) = -oo. This implies that lim 17(x) = 0 since 17(x) = r7(xo) :-+co :-+oo

x exp Vo

This contradicts (c) of Theorem VI-1-5. O J

The converse of Theorem VI-1-6 is not true, as shown by the following example.

Example VI-1-7. Solutions of the differential equation

(VI.1.7)2 +

A

= 0,

1 z2

where A is a constant, have infinitely many zeros on the interval -oo < x < +oo if and only if A > 4 Proof

Look at (VI.1.7) near x = oo. To do this, set t = r

(VL1.8) L

462 + 26 + 1 + t,

x 0,

to change (VI.1.7) to

VI. BOUNDARY-VALUE PROBLEMS

148

where b = t. Note that t = 0 is a regular singular point of (VI.1.8) and the indicial equation is 4s2 + 2s + A = 0 whose roots are s = - 2 f

4 - A. These

roots are real if and only if A < 4 This verifies the claim. 0 +00

Note that J 0

1 + x2

dx

2*

The following theorem shows some structure of solutions in the case when +00

ig(x)ldx < +oo. 0

Theorem VI-1-8. Let g(x) be a real-valued and continuous function on the interf+C0

valA. If 1

jg(x)ldx < +oo, then there exist unbounded solutions of differential

0

equation (VI. 1. 1) on Ia. Proof.

The assumption of this theorem and (VI.1.1) imply that lim dq(x) = -Y exists :-+oo dx for any bounded solution n(x) of (VI.1.1). If 7 0, then i(x) is unbounded. Hence, z

lim odd) = 0. Therefore, calculating the Wronskian of two bounded solutions

of (VI.1.1), we find that those bounded solutions are linearly dependent on each other. This implies that there must be unbounded solutions. 0

Remark VI-1-9. Theorems VI-1-1 and VI-1-2 are also explained in ]CL, §1 of Chapter 8, pp. 208-211] and [Hart, Chapter XI, pp. 322-403]. For details concerning other results in this section, see also [Cop2, Chapter 1, pp. 4-33] and [Be12, Chapter 6, pp. 107-142].

VI-2. Sturm-Liouville problems A Sturm-Liouville problem is a boundary-value problem dd-x (p(x) Ly) + u(x)y = f (.T),

(BP)

y(a) cos a

-I p(a)y'(a) sin or = 0,

y(b) cos Q - P(b)y(b) sin )3 = 0,

under the assumptions: (i) the quantities a, b, a, and $ are real numbers such that a < b, (ii) two functions u(x) and f (x) are real-valued and continuous on the interval

Z(a,b) = {x: a < x < b}, (iii) the function p(x) is real-valued and continuously differentiable, and p(x) > 0 on 1(a, b). In this section, we explain some basic results concerning problem (BP).

149

2. STRUM-LIOUVILLE PROBLEMS

Let 4(x) and O(z) be two solutions of the homogeneous linear differential equation (VI-2.1)

-dx

(p(x)A) + u()y = 0

such that (VI.2.2)

m(a) = sins,

p(a)d'(a) = cosa,

,p(b) = sin f3,

p(b)t'(b) = cosfi,

respectively. Then, these two solutions satisfy the boundary conditions (VI.2.3)

¢(a) cos a - p(a)4,'(a) sin a = 0,

i (b) cos /3 - p(b)0'(b) sin /3 = 0.

The two solutions 4(x) and /;(x) are linearly independent if and only if (VI.2.4)

4(b) cos Q - p(b)©'(b) sin /3 # 0 or

'(a) cos a - p(a)rb'(a) sin a # 0.

The first basic result of this section is the following theorem, which concerns the existence and uniqueness of solution of (BP).

Theorem VI-2-1. If the two solutions 4(x) and ip(x) of (VI.2.1) are linearly independent, then problem (BP) has one and only one solutwn on the interval I(a, b). Proof

Using the method of variation of parameters (cf. Remark IV-7-2), write the general solution y(x) of the differential equation of (BP) and its derivative y'(x) respectively in the following form: (VI.2.5) 6

VW = CIOW + C20 W + 4(x)

J

'j'(_)f (_) 4

P(f)WW

Y' (X) = Clo'(x) + C21k'(x) + 0'(x) I s P(A)W

+ i,&(x)1= o(f)f a

4

(x) J

JJJu P(A)W V)

A,

where cl and c2 are arbitrary constants and W (x) denotes the Wronskian of Now, using (VI.2.3) and (VI.2.4), it can be shown that solution (VI.2.5) satisfies the boundary conditions of (BP) if and only if cl = 0 and c2 =

0. 0

Observation VI-2-2. It can be shown easily that p(x)W(x) is independent of x. Observation VI-2-3. Under the assumption that the two solutions 40(x) and tfi(x) of (VI.2.1) are linearly independent, the unique solution of (BP) is given by rb

y(x) = d(x) (VI.2.6)

Jx P(A)W (O

Y 'W = O(x)

rb owfwdC 1r

W

10(x) I

i

(- ))- W(f )

+ V,'(x) r= 4(W W 4. Ja P(OW (O

VI. BOUNDARY-VALUE PROBLEMS

150

Setting

4(x)1() G(x, ) =

(VI.2.7)

p(OW (c) O WOW


if

x <

C

if

a <

t<

,

x,

p(f)W(c)

we can write (VI.2.6) in the form b aG

b

y(x) =

J

G(x, )f

y '(x) =

J

8x (x, Of (Ode.

The function G(x,l;) is called Green's function of problem (BP). The following theorem gives the characterization of Green's function. Theorem VI-2-4. The function G(x, C) given by (VI. 2.7) satisfies the following conditions:

(i) G(x, l:) is continuous with respect to (x, e) on the region D = {(x,) a < x <

b, a< £
a<

(ii) 8 (x, l:) is continuous with respect to (x, C) on the region D C < b},

(iii) at ({, l;), we have

8 (tr + 0. 0 -

( - 0, 0 =

p(0

for

a<

< b,

(iv) as a function of x, G(x, l:) satisfies homogeneous linear differential equation (VI. 2. 1) if x 0 4, (v) as a function of x, G(x, 4) satisfies the boundary conditions of (BP), i. e.,

G(b,Z:)cosf3-p(b)8G(b,{)sinf3=0 for a:5 < b. Furthermore, the function G(x, f) is uniquely determined by these five conditions. Proof.

It is evident that the function G(x, l;) satisfies these five conditions. Suppose H(x,t) satisfies the five conditions. Conditions (iv) and (v) imply that Cl(l;)4(x)

for

{ C2(t;)V(x)

for

a < x < l;, {
for (x,l;') E D, where Ci(1;) and C2(C) must be determined by conditions (i), (ii), and (iii). This means that C,({) and must be determined by 0

and

-PW

2. STRUM-LIOUVILLE PROBLEMS

151

OW and C2({) = 4(f) Since {(x,e) : a < x < } This yields Cl(a;) = p(OW(O p(OW(O

- x
(x, ) E V, we obtain H(x,1;) = G(x, l;).

The second basic result co//ncerns self-adjointness of problem (BP). Define a dif-

ferential operator L[y] _

(p(x)

real number field 1R by

\

I + u(z)y and a vector space V(a, b) over the

/

V(a, b) _ { f r C2(a, b) : f (a) cos a - p(a) f'(a) sin a = 0,

f(b)cos5 - p(b)f'(b) sin,3 = 01,

where C2(a, b) denotes the set of all real-valued functions which are twice contin-

uously differentiable on the interval I(a, b) = {x : a < x < b). Define also an inner product (f, g) for two real-valued continuous functions f and g on I(a, b) by

(f,9) =

f

Since

a

(f, 4191) =

(()) J6f()

[

4

+

f

= p(b)f (b)9 (b) - p(a)f (a)9'(a) +

b

dl;

a

for f E V(a, b) and g E V(a, b), we obtain

(f,L[g) - (L(f1,9) ={p(b)f(b)9'(b) - p(b)g(b)f'(b)} - {p(a)f(a)9 (a) - p(a)9(a)f'(a)}. Also, since (

t

f (a) cos a - p(a) f'(a) sin a = 0, 9(a) cos a - p(a)9'(a) sin a = 0,

f (b) cos Q - p(b) f'(b) sin O= 0, g(b) cos, - p(6)9'(b) sin /3 = 0

for f E V(a. b) and g E V(a, b), we obtain p(a)f(a)9'(a) - p(a)g(a)f(a) = 0,

p(b)f(b)9'(b) - p(b)g(b)f(b) = 0.

Thus, we proved the following theorem.

Theorem VI-2-5 (self-adjointness). The operator L has the following property: (f, 0191) = (L(f], 9)

for

f E V (a, b)

and g E V(a, b).

VI. BOUNDARY-VALUE PROBLEMS

152

Observation VI-2-6. In Theorem VI.2.1, it was assumed that the two solutions ¢(x) and '(x) are linearly independent. Consider the case when this assumption is not satisfied. So, assume that ¢(x) and ?P(x) are linearly dependent. This means that 0(a) cos a - p(a)0'(a) sin a = 0 and ¢(b) cos ji - p(b)5'(b) sin fi = 0. In other words, (VI.2.8)

lr[o) = 0

and

0 E V(a, b).

The boundary-value problem (BP) can be written in the form (VI.2.9) G[yl = f and y E V(a, b). Using (VI.2.8) and (VI.2.9), we obtain (f, 0) = (C [y], 0) = (y, x[01) = 0,

if there exists a solution y of problem (VI.2.9). The converse is also true, as shown in the following theorem. Theorem VI-2-7. Assume that O(x) satisfies condition (VI.2.8). Then, if a realvalued continuous function f (x) on T (a, b) satisfies the condition (f. 0) = 01

(VI.2.10)

problem (BP) has solutions depending on an arbitrary constant. Proof.

Using the method of variation of parameters, write the general solution y(x) of the differential equation of (BP) and its derivative y(x) respectively in the following form: (VI.2.11) b

y(x) = cjW(x) + c2p(X) +

+ (x)J

0(x)j

__

11

o P(A)W (E)

P(A)W f )

_),

J p_ { 1-v + F' (r) u P(t) r P(f)ww where u(x) is a solution of the linear homogeneous differential equation (VI.2.1) such that 0 and p are linearly independent, two quantities ca and c2 are arbitrary constants, and W(x) denotes the Wronskian of {0(x),p(x)}. Note that p(x)W(z) is independent of x and that ?% (z) =

(VI.2.12)

p(a)cosa-p(a)u'(a)sina,-40, p(b)cosQ-p(b)p'(b)sinfl,40.

From (VI.2.10) and (VI.2.11), we derive b

y(a) = ctm(a) + c2p(a) + 0(a)

Jc P(4)W(E)

y'(a) = ca0'(a) + c2 (a) + 0'(a)

r° a

A,

-N)

y'(b) = c,O'(b) + c2p'(b) y(b) = ciO(b) + c2p(b), The condition that y E V(a,b) and (VI.2.12) imply that ca is arbitrary and c2 = 0 0.

153

3. EIGENVALUE PROBLEMS

Remark VI-2-8. The materials of this section are also found in (CL, Chapters 7 and 11] and [Har2, Chapter XI].

VI-3. Eigenvalue problems In this section, we consider the eigenvalue problem (P(x) dx ) + u(x)y = Ay,

(EP)

y(a)coso - p(a)y'(a)sina = 0, y(b) cos /3 - p(b)y'(b) sin /3 = 0,

under the assumptions: (i) the quantities a, b, a, and 0 are real numbers such that a < b, (ii) u(x) is a real-valued and continuous function on the interval Z(a, b) = {x :

a
(iii) the function p(x) is real-valued and continuously differentiable, and p(x) > 0 on I(a, b). The quantity A is the eigenvalue parameter.

Let 6(x,,\) and ti(x, A) be two solutions of the homogeneous linear differential equation

(px)

(VI.3.1)

+ u(x)y = Ay

such that (VI.3.2)

{

¢(a, A) = sin a,

p(a)4'(a, A) = cos a,

,O(b, A) = sin /3,

p(b)t,V(b, A) = cos f3,

respectively. Then, -O(a, A) cos a - p(a)6'(a, A) sin a = 0, tp(b, A) cos /3 - p(b) 0'(b, A) sin /3 = 0.

(VI.3.3)

These two solutions 6(x, A) and tp(x, A) are analytic in A everywhere in C. Also, they are linearly independent if and only if (VI.3.4)

f

0(b, A) cos (3 - p(b)¢'(b, A) sin /3 i4 0

or

t'(a,A)cosa - p(a)ty'(a,A)sina # 0.

Therefore, we obtain the following result.

Theorem VI-3-1. In order that A be an eigenvalue of (EP), it is necessary and sufficient OW ,\ satisfies the equation (VI.3.5)

¢(b, A) cos i3 - p(b)cp'(b, A) sin 0 = 0

or, equivalently,

+'(a, A) cos a - p(a)0'(a, A) sin a = 0.

VI. BOUNDARY-VALUE PROBLEMS

154

Observation VI-3-2. All roots of (VI.3.5) are simple, since, if A is a root of (VI.3.5), we can prove 80(b, A) 8A

cos j3 - P(b)

8¢'(b, A) 8A

sun f # 0

in the following way. Let A be a root of (VI.3.5). Then, notice that z = is a solution of the differential equation

84(x, A) 8A

CP(x)L1 + u(x)z = Az + ¢(x,A). This implies that (VI.3.6)

r_

' , A)2 de, Ja P(O1't'(O z'(x) = clOx, A) + C21L (x) - 4'(x, A) f pW¢(f, A)dC + p '(x) = w(), A)2 dd, A) dC + p(x) f =

Z(X) = c16(x, A) + c2p(x) - O(x, A)

J

1n

j

PwW (f)

P(O-N)

where p(x) is a solution of homogeneous linear differential equation (VI.3.1) such that 4(x, A) and p(x) are linearly independent, two quantities ct and c2 are constants, and W(x) denotes the Wronskian of {¢(x,A),p(x)}. Note that p(x)W(x) is independent of x and that (VI.3.7)

p(a) cos a - p(a)p'(a) sin a

0,

p(b) cos,3 - p(b)1 (b) sin Q # 0.

Formulas (VI.3.6) imply that (VI.3.8)

z(a) = clo(a,A) + c2p(a),

z'(a) = ci4'(a,A) + c2,u (a),

fb

z(b) = cj¢(b, A) + c2p(b) - 4(b, A)

p(b)

Jn

z'(b) = c1 ¢'(b, A) + c2p'(b) - 0'(b.,\) j

+ p'(b)

w

Pwww

rP(u)W o P(4)W (0

Since (VI.3.3) is true for all values of A, it follows that z(a) cos a - p(a)z(a) sin a 0. Hence, from (VI.3.7) and (VI.3.8), we conclude that c2 = 0. Therefore, z(b) cos

- P(b) z'(b) sin,3 = (,u(b) cos$ - P(b) (b) sin 0)

° 0(Td # 0.

Ia a P(A)W w

0

Remark VI-3-3. If A = 0 is not an eigenvalue of (EP), we can define Green's function G(x,t) of problem (BP of §V-2 so that (EP) is changed equivalently to the integral equation y(x) = A

G(x,

To prove that all eigenvalues of (EP) are real, it is convenient to extend Theorem VI-2-5 (self-adjointness) to complex-valued functions f and g. Define a differential

3. EIGENVALUE PROBLEMS

155

((x)) + u(x)y and a vector space U(a, b) over the complex

operator C[y] = number field C by

U(a,b) _ If E C2 (a, b; C) : f (a) cos a - p(a) f'(a) sin a = 0, f (b) cos a - p(b) f'(b) sin,3 = 0},

where C2(a, b; C) denotes the set of all complex-valued functions which are twice continuously differentiable on the interval I(a, b) = {x : a < x < b}. Also, define an inner product (f, g) for two complex-valued continuous functions f and g on 2(a, b) by b

(f, 9) =

f (09(041

Jn

where g(1;) denotes the complex conjugate of g(f ). Since b

(f,'CWi) = 1

a

f

u(C)9(C)] dC

\p(e)
= P(b)f(b)9'(b) - p(a)f(a)9'(a) +

J

e [-Pwfv)9'(o + uh)fw9(-j <'

for f E U(a, b) and g E U(a, b), we obtain (f, C(91) - (C[f ], 9) = {p(b)f(b)9'(b) - p(b)9(b)f'(b)} - {p(a)f(a)g(a) - p(a)9(a)f'(a)}.

Also, since

.

f (a) cos a - p(a) f'(a) sin a = 0, 9(a) cos a - p(a)9'(a) sin a = 0,

f (b) cos Q - p(b)f'(b) sin /3 = 0, 9(b) cosQ - p(b)91(b) sin

p=0

for f E U(a,b) and g E U(a,b), we obtain

p(a)f(a)9'(a) - p(a)9(a)f'(a) = 0,

p(b)f(b)9'(b) - p(b)9(b)f'(b) = 0.

Thus, we extended Theorem VI-2-5 as follows.

Theorem VI-3-4 (self-adjointness). The operator C has the following property: (f, C[9]) = (C[f], 9)

for

f E U(a, b)

Theorem VI-3-4 implies the following conclusion.

and g E U(a, b).

VI. BOUNDARY-VALUE PROBLEMS

156

Theorem VI-3-5. All eigenvalues of (EP) are meal. Proof.

Let A be an eigenvalue of (EP) and let y(x) be an eigenfunction corresponding to A. We may assume that (y, y) = 1. Then,

(y,Cjy]) = (y, Ay) = X(y,y) = X,

(hjyl,y) = (Ay,y) = A(yy) = A, and Theorem VI-3-4 imply that J = A. 0 In order to explain distribution of eigenvalues on the real line IR, look at the basic comparison theorem (Theorem VI-1-1) more closely. Let us rewrite the differential equation

(P(X) dy) + g(x)y = 0

(VI.3.9)

in the form = (1 - p(x)g(x))rsin(O)cos(9)

p(x)

and p(x)dO = I + (p(x)g(x) - 1) sin 2(a)

(VI.3.10) by setting

y = r sin(O),

p(x) Li = r cos(B).

The right-hand side of (VI.3.10) is independent of r. Therefore, (VI.3.10) is a first-

order nonlinear differential equation on 0. Our main concern is the behavior of solutions of (VI.3.10).

Remark VI-3-6. Set w =

! ((x) L)

.

Then, differential equation (VI.3.9) be-

comes p(x) dx + w2 + p(x)g(x) = 0. This equation is further changed to (VI.3.10) by the transformation w = cot(8).

Remark VI-3-7. If the function g(x) is continuous on the interval 2(a, b), then every solution 0(x) of (VI.3.10) exists on T(a,b). Now, we prove the following basic lemma.

Lemma VI-3-8. Assume that (1) four functions gl(x), g2(x), pi (x), and p2(x) are continuous on the interval 2(a, b), (2) 92(x) ? gi(x) and pi(x) ? P2(x) > 0 on T(a,b),

!

(3) 01(x) (j = 1,2) are solutions of the differential equations Pi (x)

= 1 + [pi (x)gg (x) - 1J sin2(01)

(j = 1, 2),

157

3. EIGENVALUE PROBLEMS respectively.

Then, 02(x) > 61(x)

on

I(a,b)

02(a) > 01(a).

if

Proof.

Set 9(x) - 1

stn2(02) - sin2(0)

(1x,61,62()1p()

/ l

02

l

01 1

WS2(02)

h(x,02) _ (92(x) - 91(x))Sin2(62) +

1

P2(x)

P1 (x)

Then, the function w(x) = 02(x) - 01(x) satisfies the differential equation dw dx

- A(x,01(x),02(x)) w = h(x,02(x)) > 0. -

Therefore,

w(x)eXP

L-J a L

w(a) + on T(a, b).

I

r

Ja

x

ds > 0

h(s,02(s))eXP [- f

0

Remark VI-3-9. The proof given above also showed that if pi (x) > p2(x) > 0 on Z(a, b), then we obtain (a) 02(x) > 61(x) on I(a,b) if 02(a) > 01(a),

02(x) > 01(x) for a < x < b if 02(a) > 01(a) and g2(x) > g1(x) on a < x < b, ( y) in the case when g2(x) > g1(x) on a < x < b, if 01(a) = 0 and 61(b) = 7r and

if 0<02(a)a.

Also, y=rsin(0)>0 if and only if 0<0
= 1 > 0 at

0 = 0 (mod jr). Theorem VI-1-1 follows from (y). Setting 9(x,,\) = u(x) - A, apply Lemma VI-3-8 to problem (EP).

Lemma VI-3-10. Let 0(x, A) be the unique solution to the initial-value problem

Ax)

= 1 + (P(x)(u(x) - A) - 1)sin2(0),

0(a,A) = a

on the interval Z(a, b), where u(x) is continuous on Z(a, b), p(x) is continuous and positive on I(a, b), A is a real parameter, and at is a fixed real number such that 0 < a < rr. Then, for every c E 2(a, b) such that c > a, (i) 0(c, A) is a continuous and strictly decreasing function of A for -oo < A < +00,

(ii) lira 0(c, A) = +oo, a

(iii)

lim 0(c, A) = 0. A

+00

Proof.

We prove this lemma in three steps.

VI. BOUNDARY-VALUE PROBLEMS

158

Step 1. To prove (i), apply Lemma VI-3-8 to pi(x) = 12(x) = p(x), 91(x) = g(x, A1), and 92(x) = g(x, A2). If A2 < A1, then g2(x) > 91(x) on Z(a, b). Hence, 9(x, A2) > 9(x, A1) on a < x < b, since 9(a, A2) = 9(a, Al) = a.

Step 2. To prove (ii), choosing a real number m and a positive number P so that u(x) > m and p(x) < P on I(a, b), determine 0o(x, A) by the initial-value problem

P

= 1 + (P(m - A) - 1) sin2(8o),

9o(a, A) = 0.

Then, since g(x, A) > m -A and P > p(x) > 0 for x E I(a, b), -oo < A < +oo and a > 0, it follows from Lemma VI-3-8 that 9(c, A) > 9o(c, A)

for

a
and

- oo < A
Observe that v = P tan(Oo) satisfies the differential equation Hence, tan(9o(x, A)) =

P(m - A)

mP A (x - a) f for A < m. Note that

tan (

tan(9o(x, A)) = 0 if and only if tan I VmP m-A

tan(8o(x,A)) = 0

= 1 + ( MP A ) v2.

at

A

(x - a)) = 0, i.e.,

(x - a) = na

for

n=1,2....

P and that 9o(x, A) is strictly decreasing with respect to A if x > a (cf. Step 1). Furthermore, since tan(9o(x, m)) = x a, we must have 0 < 9o(x, m) < 2 for

P

a < x. Therefore,

m-P

9

Hence,

r

9 I x, m -

()2) = ntr

for n = 1, 2, ... .

(.)2) > na

for

x-a

n = 1, 2, ... .

acc

Thus, we conclude that lim 9(c, A) = +oa if c > a.

Step 3. To prove (iii, first notice that 0(c, A) > 0 for -oo < A < +oo, since P(IX) > 0 if 0 = 0. Choose a positive number M so that 0(a, A) = a > 0 and = 2(e)

lu(x) sin2(9) +

1

Ax) si n

I


on I(a, b).

Then,

dO < M - A sin2(0)

on Z(a, b).

159

3. EIGENVALUE PROBLEMS

For any positive number 6 such that 0 < a < r - 6, fix another positive number A(6) so that

M - A sin2(6) < -

c-10a

for

--

6<0< r - b

and A > A(6).

-

This implies that dO < 0 if 9 = 6. Hence, 9(x, A) > 6 for a < x < c if 9(c, A) > 6,

whereas9(c,A) 6fora A(6). Thus, we conclude that lim 9(c, A) = 0. 0 +00

Now, using Lemma VI-3-10, we prove the following theorem concerning problem

(EP).

Theorem VI-3-11. Assume that u(x) is continuous on the interval I(a.b) = (x: a < x < b} and that p(x) is continuously differentiable and positive on I(a,b). Then,

(1) problem (EP) has infinitely many eigenvalues:

AI >A2>...>An >..., (2)

n

lira An = -oo,

(3) every eigenfunction Q,,(x) corresponding to the eigenvalue An has exactly n-1 zeros on the open interval a < x < b. Proof.

Assume without any loss of generality that 0 < a < r and 0 < 0 < r. Define 9(x, A) by the initial-value problem (VI.3.11)

= 1 + (p(x)(u(x) - A) - 1)sin2(9),

p(x)

0(a, A) = a,

Tx-

and, then, define r(x, A) by (VI.3.12)

p(x)d. = (1 - p(x)(u(x) - A))rsin(9) cos(9),

r(a,A) = 1.

Then, (VI.3.13)

y = i(x, A) = r(x, A) sin(9(x, A))

is a nontrivial solution of the differential equation (VI.3.14)

d ds (P(x)±y) d.T

+ u(x)y = Ay

that satisfies the condition y(a)cos (a) - p(a)LY(a)sin (a) = 0. Note that from (VI.3.11), (VI.3.12), and (VI.3.13), it follows that p(x)

A)

= r(x, A) cos(9)(x, A).

VI. BOUNDARY-VALUE PROBLEMS

160

The eigenvalues of (EP) are determined by the condition 0(b,,\) = f3 ( mod sr). Since 8(b, A) is strictly decreasing as A +oo and since 8(b, A) takes all positive values (cf. Lemma VI-3-10), the eigenvalues A. are determined by

n = 1,2,... .

0(b,,\.) = Q + (n - 1)a, Observe that

(I) ¢(x, A) = 0 if and only if 8(x, A) = 0 (mod a), 8=0(mod

(II) 2 =p(x) >0if

a).

This implies that 4(x, A) has exactly k zeros on the open interval a < x < b if and only if kar < 0(b,,\) < (k + 1)7r. Observe also that

(n - 1)-,r < ,3 + (n - I)a < nn. Hence, the eigenfunction

has n - I zeros on the open interval

45(x,

a
Observation VI-3-12. As shown earlier, if u(x) > m and p(x) < P on Z(a.b), then

P

na

> n'rr

b-a

for

n = 1, 2, ... .

This implies that (VI.3.15)

/ an>m - P( bn,)z - a

n = 1, 2,....

Observation VI-3-13. If u(x) < p and p(x) > p > 0 on I(a.b), determine 81(x, A) by the initial-value problem

p dx = 1 + (p(p

sin2(81),

81(a, A) _ ?r.

Then, since u(x) - A < p - A and p(x) > p > 0 for x E T(a, b), -oo < A < +oo, and a < s, it follows from Lemma VI-3-8 that

0(c,,\) < Oi(c,A)

for

a
and

-00 < A < +oo.

Observe that v = p tan(81) satisfies the differential equation L = 1 + (1f Hence,

tan(81(x, A)) =

1

Pµ-

tan P

(x - a)

-P A) v2.

for a < µ.

161

3. EIGENVALUE PROBLEMS

(x - a) = nrr for n = 1,2,... and

Note that tan(91(x,A)) = 0 at

AP2 that 01(2, A) is strictly decreasing with respect to A if x > a.

2 for a < x since tan(01(x, {e)) = x p a 7r < 91(x, p) < 91

(X,

p xa (-na \2 I

Furthermore,

Therefore,

.

= (n + 1)ir

for

n = 1, 2, ... .

< (n + 1);r

for

n = 1, 2,...

Hence, P

)2)

( nTr

x-a This implies that

An+2 < p - p

(VI.3.16)

2

nrr

n = 1,2,... .

for

G-a

If Al and A2 are determined by

(VI.3.17)

01(b,A2) = 0 + it,

and

01(b,A1) = Q

then Al < Al and A2 < A2. From (VI.3.15) and (VI.3.16) we derive the following result concerning distribution of eigenvalues of (EP).

Theorem VI-3-14. For n > 3, eigenvalues An satisfy the following estimates:

< Iml (i)

if

n2

An < n2

2

P

Cb

- a)

'

and 2

n2 +

< IµI +

P(b- a)

n2

rr

2

4PCb-a) C1

if

_An> n2

1

1

n

n

-P Wa

where in, p, P and p are real numbers such that

P > p(x) > p > 0

and

p > u(x) > m

for

x E Z(a, b).

VI. BOUNDARY-VALUE PROBLEMS

162

Remark VI-3-15. Lemma VI-3-8, Lemma VI-10, and Theorem VI-3-11 are also found in (CL, §§1 and 2 of Chapter 8] and [Hart, Chapter XI].

VI-4. Eigenfunction expansions Let us define a differential operator L[y] = dx

(p(z)) + u(x)y and the vector !LV

space V(a, b) over the real number field R in the same way as in §VI-2. Also, as in §VI-2, define the inner product (f, g) for two real-valued continuous functions f b

and g on Z(a, b) _ {x : a < x < b} by (f, g) = operator L has the following property:

(f,L[g]) = (L[f],g)

for

It is known that the

J

f E V(a,b) and g E V(a,b)

(cf. Theorem VI-2-5). Define a norm of a continuous function on X(a, b) by 11f 11 =

(f, f). Then, 1(f, g)f < Ilf II llgll. The following theorem is a basic result in the theory of self-adjoint boundary-value problems. Theorem VI-4-1. Let Al and A2 be two distinct eigenvalues of the problem L[y] = Ay,

(EP)

y E V(a, b).

Let r11(x) and r}2(x) be eigenfunctions corresponding to Al and A2, respectively.

Then, (rll,rh) = 0. Proof.

Note that (r71,L[r12]) = A2(rh,g2) and (Ljrll1,rh) = Al(rl1,rn). This implies that A2(r11, r12) = AI (r11, r12). Therefore, (r1i, r12) = 0 since Al 0 A2 0

It is known that problem (EP) has real eigenvalues ( lien

Al > A2 > ... > An > ...

Let r)1(x),-. *2(x),

.

n-++oo

An = -co).

be the eigenfunctions corresponding to Al, A2,...

,

such that 11,11 = 1 (n = 1, 2, ... ). Theorem VI-4-1 implies that (' h,''m) = 0 if h # k. From these properties of the eigenfunctions rjn, we obtain k

k 11f

-

>(f,vh)T1hII2 = Ilfll2 h=1

and hence

-

E(f.rlh)2 > 0 h=1

+oo E(f,7,lh)2

< IIffl2

(the Bessel inequality)

h=1

for any continuous function f on Z(a, b). In this section, we explain the generalized Fourier expansion of a function f (x) in terms of the orthonormal sequence {rl (x) : n = 1,2,... }. As a preparation, we prove the following theorem.

163

4. EIGENFUNCTION EXPANSIONS

Theorem VI-4-2. Let µo not be an eigenvalue of (EP) and let f (x) be a continuous function on Z(a, b). Then, the series (f, 1h)t1h(x) h=1

Ah - Jb

is uniformly convergent on the interval Z(a, b).

Proof Define a linear differential operator Go by £o[yl = G[yl - poy. Then, there exists Green's function of the boundary-value problem (BP)

y E V (a, b)

Go [y1 = f (x),

such that the unique solution of problem (BP) is given by b

y(x) =

J.

(cf. §VI-2; in particular, Observation VI-2-3). Define the operator G[f[ by b

G(f 1 = (G(x, ), f) =

J

G(x, Of WA

for any continuous function f(x) on 1(a, b). Since Go[g[f]l = f, we obtain (I,G[.1) = (4[9[f)), 9191) = (9[f], 4191911) = (9[fl,9)

for any continuous functions f (z) and g(x) on Z(a, b). Also, since Go[rlh1 = c[nh1 Ahr1hUO . Hence, A01% = (Ah - l'o)llh, we obtain 9['7h] =

(f, rlh)ghI lz

< K

lz

(f, '1017h

=K

h=l,

(f, ih)2, h=11

where K is a positive constant such that J!G(x, ) Il < K on 1(a, b). I

the Bessel inequality implies that T(a, b).

0

Jim

Finally,

12

li,ls-+oo h=l ,j

(f' nh)nh(x) l = 0 uniformly on - 110

Now, we claim that the limit of the uniformly convergent series of Theorem VI-4-2 is equal to 9[f].

164

VI. BOUNDARY-VALUE PROBLEMS

Theorem VI-4-3. For any continuous function f (x) on Z(a, b), the sum +00 (h,nh) is equal to 9[f]. Proof Set 91 (f ] = 9 If, - 1: (f, TO 17h . Then, it suffices to show Ah - PO h=1=1 (VIA. 1)

lim IIGe[f111 = 0

for every continuous function f (x) on Z(a, b). We prove (VIA.1) in four steps. Without any loss of generality, we assume that Ah - uo < 0 for h > t . Also, note that (VI.4.2)

(f,9e[9]) = (94f1,9)

for any real-valued continuous functions f and 9 on Z(a, b). Step 1. It can be shown easily that Gr[>jh] = chrjh, where

Ch =

for

h = 1,2,... ,e,

for

h > t.

Let y be an eigenvalue of 9t and let ¢(r) be an eigenfunction of 9t associated with

y. Then, ¢ E V(a,b), 11011 # 0, and y¢ = 9[41 -

(0, 17017h

h=1 An

- go

.

Also, (VI.4.2)

implies that (0,9471h]) = (91'101,170. Hence, ch(¢, nh) = y(¢, nh) Consequently,

(¢, rjh) = 0 if y 0 ch. Therefore, (¢, rjh) = 0 for h = 1, ... , I and y¢ = G[¢] if y 96 0. Hence, either y = O or y = ch (h > P). Step 2. In this step, we prove that (VI.4.3)

sup 1190JI1 = sup

nfa=1

uni=1

1(911f1.f)I-

In fact, the inequality I(9e[f),f)1

5 11Ge[f]IIIIfII implies sup 1I9t[f111 >ufu=1 sup 1(91[f],f)I Also, since (Ge[f ±9],f ±9) = (911f],f)+(9t[9],9)±2(9t[f],9),

Of H=1

we obtain

4(91[f],9) = (91 [f+9) ,f+9) - (9e[f-9],f-9) <

( sup l(Ge[w], w)I lH'°t=1

=2

sup N++ N=1

(IIf + 9II2 + If - 9112)

I(9e[wj,w)l

(11f112+119112).

165

4. EIGENFUNCTION EXPANSIONS

0 and IIfIII = 1, and set g = Gt[f]

Suppose that 9t[f]

IIGt[f]II'

sup I (Gt[w], w)I. This shows that sup 119t[f]II IIwIl=1

Then, IIGe[f]II

sup I (Gt[f]+f )I

-

Thus, the

I1I11=1

11!11=1

proof of (VI.4.3) is completed.

Step 3. In this step, we prove that - sup

I (Gt If], f) I

is an eigenvalue of Gt. To

IlfI1=1

do this, set c = sup I(Ge[f], f)I. Note first that c > I(Gt[ne+I],ne+i)I = Ice+II > Ill 11=1

0.

Also, note that c = sup (911f], f) or -c = inf (Gt[f], f). Suppose that [if U=1

Iif11=1

c = sup (9t If], f ). Then, there exists a sequence f (j = 1, 2.... ) of continuous 11111=1

functions on T(a, b) such that lim (Gt[ fi], fi) = c and IIf, II = 1(j = 1, 2, . . . ) Since {Gt[ fj] : j = 1,2.... } is bounded and equicontinuous on T(a, b), assume without any loss of generality that lim Gt[ f)] = g E G(a, b) uniformly on T(a, b). -.+00

Let us look at (VI-4.4)

IIGt[fjI - cf,

II2

= I1Gt[fj]II2 + c2 - 2c(Gt[f,], f,) <-

2c2

- 2c(Ge[fjl , fi).

This implies that lim

- cfj II = 0 and, hence, Um 119 - c f j II = 0. Thus, j- +00 Gt[g] = cg. Also, from (VI.4.4), it follows that IIgII2 = c2 > 0. Therefore, c must be an eigenvalue of Gt. However, since all eigenvalues of Gt are nonpositive, this is a contradiction. Therefore, -c = inf (91 [fit f ). We can now prove in a similar way

)+oo 11 9t [f, I

Illil=l

that 1

-c = ct+1 =

),ttl - µ0

Step 4. Since sup IIGt[f]II =

, we conclude that lien sup I191[f]II = 0 1 !Lo - t+1 t-'+0°1!11!=l and the proof of (VI.4.1) is completed. The following theorem is the basic result concerning eigenfunction expansion. 11111=1

+00

Theorem VI-4-4. For every f E V(a, b), the series f = 1: (f, nh)nh converges to h=1

f uniformly on T(a, b). Proof.

Set g = .Co[f], then f = g[g]. Therefore,

f=

+00

h=1 +00

_ h=1

(9+17 h)17h

+C0 = 1: (GO[J]+nh)nh

Ah - {b (1 £ofrlh])r)h

ah _

h=1

Ah - 110 +00

_ >(f, rlh)rlh h=1

VI. BOUNDARY-VALUE PROBLEMS

166

For every continuous function f on 1(a, b) and a positive number e, there exists an f, E V(a, b) such that 11f - fe II <_ e. Since t

t

f - (f, 77h)rh = f /

t

ff + fe - 1: (ff,17h)nh + 1: (ff - f, 17077h, h=1

h=1

h=1

C

(f , 7nh )T7h = 0. Therefore, f = 0 if (f, Tjh) = 0 for h =

we obtain lim I f t-.+00

l

h=1

1, 2,.... This proves the following theorem.

Theorem VI-4-5. If a continuous function f on 1(a, b) satisfies the condition U, 17h) = 0 f o r h = 1, 2, ... , then f is identically equal to zero on the interval 1(a, b).

Remark VI-4-6. Also, we have +00

j11112 = >(f, rth)2

(the Parseval equality).

h=1

Furthermore, this, in turn, implies that if f (x) and g(x) are continuous on 1(a, b), then h=1

= Ay, y'(0) = 0, y(a) = 0, Example VI-4-7. Using the eigenvalue problem dX2 we construct an orthogonal sequence {cos(nx) : n = 0, 1, 2.... } of eigenfunctions. 00

This yields the Fourier cosine series a-° 2 + Ean cos(nx) of a function f (x), where n=1

an =

2

r*

a oJ

f(x) cos(nx)ds. By virtue of Theorem V1-4-4, this series converges

uniformly to f (x) on 1(0, 7r) if f (x) is twice continuously differentiable on 1(0, zr)

and f'(0) = 0 and f'(Tr) = 0.

Example VI-4-8. Using the eigenvalue problem d = Ay,

y(O) = 0, y(7r) = 0, we construct an orthogonal sequence {sin(nx) : n = 1,2,...l of eigenfunc00

tions. This yields the Fourier sine series >bn sin(nx) of a function f (x), where n=1

bn =

f/ f (x) sin(nx)ds. By virtue of Theorem VI-4-4, this series converges

ao

uniformly to f (x) on 1(0, ir) if f (x) is twice continuously differentiable on 1(0, ir) and f (0) = 0 and f (a) = 0. Example VI-4-9. If f (x) is twice continuously differentiable on Z(-ir, ir), f (-vr) _

f (ir), and f'(1r) = f'(-7r), set fe(x) = 2 if (x)+f (-x)] and fo(x)

[f (x)-f (-x))

167

4. EIGENFUNCTION EXPANSIONS

Then, fe satisfies the conditions of Example VI-4-7, while fo(x) satisfies the conditions of Example VI-4-8. Thus, we obtain the Fourier series cc

+

ao

[an cos(nx) + bn sin(nx)], n=1

where an =

7r

J

* f (x) oos(nx)ds and bn =

!

r J

f (x) sin(nx)ds. The Fourier

series converges uniformly to f (x) on Z(-7r, 7r).

Remark VI-4-10. The Fourier series of Example VI-4-9 can be constructed also from the eigenvalue problem L2 = Ay, y(-7r) = y(7r), y'(-7r) = y'(ir). Including this eigenvalue problem, more general cases are systematically explained in [CL, Chapters 7 and 11]. For the uniform convergence of the Fourier series, it is not necessary to assume that f (x) be twice continuously differentiable. For example, it suffices to assume that f (x) is continuous and f'(x) is piecewise continuous on 1(-7r, 7r), and f (7r) = f(-7r). For those informations, see [Z]. Remark VI-4-11. The results in §§V1-2, VI-3, and VI-4 can be extended to the case when p(x) is continuous on 1(a, b), p(x) > 0 on a < x < b, and p(a)p(b) = 0 under some suitable assumptions and with some suitable boundary conditions. Although we do not go into these cases in this book, the following example illustrates such a case.

Example VI-4-12. Let us consider the boundary-value problem

d f_dy

.

_

(VI.4.5)

and y(l) = 0,

y(x) is bounded in a neighborhood of x = 0,

where u(x) and f (x) are real-valued and continuous on the interval 1(0, 1).

Step 1. Consider the linear homogeneous equation (VI.4.6)

(X2idv)

d

+ u(x)v = 0

on the interval 1(0,1). Since 1 and logs form a fundamental set of solutions of

the differential equation ± (xdx

= 0, a general solution of (VI.4.6) can be

constructed by solving the intergral)equation v(x) = c1 + C2 109X +

10

(logs)10 u(C)V(4)dC, 0

where c1 and c2 are arbitrary constants. In this way, we find two solutions O(x) and

-i(x) of (VI.4.6) such that O(x) and 4'(x) are continuous on 1(0,1), lim O(x) = 1, and lien x¢'(x) = 0, whereas tI'(z) and tY(x) are continuous for 0 < x < 1, r_0+

lim (O(z) - logs) = 0, and urn (xo'(x) - 1) = 0. In general, this step of analy0+

z-+0+

sis is very important. The behavior of solutions at x = 0 determines the nature of eigenvalues of the given problem. Denote also by p(x) the unique soltion of (VI.4.6)

such that p(1)=0andp'(1)=1.

VI. BOUNDARY-VALUE PROBLEMS

168

Step 2. Assuming that 0(l) # 0, set ¢(x AO G(x,

WW

for

¢(F)p(x)

for

0< x < e <1 ,

where W(x) is the Wronskian of 10,p) and xW(x) = 0(1). Then, G(x,t) is Green's function of problem (VI.4.5). The unique solution of (VI.4.5) is given i

by y(x) = 1

Jo

G(x, )2dxd{ < +oo. 0

Step 3. It is easy to prove the self-adjointness of problem (VI.4.5). Hence, all eigenvalues are real, and the orthogonality of eigenfunctions follows. By virtue of (VI.4.7), eigenfunction expansions can be derived in the exactly same way as in §VI-4.

Step 4. We can also derive Theorem VI-3-11 in the exactly same way as in §VI-3. To do this, consider the differential equation (VI.4.8)

0-0 + u(x)y = )y.

For every value of A, there exists a unique solution O(x, A) such that lim O(x, A) = 1

xo+

and lim x¢'(x, A) = 0 (cf. Step 1). It is easy to see that p(x, AT and x*'(x, A) X--o+

are continuous in (x, A) for 0 < x < I and -oo < A < +oo. Define 6(x, A) A). >r by cot (( B x, A )) = A) We fix B(0, A) = . Using the same method as in

4(x, §VI-3, we can verify that ¢(1,A) is strictly decreasing and takes all values between 0 and -oo. Eigenvalues of problem (VI.4.8) are determined by the equations 9(1, A) = mar (m = 1, 2.... ). Theorem VI-3-14 cannot be extended to the present case, since there is no positive lower bound of x on 1(0,1).

VI-5. Jost solutions So far, we have studied boundary-value problems on a bounded interval on the real line R. Hereafter, we consider the scattering problem, which is a problem on the entire real line. To explain the essential part of the problem, let us consider

a homogeneous linear differential equation Ly + ((I - u(x))y = 0, where ( is a complex parameter and u(x) is a real-valued continuous function of x such that u(x) = 0 for all sufficiently large values of ext. It is evident that e_ttx and e'(' are two linearly independent solutions of this equation if 1xI is large. Therefore, if a

5. JOST SOLUTIONS

169

positive number M is sufficiently large, the solution y = e-'(= for x < -M becomes a linear combination a(()e-'t=+b(()e't= of two solutions a-K= and eK' for x > M. The main problems are (i) the properties of a(() and b(() as functions of ( and (ii) construction of u(x) for given data {a((), b(()). Keeping this introduction in mind, let us consider a differential operator

G = - D2 + u(x),

(VI.5.1)

and u(x) is real-valued and belongs to C°°(-oo, +oo) such that

where D =

aj

+o°

(VI.5.2)

J

(1 + jxj)lu(x)jdx < +oc.

We study the differential equation

Gy = (2y,

(VI.5.3)

where (is a complex parameter. First, we construct two basic solutions which are called the Jost solutions of (VI.5.3). Theorem VI-5-1. There exist two solutions f+(x, () and f_ (x, () of (VI.5.S) such that

(i) ft are continuous for -oc < x < +oo, are analytic in ( for £ (() > 0,

(VI.5.4)

(ii) ft

0,

(iii) e:

°

jf+(x, () - e'`=I <_ C(x)P(x) 1 +- 1(I

f- (x, () -

e-K=I

e"x

< C(-x)P(x)1 +

I(1

for (VI.5.4),, where q = Q3'((). C(x) is non-negative and nonincmasing, t00

P(x) = I

(1 + IrI)Iu(r)Idr,

and

O(x) =

J

(1 + Irj)ju(r)jdr. o0

Proof.

Step 1. Construction of f+: Solve the integral equation

y(x, () = e"z

sin(((s - r)) u(r)y(r,

()dr

by setting yo(x,() = eK=,

+00 sin((((- r)) u(r)ym(r,

ym+1(x, () x

and 00

f+(x, () = E Ym(x, () m-o

()dr (m > 0),

VI. BOUNDARY-VALUE PROBLEMS

170

Step 2. Proof of (i) and (ii): To prove (i) and (ii), it suffices to prove that if

( 21x1 + 2

e(x) -

2

for

x < 0,

for

x > 0,

then

(VI.5.6)

(C(x)P(x))me-1,

Iym(x> }l < mj

m= 0,1,2....

for (V1.5.4).

Inequality (VI.5.6) is true for m = 0. Assume that (VI.5.6) is true for an m; then,

r+ Isin(((x - r)) I

Iym+i(x,C)J < mi Z

Note that

sin(((x - r))

a

2i(

(1

-

e-2i(z-7)() =

z-r eft(Z-r)

1

e-2it=dz.

Hence,

sln(((x - r)) I < e-q(z-T)(r - x) for r1 > 0. C

Therefore, +00

I ym+1(x,C)I <

(r - x)tu(r)IP(r)mdr.

f

(C(x)me-nx

z

Since

j

+00

(T - x)Iu(T)IP(r)mdr < C(x)f.

+00 (1

m+1

+ I rI )I u(r)I p(r)mdr

C(x)P(x)m+

we obtain I ym+1(x, ()I <

(C(x)P(x))m+ie-nZ

1)!

(m

Remark VI-5-2. At the last estimate, the following argument was used:

(a) r-xO,

(b)

= Jz

for x < 0.

( z

+00

+

J

< 2Ixl1 z

z

I u(r){P(r)mdr +

2fz

TI u(r)I P(T)mdT

5. JOST SOLUTIONS

171

Step 3. Proof of (iii): First, the estimate 'o

I f+(x, 0-

I

I

<- E n,, (O(x)v(x))me-''z m=1

(VI.5.7)

00

< C(x)p(x)

I

l i

m=1

(C(x)p(x))m-1 I

e-"z

follows from (VI.5.6). However, this estimate is not enough to prove (iii). So, let us derive another estimate for large I(I. To do this, we shall prove that (VI.5.8)

I ym(x, ()I

rn!

(k)

or(x) = fx

"

m

m=0,1,2....

e-?=,

iu(r)jdr.

Inequality (VI.5.8) is true for m = 0. Assume that (VI.5.8) is true for an m. Then, Iym+1(x, ()1

-

+ao

M! I(I 1

Jz

I sin(((x - r))IIu(r)I e-"'a(r)mdT.

Note that

sin(((x -

r))e-+,zl = e 2 11

-

e-'

e-2it(z-7)1

if

3(() 2!

0.

Hence, 1

Iym+1(x,()I

m!

+00

e-'7= KIm+1I J.

1

m+1

iu(r)io(r)mdr = (m+ 1)! (!!-(X) 1(i}

a-'

.

This establishes (VI.5.8). Therefore, (VI.5.9)

v(x) If+(x, () - ei(zl < (i

=

1

M!

-))M-,l I e-qz.

OWC-1

The proof of Theorem VI-5-1 for f+ (x, () can be completed by using (VI.5.7) and (VI.5.9).

For f-(x,(), change x by -x to derive (VI.5.10)

-LY + u(-x)y = (2y.

The solution f+ for (VI.5.10) is the solution f- for (VI.5.3). 0

VI. BOUNDARY-VALUE PROBLEMS

172

Remark VI-5-3. The solutions f± are uniquely determined by the conditions given in Theorem VI-5-1. Also, (VI.5.11)

j+00 mi(((x

d

(x,

- r))u(r)f+(r, ()dr

and Id'

(x,() - i(e{C2 +

f

cos(((x - r))u(r)eK=dr

(VI.5.12)

-

< C(x)p(x)a(x)

1+I(I

for (VI.5.4).

Remark VI-5-4. If u(x) = 0 identically on the real line, then f f (x, () = et't'. If u(x) = 0 for Ixl _> M for some positive constant M, then f+(x, () = e'(= for x > M, while f- (x, () = e-K= for x < -M.

VI-6. Scattering data For real (, 1+ (x, () = f+ (x, -(), where f denotes the conjugate complex of f . Both f+ and f+ are solutions of Gy = (2y and

I f+(x, f) - e

ZI

< C(x)P(x)

dl (x, + i(e-+ dx

1 + ICI

r+

C() +

r))u(r)e(rdrl

(x) 16

for -oo < x < +oo and -oo < f < +oo. Let W(f,gj denotes the Vlwronskian of {f,g}. Then,

W If+, f+l - I f+(x, ()

f+, (x, ()

-2i(

(-oo <

< +oo).

J

This implies that I f+, f+} is linearly independent if 36 0. This, in turn, implies that the solution f_ is a linear combination of I f+, 1+}. Set

f-(x,() = a(()f+(x,() + b(f)f+(x,()

(VI.6.1)

It is easy to see that (VI.6.2)

i

i

f+(x+

a(() =

f-(x,()

and (VI.6.3)

b(() = Wif+,f-j = I

f+(x,() f .(x,()

I.

The function a(() is analytic for %() > 0 and (VI.6.4)

a(() = 1 + 0((-1)

as

(-' 00

on

`3(() > 0,

whereas the function b(() is defined and continuous on -oo < ( < +oo. Furthermore, b(() = O(Ifl-1) as I(I ---. +oo. To simplify the situation, we introduce the following assumption.

6. SCATTERING DATA

173

Assumption VI-6-1. We assume that Iu(x)I <

(-oo < x < +oo)

Ae-kj=j

for some positive numbers A and k. The boundary-value problem

y E L2(-oo,+oo)

Gy = Ay,

(VI.6.5)

is self-adjoint, where L2(-00, +oo) denotes the set of all complex-valued functions f (x) satisfying the condition

r+00

J

`f (x) I2dx < +oo. The self-adjointness

of problem (VI.6.5) can be proved by using an inner product in the vector space L2(-oo, +oo) in the same way as the proof of Theorem VI-3-4. Therefore, eigenvalues of problem (VI.6.5) are real. Furthermore, all eigenvalues are negative. In fact, if t 96 0 is real, then A = £2 > 0 and the general solution ci f+ + c2f+ is asymptotically equal to cle`4=+c2e-'4= as x --. +oc. If t = 0, then f+r(x,0) is asymptotically

equal to 1 as x equal to x as x

+oc. Moreover, another solution f+J z

is asymptotically f+ +oc. Therefore, all eigenvalues are A = (ir7)2 < 0. Furthermore,

all eigenvalues of (VI.6.5) are determined by

a(iry) = 0.

(VI.6.6)

This implies that all zeros of a(() for 3(() > 0 are purely imaginary.

Under Assumption VI-6-1, ft(x, () are analytic for 9'(() > - 2. Hence, a(() has only a finite number of zeros for 3(() > 0 (cf. (VI.6.4)). Let S = irlj 0 = 1, 2, ... , N) be the zeros of a(() for (() > 0. Then, f+(x, ir73) are real-valued and f+(x, ins) E L2(-oo, +oo). Set

c) _

r+ao

(j = 1, 2, ... , N),

1

f+(x,n,)2dr

(VI.6.7)

(-oc < { < +oo). Observation VI-6-2. Every eigenvalue of (VI.6.5) is simple, i.e., da(C)

4

;f 0

if

a(() = 0.

Proof.

From a(t;) = 0, it follows that `ia(() d(

i d W[ f+, f-[ 2S d(

- 2 2 W [f+, f-I = 2Zr (u'[f+(, f-] + N'[f+, f-
VI. BOUNDARY-VALUE PROBLEMS

174

where ft denotes 4f . Also, two relations

`f+S + of+C = (2f+C + uf+

and

imply that d

d w(f+(, f-1 = 2-f+f-

d

and

W(f+, f-cl = -2f+f_

Since a(() = 0, there exists a constant d(() such that f_(x,() = d(()f+(x,C). Therefore, -2S+00

W(f+t, f-) =

+

f+f-dr

and

W(f+, f-tl = -2C

f

o0

f+f-dr.

Thus, we obtain da(()

+oo

-i

d(

f+f-dr = -id(()f

00

+oo

f+dr = -id(()

# 0.

Ci

oo

Observation VI-6-3. The quantities a(() and b({) satisfy the following relation: (VI.6.8)

Ia(t)12 - Ib(E)12 = 1

- oo <

for

+00.

Proof.

From

f_(x, -{), it follows that

aW = a(-4),

b(() = b(-(),

W(f-,f-l = 2i,.

and

Therefore, (VI.6.8) follows from

f- = af+ + b(af- - bf_) = a(f+ + bf-) - Ib12f_ _ aaf- - Ibl2f_ = {1a12 - Ibl2}f

.

Observation VI-6-4. The formula (VI.6.9)

a(() _

(- iq, exp [-L f+°° log(1 i1h

2ri

ao

(- S

for %() > 0 shows that the quantities q, and r({) determine a((). Proof

Set f (() = a(()j

Then,

J

7. REFLECTIONLESS POTENTIALS

175

(i) f (() is analytic for $(() > -2 and (94 0, (ii) ((f(() - 1) is bounded for 3(() >

-2,

(iii) f (() & 0 for Qr(() >- 0 and (54 0. O(log(()) near = 0 and F(() _ Observe that f(() - 1 = O((-'). From this 0((-1) as ( -, 00 on 3(() > 0.

Set also F(() = log(f (()). Then,

observation, it follows without any complication that 1 /'}Oc 2log if (01 F(() - Trif_. -(

for

£(() > 0

(cf. Exercise VI-18). Now, (VI.6.9) follows from If(t) = Ia(()j and loBja(()f2 =

log(Ta(f)22/

Definition V1-6-5. The set {r((), (n,,n2,... ,nN), (cl,c2,... ,CN)} is called the scattering data associated with the potential u(x).

VI-7. Reflectionless potentials is called the reflection coefficient. If this coefficient is zero, the The function potential u(x) becomes a function of simple form. Let us look into this situation.

Observation VI-7-1. If r(() = 0, then b(() = 0. This means that f_(x,t:) = a(t) f+(x, £) = a(t) f+(x, -t) (cf. (§VI-6) ). Therefore, using the relation f-(x, -C)

f+(x,0) =

a(-()

for

3(() !5

0,

we can extend f+ for all ( as a meromorphic function in (. Furthermore, if irlj (y =

1, 2,... , N) are zeros of a(() in 3'(() > 0, then -inj (j = 1, 2,... , N) are simple poles of f+(x, () in ( and = C)f-(x,irli) = -icj f+(x, ins). Ftesidueoff+at -inj = - f-(x,ir,) a<(inj) id(irtj)

Observation VI-7-2. Set gJ (x) = e-" cj f+(x, iii ). Then, from the fact that e_,<= f+(x, () - I as I(I - +oo, it follows that

1 - i (+9,(x)ins

f+(x,0) = e'S=

(VI.7.1)

=1

Setting ( = int in (VI.7.1), we obtain e2ne=

(VI.7.2)

N

9t(x) + C-1

J=1

1

nt+n'gj(x) = 1

(t=1,2,...,N).

VI. BOUNDARY-VALUE PROBLEMS

176

Observation VI-7-3. If we set F(x,() = e-'t= f+(x, (), then -F" - 2i(F'+uF =

'` 91 +(x) , it follows that 0. Since F(x, ) = 1 - i-1 N u(x) = 2E gj,(x).

(VI.7.3)

.7_1

Observation VI-7-4. Let us solve (VI.7.2) for the g,(x). First, set

hi = gie'''=

(j = 1,2,... ,N).

Then, (VI.7.2) becomes (VI.7.4)

c

ht +

h = cte-''i:

rJt + I?j

J=1

(t

Write the coefficient matrix of (VI.7.4) in the form IN + C(x), where IN is the N x N identity matrix. Since N

E 717t )t°1 'b

+op

=

L

+ 171

2

N

E 7 e-n,,= i

dx

p

and 77, (j = 1, 2,... , N) are distinct, the matrix IN + C(x) is invertible for -oo < x < +oo. Set L(x) = det(IN + C(x)). Then, manipulating with Cramer's rule, we

can write g, (j = 1, 2,... , N) and (VI.7.5)

dL da

in the following forms:

9j (x) = A(X)

(j = 1, 2,... , N)

and (VI.7.6)

dL(x) dx

N

J=1

Thus finally, from (VI.7.3), it follows that N

u(x) =

2E g,.;(x)

= -2 a(x)

or

(VI.7.7)

u(x) = -2z(log(0(x))}.

177

7. REFLECTIONLESS POTENTIALS

This implies that u(x) is a rational function of exponential functions and satisfies Assumption VI-6-1 of §VI-6. To see this, the following remarks are also useful: (a) we have the identity N

N

?1 e2n'x9s(x)2,

2

gg(x)

j=1

j=1

i

(b) g j (-oo) = = limo g . (x) ( j = 1, 2, ... , N) exist and N

E

1

g,(-oo) = 1

(e = 1, 2,... , N).

1=1171+n,

To show (a), derive e2nez

cr

9t(x) +

N =1 1h + n

-

g, (x) =

2cl[

e2arxgt(x)

((= 1, 2, ... , N)

from (VI.7.2). Multiplying both sides by ge, adding them up over f, and interchanging the orders of summation, we obtain (a). To show (b), calculate the inverse of the coefficient matrix of (VI.7.2). Using Cramer's rule on (VI.7.4), it can be shown that he(x) -, 0 exponentially as x -+ oo. Also, (b) implies that e2°,, xg,(x)2 is expo-oo. Therefore, (a) implies that u(x) satisfies Assumption nentially small as x VI-6.1.

Example VI-7-5. In the case N = 1, if we determine g(x) by

e2+ gz

1

g

c' we obtain u(x) = -- e2'rzg(x)2.

1, then u(x) = 2g'(x). Since g(x) = 2c e+7x +

Qx

Also,

FF17e-nx

In

xsech(11(x + p)), where p =

g(x) _

/

C

g(x) = 1 implies that g(x) =

2 e-n=

(211

`c

,

and, hence,

211

(VI.7.8)

u(x) = -2112 sech2(g(x + p)). 4

In particular, formula (VI.7.8) yields u(x) = -8 sech (2 (x + In (s) )

I

in the

case when N = 1, n1 = 2, and c1 = 5. On the other hand, a\straightforward calculation using formula (VI.7.7) yields u(x) _

(5+4)2. Also, u(s) = -2sech(x)

is the reflectionless potential corresponding to the data r1= 1 and c = 2.

Example VI-7-6. If N = 2, ill = 2, rh = 3, c1 = 5, and c2 = 2, we obtain 40e42(16 +

135e2-- + 600e6x + 2160e10s + 3600e12=)

U(X) (1 + 20e4x + 75e6. + 60ebox)2

by calculating (VI.7.7).

VI. BOUNDARY-VALUE PROBLEMS

178

Remark VI-7-7. If the reflection coefficient r(t) = 0 and if there is no eigenvalue for the problem

-

d2 y

+ u(x)y = Ay,

y E L2(-oo, +oo),

2

then u(x) = 0 identically for -oo < x < +oo. In fact, f+(x,() is analytic in everywhere in the (-plane. Furthermore, l e-`<= f+(x, () -11 -+ 0 as - oo. Hence, f+(x,() = e'(=. This shows that u(x) = 0. Remark VI-7-8. If u(x) 0 but u(x) = 0 for }xI _> M, where M is a positive number, then the reflection coefficient r(S) # 0. In fact, if r(C) = 0, then u(x) is analytic for -oo < x < +oo. Hence, u(x) = 0. This is a contradiction. This remark reveals that the problem posed at the beginning of §VI-5 is not as simple as it looks.

VI-8. Construction of a potential for given data If scattering data {0, (71,... 7x), (cl,... ,cN)} are given, formula (VI.7.3) gives the corresponding reflectionless potential u(x) as it is shown in Examples VI7-5 and VI-7-6. However, in these two examples, we assumed implicitly that such

a potential exist. Since the existence of u(x) has not been shown yet, it must be proved. To do this, for given data

7i>0, 72>0, ..., 7N>0 and c1>0, c2>0, ..., cN>0, define g, (j = 1, 2,... , N) by (VI.7.2) and define f+ and u by (VI.7.1) and (VI.7.3). Hereafter, the first thing that we have to do is to show that y = f+ is one of the Jost solutions of Ly = (2y.

Observation VI-8-1. Let gi (t = 1, 2,... , N) be determined by (VI.7.2) and u(z) be defined by (VI.7.3). Then, (VI.8.1)

gi + 27r9t - ugt =

(=1,2,... , N).

0

Proof Write (VI.7.2) in the form N

E ae, (x)9, (x) = 1

(VI.8.2)

(t = 1, 2,... , N)

3=1

and differentiate both sides with reaspect to z. Then, (VI.8.3)

9j(x) = 0

Eat, (x)9f (x) + 2t,

(t = 1, 2,... , N)

}=1

and, hence, N

Eata(x)9j'(x) + 27t 11 -

1=1

N 1

i=1 7e+7)

g, (x)

=0

(t = 1,2,... ,N).

8. CONSTRUCTION OF A POTENTIAL FOR A GIVEN DATA

179

From (VI.8.2), it follows that N N E ajj(x){g,(x) + 2gegj(x)} - 237tE

g, (x) = 0

j=1 ge +T7j

j=1

(e = 1, 2, ... , N) or

N

N

N

3=1

j=1

j=1

g, Eatj(x){gf(x) + 2gtgj(x)} - 2>9,(x)+ 2E 711+17., 92(x) = 0

(e = 1, 2, ... , N).

Differentiating again, we obtain N

2o[s

E ata (x){gj"(x) + 2r7eg, (x) } + 2ge

(9e(x) + 2gtgt(x)) - u(x)

,=1

N

+ 2E 3=1

gj

ge+g,

9'(x) = 0

(e = 1,2,... N)

or

N

N

Eat,j(x){g,"(x) + 217eg,(x)) - Eaea(x)u(x)9J(x) ,=1

f=1 N

e2nas

aea(x)2%g,'(x) + 2171

+ ,=1

C27jt

ct

)

=

0

(e = 1,2,N). ... ,9e(x)

Thus, we derive N

F'ae,,(x){9,(x) + 217,g,(x)

- u(x)g,(x)}

j=1

N 2171 E atj (x)9, (x) + 217t ct 91(x) ,=1

=0

(t = 1, 2, ... , N),

and (VI.8.3) implies that N

F'at,1(x){9;'(x) + 2s7,g,,(x) - u(x)g,(x)} = 0 ,=1

Therefore, (VI.8.1) follows.

(e= 1,2,... ,N).

VI. BOUNDARY-VALUE PROBLEMS

180

Observation VI-8-2. If we further define f+ by (VI.7.1), then £f+ = (2f+. Proof.

(e-"=f+)11 +

-i

u(e_" f+)

( ,=1

+ 2i(g - ttgJ

g1'

2

N

+i

J=1

-i N

-

g;,

9J

+ 2rligl' - ugJ = 0.

E j=1

(+ tnJ

Observation VI-8-3. Since N

f+(x,it7t) = e-fez 1 J=1

x

g., (x) 711+17.;

en," _ -gt(x) E ct

L2(-oo,+oo),

N numbers -1l,2 (j = 1, 2,... , N) are eigenvalues. N

Observation VI-8-4. Set a

(cf. (VI.6.9) with r

0) and

J=1

N f-(x,() = a(()f+(x, -() =

a(S)e-Cz

1 + iE

g3 (x)

J=1(-ig,

(cf. Observation VI-7-1). Then, £f_ _ (2f_ . Furthermore, since

E gJ (-oo) = 1

(e = 1, 2, ... , N)

J=1171+17J

(cf. (VI.7.2)), we obtain

a(() = 1 - i1 ____

(VI.8.4)

)

j=1(+nJ

In fact, this follows from the fact that both sides of (VI.8.4) are rational functions in ( with the same zeros, the same poles, and the same limits as ( , oo. Observation VI-8-5. The functions f±(.x, () are the Jost solutions of Ly = (2y. Proof.

Note first that lim f+(x. ()e-'t= = 1 (cf. (VI.7.1)). Also, we have

s+oo

lim

gg(-o) = a(()

f+(x,()e-`
(cf. (VI-7.1) and (VI.8.4)).

i=1

This implies that

lim f+(x,-()e"x = a(()a(-() = 1. lim f-(x,()e'S= = a(() x---oo t--oo

O

9. DIFF. EQS. SATISFIED BY REFLECTIONLESS POTENTIALS

Observation VI-8-6. Note that f_ (x, ) = a(() f+(x, -{) =

181

f+(x, ) for ( =

0. Hence, r(C) = 0. Thus, we conclude that u(x) is

C real. This means that reflectionless.

Remark VI-8-7. For the general r((), the potential u(x) can be constructed by solving the integral equation of Gel'fand-Levitan: +00 +

K(x, r) + F(x + r) +

F(x + r + s)K(x, s)ds = 0,

0

where

F(x) = 1R J

+

2cL` cle-2n,=

oo

J=1

Find K(x, r) by this equation. Then, the potential is given by u(x)

8x

(x,0).

If we set r({) = 0 in this integral equation, we can derive (VI.7.2) and (VI.7.3). Details are left to the reader as exercises (cf. (Ge1LJ ).

V1-9. Differential equations satisfied by reflectionless potentials Suppose that u(x) is a reflectionless potential whose associated scattering data are given by 0, (r11, r12, ... flN ), (cl, c2,... , cN) },where 0 < ril < ]2 < < rily. It was proven that one of the Jost solutions is given by N

f+(T,() = e`t=

1 - j_1i (+

Furthermore,

9j (T)

N f+(x, -() = e`= 1 -

g1 (r)

-( + trj7

is also a solution of

dX2

- (u(x) - A)y = 0, where J = (2.

Therefore, N

(VI.9.1)

P(x, A) = 11 ! 1+11- f+(T,C)f+(x, -() A 3=1

)

satisfies the differential equation 3

(E)

2 dz3

+ 2(u(x) - A) dP +

ddx

)P = 0.

VI. BOUNDARY-VALUE PROBLEMS

182

Let

+oo

P = E an

(S)

(PO 3

0)

n=0

be a formal solution in powers of A-1 of differential equation (E). Then,

0 = P [-- + 2(u(x) - A)P + u'(x)P P2

,

+ (4)2 + (u(x)

- A)P2 J

and, hence,

A(P2)' = I - PZ it + (4)2 + u(x)P2 }, . Therefore, (i) the coefficients pn can be determined successively by

po = ca, 2C Pn+l =

where co is a constant, n

n

n

t=o

t=o

c=o

uE pip.-t 2 E PtFri-1 + 4E Ptpn-t +

nPtPn+1-t r

1=1

(n = 0,1,2,...), (ii) the coefficients pn are polynomials in u and its derivatives with constant coefficients,

(iii) the formal power series (S) is uniquely determined by the condition P = 1 for u = 0, +00 G"( u) ( Go = 1), then the ()lv i f we denote the unique q P of (iii) by G u, A ) = An Y

(

n=0

general formal solution of (S) is given by P(u, A) = P(0, A)G(u, A).

Example VI-9-1. A straight forward calculation yields

Gl =

u Z'

G2 =

3u2 - u" 8

G3 =

10u3 - lOuu" - 5(u')2 + u(4) 32

Observe that P(x, A) of (VIA 1) is a polynomial in the potential u(x) satisfies a differential equation (VI.9.2)

of degree N. Therefore,

GN+1(u) + a1GN(u) + a2GN-1(u) + ... + QNG1(u) = 0

for some suitable constants a,, a2, ... , 0N. More precisely speaking, it is shown above that P(u, A) = P(0, A)G(u, A). Compute the coefficients of A-(N+1) on both sides of this identity. In fact, N

N

u(x) = 2

g}(x)

4 1`

e2,7ixgj(x)2 171

C.,

i=1

1=1

183

10. PERIODIC POTENTIALS

implies that u = 0 if and only if gl = 0 (1 = 1, 2, that P(0, A) = 11

C1 +

-

,

N). This, in turn, implies

A

and

Hence, ao = 1 and

EN

1 +

3=1

A=C2.

where

G(u, A) = f+(x, C)f+(x, N

+

2

= 1 =1

Example VI-9-2. The function u(x) = -2q2 sech2(17(x /+ p)) is a reflectionless

1c)

(cf. Example VIpotential corresponding to the data {r7, c}, where p = 2 In 7-5). In this case, G2(u)+172G1(u) = 0. Also, G3(u)+(n1+7)2 2(u)+rI14Gi(u) _ 0 in the case when N = 2 and {r(C) = 0, (i71, n2), (cl, c2)} are the scattering data. In particular, G3 (U) + 13G2(u) +36G1(u) = 0 if i1 = 2 and '12 = 3.

Remark VI-9-3. The materials of §§VI-5-VI-9 are also found in (TDJ.

VI-10. Periodic potentials In this section, we consider the differential equation

2 + (A - u(x))y = 0

(VI.10.1)

under the assumption that (I) u(x) is continuous for -co < x < +oo, (II) u(x) is periodic of period 1, i.e., u(x + e) = u(x) for -oo < x < +oo. The period a is a positive number and A is a real parameter. Denote by 01(x, A) and 02(x, A) the two linearly independent solutions of (VI.10.1) such that (VI.10.2)

1(o

01(0,A) = 1,

A) = 0,

02 (0, A) = 0,

01(e, A)

02(t, A)

d2(0,A) = 1.

Set .D(A)

=

O1 (t"\)

(e A)

The two eigenvalues Z1(A) and Z2 (A) of the matrix periodic system (VI.10.3)

d dx

are the multipliers of the

[Y2J = [u(--O-A 0, [y21

VI. BOUNDARY-VALUE PROBLEMS

184

(cf. Definition IV-4-5).

It is easy to show that det [1(A)] = 1 for -oo < A <

+oo. Hence, the two multipliers Z1(A) and Z2(A) are determined by the equation Z2 - f (,\)Z + 1 = 0, where (VI.10.4)

2(t,A)

f(A) =

Note that f (A) is continuous for -oo < A < oo (cf. Theorem 11-1-2). We derive first the following conclusion.

Lemma VI-10-1. The two multipliers Z1(A) and Z2(A) of system (VI.10.3) are

2l

f(,\) +

f (A)2 - 4, , f (A)2 - 4,

2 l f (A) -

,

where f (A) is given by M. 10.4). Therefore, (i) if I f (A) I > 2, two multipliers are real and distinct, i.e.. ZI(A) > 1, -1
0 < Z2(A) < 1

Z2(A) < -1

if f(A) > 2, if f(A) < -2,

(ii) if I f (A) I < 2, two multipliers are complex and distinct, and I Zt (A)

I

=1

and

I Z2(A) 1

= 1,

(iii) if f (A) = 2, then Z1(A) = Z2(A) = 1.

(iv) if f(A) _ -2, then Z1(A) = Z2(A) = -1. Observation VI-10-2. If f (A) = 2, let

ICCl2

be an eigenvector of 4;(A) associated J

with the eigenvalue 1. This means that cl and c2 are two real numbers not both zero and that Cl d ' (t, A) + c2 d-i (t, A) = C2 -

c141(t, A) + C202(1, A) = c1,

Set 6(x,,\) = cl o1(x, A) + c202(x, A). Then, d(x, A) is a nontrivial solution of (VI.10.1) such that .0(t, A) = 0(0,A)

and

LO (t, A) =

(0, A).

This implies that O(x, A) is a nontrivial periodic solution of (VI.10.1) of period t.

Observation VI-10-3. If f (A) = -2, it can be shown that equation (VI.10.1) has a nontrivial solution O(x, A) such that

0(t, A) = -0(0,A)

and

(t, A) = -

(O, A)

10. PERIODIC POTENTIALS

185

This implies that ¢(x, A) satisfies the condition

¢(x + e, A) = -&, A)

for - oo < x < +oo. Observation VI-10-4. Cconsider the eigenvalue problem (VI.10.5)

2 + (A - tL(x) y = 0,

y(o) = 0,

y(e) = 0.

Applying Theorem VI-3-11 (with -A instead of A) to problem (VI.10.5), it can be shown that (VI.10.5) has infinitely many eigenvalues (VI.10.6) µI < i2 < which are determined by the equation (VI.10.7) t2(£, A) = 0. The eigenfunction 02 (x, has exactly n -1 zeros on the open interval 0 < x < e.

<µ<

Since

.2 (0, A) = 1 > 0, it follows that dO2

dx

<0 (e'µ") I > 0

for n is odd, for n is even.

A) = 1.

Furthermore, condition (VI.10.7) implies that det t I (A)] = Thus, f (A) = 01(e, A) +

1

follows from (VI.10.4). Therefore,

Z V. A)

(e, A) >0 and f(A) :5-2 if

f(A) > 2 if

(e, A) < 0,

if n is even,

>2

f (µ") { < -2

if n is odd

(cf. (VI.10.7)). Here, use was made of the fact that jaj+ 1a1,

ifa76 0. Observation VI-10-5. If (VI.10.8) A < min{u(x) : -oo < x < +oo}, then f (A) > 2. In fact, dX21 (x, A) = (u(x) - A)01 (x, A) > 0 as long as 01(x, A) > 0 dol and, hence, (x, A) > 0. Thus, we obtain s1(£, A) > 1. Similarly, 2 (P, A) > 1 if (VI.10.8) is satisfied. In this way, a rough picture of the graph of the function f (A) is obtained (cf. Figure 1). At

A2

44

3

T

3

W Ul

A3=

l4 -F

I

f=2

lA

U2

U3=N3U4

u5

FIGURE 1.

Actually, we can prove the following theorem.

U6

f= -2

VI. BOUNDARY-VALUE PROBLEMS

186

Theorem VI-10-6. Assuming that u(x) is continuous and periodic of period t > 0 on the entire real line R, consider three boundary-value problems:

2 + (A - u(x)) y = 0,

(A)

($)2 + (A - u(x)) y = 0, d2 + (A - u(x)) y = 0,

(C)

y(0) = 0,

W) = y(0),

y(t) = -y(0),

y(t) = 0, y

dx (t) _ dY(t)

(0),

_ L(0).

Then, each of these three problems has infinitely many eigenvalues:

µI

<µ2
A0<\1
(VI.10.9)

V1
and 7n(x) the eigenfuncrespectively. Furthermore, if we denote by an(x), tions associated with the eigenvalues µn, An, and vn, respectively, then

(1) A0 < v1, (2) v2n-1 :5 92n-I

n = 1,2,...,

1'2n < A2n-1 !5 µ2n < A2n < V2n+1 < µ2n+1 < i/2(n+l) for

(3) 02n_1(x) and /32n(x) are linearly independent if A2,,-1 = A2n, (4) 72n-1(x) and 72n (x) are linearly independent if v2n-1 = v2n, (5) 3D(x) does not have any zero on the interval 0 < x < t,

(6) 02._1(x) and /32,,(x) have exactly 2n zeros on the interval 0< x < t, (7) 72n-1(x) and 72n (x) have exactly 2n - 1 zeros on the interval 0 < x < t. Proof.

We prove this theorem in five steps. Note first that (VI.10.9) is obtained from Figure 1. Step 1. If ¢(x, A) is a solution of differential equation (VI.10.1), then a solution of the initial-value problem

(x, A) is

(VI.10.10) d2w dx2

i

w(0, A) =

+ (A - u(x)) w + O(x, A) = 0,

i

(0,,\) =

(0, )L),

TA_

( dx

(cf. §lI-2). Therefore, using the variation of parameters method, we obtain (VI.10.11)

_

-1(x, A) _ d z (x,,\)

(x, A) fT 01(t, A)02(t, A)dt

= 01(x"\) 102(t, A)2dt

¢2(x, A)

f01(t"\)2dt'

- 02(X,,\) f -01(t, A) 02(t, A)dt

10. PERIODIC POTENTIALS

187

and, hence, 1(x, A) f ; m2(t, A)2dt

da (dd-x (x, A)) =

Therefore,

df dA

Q(Y1,Y2)

-

(_, A)

la

mi (t, A)yi2(t, A)dt.

rr Q(01 (t, A), ¢2(t, A))dt follows from (VI. 10.4), where

=

0

= -02(e,A)Y2 + dTl(e,A)Y2 + [1(t1A)

-

(e,A)Jr1Y2.

Step 2. The discriminant of the quadratic form Q is f(,\)2 - 4. In fact, 2

1(t"\)

A)]

xd

+4

l (e, A)o2(t, A)

2

+ 401(e, A)

2 (e, A)]

dO2

(e, A) - 4 = f(,\)2 - 4.

Note that

41(e, )

(e, A) -

A) = det [ I (A)] = 1.

' (e,

Thus, Q(01 (X, A), ¢2(x, A)) does not change sign for 0 < x < e if If (A)f < 2. It follows that (I)

df(A)

If(A)f < 2.

if

0

dA

Step 3. Also, it can be proven that df(A) d,\

<0

if

A < pl and f(,\)2 = 4,

(-1)1 d('\) < 0 if pl < A < pi+1 and f(A)2 = 4. To prove this, notice that Q is a perfect square if f(,\)2 = 4 and that 02(1,.\) # 0 if A 0 pj for every j. Hence, -A instead of A), we obtain

dA\) ¢2(e,

02 (1, A) > 0

A) > 0 Thus, (1) is verified.

A) < 0. Also, using Lemma VI-3-11 (with

if if

A < p1,

p., < A < pj+1.

VI. BOUNDARY-VALUE PROBLEMS

188

Step 4. Let us prove that dzf

(a)

J< 0 if

f (a) = 2 and

>0

f (A) _ -2 and

dal

if

d(\)

= 0,

df(A) = 0. d,\

First observe that Q is a perfect square if (f(A)l = 2. Hence,

=±

71K (A)

f

t

0

[c101(t, A) + c202(t, \)12 dt

= 0,

where cl and c2 are some real numbers. Therefore, cl = 0 and c2 = 0, since 01 and 02 are linearly independent. This means that

02(f,,\)=0,

(IV)

' (e, A) = 0,

ax

UX

1, we conclude that

Since det

12

- I2

d (a) = 0,

if

f (a) = 2 and

if

f (A) = -2 and -(A) = 0,

fi(A) =

(V)

f

d

where 12 is the 2 x 2 identity matrix. Furthermore, z

e

-

2 (a) = Jo

(t, A)0 (t, A)2 + da

d

+

Also, if f (A) = 2 and

d

a)2

) (e, A) } m (t, A)m2(t, A)1 dt.

(A) = 0, it follows from (VI.10.11) and (IV) that

d z

d

( d ') (e,

I (e, a) =

{e, ) = j42(s,A)2ds, t 0i (s, A)2 ds, 0

f 0t 0r (s, A)02(s, A)ds.

da ( Hence,

d

(A) _ - f

tf

t(0f(t, A)02{s, A) - 0e(s, A)02(t,A)j2dtds < 0

10. PERIODIC POTENTIALS

if f (A) = 2 and

d2f 2

I

(A) =

189

(A) = 0. Similarly,

ft f t

[Ol (t, A)02(s, A) - 41(s, A)4 2(t, A)J2dtds > 0

if f (A) = -2 and df (A) = 0. Note that, if A2,,-1 = A2,,, it follows from (VI. 10.2), (IV), and (V) that 01(x, A) and 02(x, A) are two linearly independent solutions for problem (B). Similarly, if w2r_1 = v2,,, it follows from (VI.10.2), (IV), and (V) that 01(x, A) and 62(x, A) are two linearly independent solutions for problem (C). Thus, (3) and (4) are verified.

Step 5. The functions

(respectively -yn(x)) have an even (respectively odd) number of zeros on the interval 0 < x < 1, since /3, (0) = Qn(1) and 7n(0) Yn(1) As can be seen in Figure 1,

112n-i < A2n-1

A2n < 12n+l-

Therefore, by virtue of Theorems VI-3-11 and VI-1-1, we conclude that fit, 1 and per have more than 2n - 1, and less than 2n + 2, zeros on the interval 0 < x < 1. Hence, they have exactly 2n zeros there. Similarly, since (VII)

/12n-2 < V2n-1

V2n < 112n,

and -t2n have exactly 2n - I zeros on the interval 0 < x < e. The function ,8a does not have any zero on the interval 0 < x < t since Ao < 111. Thus, (5), (6), and (7) are verified. Finally, (2) follows from (VI.10.9), (VI), and (VII).

Definition VI-10-7. (I) The set {A f(A)2 < 4} is called the stability region of the differential :

equation (VI.10.1). (11) The set {A : f(A)2 > 4} is called the instability region of the differential equation (6210.1).

Example VI-10-8. A periodic function u(x) is called a finite-zone potential if the function f (A)' - 4 of the differential equation

d2y dx2

+ (A - u(x))y = 0 has a finite number of simple zeros (i.e., all other zeros are double). For example, consider the case when u(x) = a, where a is a constant. The differential equation becomes d? 2

dx+ (A - a)y = 0 and, hence, 1

if A > a, if A =a,

cosh( a --Ax)

if A < a

cos(v1'"T-_ax)

01 (X, A) =

VI. BOUNDARY-VALUE PROBLEMS

190 and

( sin(v1_T_-ax)

A-a 02(x, A) =

x

sinh(x)

if A>a, if A =a,

if A < a.

Therefore,

12 cos(\/-), --at)

AA) =

A) +

(e, A) _

if

A >a,

2

if A=a,

2 cosh( a --At)

if A < a.

Thus, we conclude that in this case, f (A)2 -4 has only one simple zero a (cf. Figure 2).

FicURE 2.

The materials in this section are also found in [CL, Chapter 8j.

EXERCISES VI

VI-1. Assume that u(x) is a real-valued continuous function on the interval 20 = {x : 0 < x < +oo} such that u(x) > rno for x >_ xo for some positive numbers "to and x0. Show that (1) every nontrivial solution of the differential equation (E)

day

&2

- u(x)y = 0

has at most a finite number of zeros on Z0, (2) the differential equation (E) has a nontrivial solution 1)(x) such that

hm q(x) = 0.

Hint. (1) Note that if y(xo) > 0, then y"(xo) > 0. Hence, y(x) > 0 for x > x0 if 1/(xo) > 0.

(2) It is sufficient to find a solution 0(x) such that 0(x) > 0 and 0'(x) < 0 for

x>xo.

EXERCISES VI

191

VI-2. For the eigenvalue-problem (EP)

L2 + u(x)y = Ay,

Y(()) = y(1),

y(0) = y'(1),

where u(x) is real-valued and continuous on the interval 0 < x < 1, (1) construct Green's function, (2) show that (EP) is self-adjoint, (3) show that (EP) has infinitely many eigenvalues.

VI-3. Let A, > A2 >

- > An >

-

be eigenvalues of the boundary-value problem

d2y

+ u(x)y = Ay,

y(a) = 0,

y '(b) = 0,

where u(x) is real-valued and continuous on the interval a < x < b. Show that there exists a positive number K such that 2

n + n2

ir

K

b-a

n

for n = 1, 2, 3, ...

.

VI-4. Assuming that u(x) is real-valued and continuous on the interval 0 < x < +oo and that lim u(x) = +oo, consider the eigenvalue-problem

r+oo

dx2 - u(x)y = Ay,

y(0) cos a - y'(0) sin a = 0,

z

lim

y(x) = 0.

where a is a non-negative constant. Show that (a) there exist infinitely many real eigenvalues AI > A2 > . . . such that lim An = n-.ioo -00, (b) eigenfunctions corresponding to the eigenvalue An have exactly n -1 zeros on the interval 0 < x < +oo. Hint. Let A1(b) > A2(b) > . . . be the eigenvalues of

dx2 - u(x)y = Ay,

y(0) cos a - y'(0) sin a = 0,

y(b) = 0,

where b > 0. Define Am by lim A,(b). See JCL, Problem 1 on p. 2541. 6

+oo

VI-5. Show that if a function O(x) is real-valued, twice continuously differen-

tiable, and ¢"(x) + e-'O(x) = 0 on the interval 10 = {x : 0 < x < +oo} and

if J

J0

O(x)2dx < +oo, then O(x) is identically equal to zero on I

.

VI. BOUNDARY-VALUE PROBLEMS

192

VI-6. Using the notations and definitions of §VI-4, show that +oo

(f,C(f)) = j>n(f,17n)2 n=1 b

=

j

{u(x)f(x)- P(x)f'(x)2}dx + P(b)f(b)f'(b) - P(a)f(a)f'(a),

if f E V(a, b).

VI-7. Assume that u(x) is real-valued and continuous and u(x) < 0 on the interval I(a, b), where a < b. Denote by yh(x, A) the unique solution of the initial-value problem 2 + u(x)y = Ay, y(a) = 0, y(a) = 1, where A is a comlex parameter. Show that (i) O(b, A) is an entire function of A, (ii) ¢(b, A) 54 0 if A is a positive real number, (iii) ¢(b, A) has infinitely many zeros A,, such that 0 > A0 > Al > A2 > and n

- - \ b r- al/ +oo n2

llm

A"

)

\

-

2 .

VI-8. Find the unique solution O(x) of the differential equation

) = y such that

Ix (i)\\\

is analytic at .x = 0 and 0(0) = 1. Also, show that

m(x)

is an entire function of x, (ii) ¢(x) A 0 if x is a positive real number,

(iii) ¢(x) has infinitely many zeros an such that 0 > \o > a1 > A2 > lim 1n = -oc, n-+00 (iv)

nx)dx = 0 if n 96 m.

J0

VI-9. Show that the Legendre polynomials 2"n1

Pn(x)

d" !dx"((x2 - 1)'J

=

(n = 0,1,2,...)

satisfy the following conditions: (i) deg P,, (x) = n (n = 0,1,2,...),

(ii)

J

1

Pn(x)Pm(x)dx = 0 if n 0 m,

(iii) j P n (x)2dx =

1

n (iv)

J

(n = 0,1, 2, ... ),

z

xkPn(x)dx = 0 for k = 0,... , n - 1,

(v) Pn(x) (n > 1) has n simple zeros in the interval Ixl < 1,

.

and

EXERCISES VI

193

(vi) if f (x) is real-valued and continuous on the interval jxj < 1, then

lim f

N-+oo where (f, Pn) =

f

2

N

1

1

f (x) - E (n+ Z) (f, PP)Pn(x))

dx = 0,

n=O

1

f (x)PP(x)dx,

(vii) the series +00 F, (n + 21 (f, PP)Pn(x) converges to f uniformly on the interval n=0

jxj < 1 if f, f', and f" are continuous on the interval jxj < 1. Hint. See Exercise V-13. Also, note that if f (x) is continuous on the interval jxj < 1, then f (x) can be approximated on this interval uniformly by a polynomial in x. To prove (vii), construct the Green function G(x,t) for the boundary-value problem

((I -x2)d/ +aoy=f(W), y(x) is bounded in the neighborhood of x = ±1, 1

1

where ao is not a non-negative integer. Show that f f G(x, )2dx< < +oo. J!

Then, we can use a method similar to that of §VI-4.

1J

1

VI-10. Assume that (1) p(x) and p'(x) are continuous on an interval Zo(a,b) _ {x : a < x < b}, (2) p(x) > 0 on Zo, and (3) u(x, A) is a real-valued and continuous function of (x, A) on the region Zo x ll = {(x, A) : x E Zo, A E It} such that

lim u(x, A) = boo uniformly for x E Zo. Assume also that u(x, A) is strictly A-too decreasing in A E R for each fixed x on I. Denote by O(x, A) the unique solution of the intial-value problem (P(x)L ) + u(x,.\)y = 0

y(a) = 0,

y(a) = 1.

Show that there exists a sequence {pn : n = 0, 1, 2, ... } of real numbers such that (i) pn < 1An-1 (n = 1, 2, ... ), and lim pn = -oo, n .+oo (ii) 4(b, pn) = 0 (n = 0,1, 2, ... ), (iii) ¢(x, .\) 0 on a < x < b for A > po, and q(x, A) (n > 1) has n simple zeros ,-

on a
strictly decreases from +oo to -oo as A decreases from pn_1

(iv) to An-

VI-11. Assume that p(x) and u(x) are real-valued and continuous on the interval

1(0,1) = {x : 0 <- x < 1} and that p(x) is also continuous on 7(0,1), p(x) > 0

for 0 < x < 1, p(0) = 0, and p'(0) < 0. Show that the differential equation

VI. BOUNDARY-VALUE PROBLEMS

194

+u(x)y = 0 has a fundamental set {0, ii} of solutions on the interval (px ) d dx)

dx

0 < x < 1 such that (i) slim 4(x) = 1 and Zli o p(x)O'(x) = 0, (ii)

sl

o

(i(z) -

11

F

0 and =li m- p(x)ii (x) = 1. )

VI-12. Set for

0
for

2
1.

Denote by 01(x, A) and 42(x, A) the two unique solutions of the differential equation y

+ (A - u(x))y = 0 satisfying the initial conditions 4i(0, A) = 1, X1(0, A) = 0, (0, A) = 1, where A is a real parameter. Sketch the graph of

42(0, A) = 0, and

the function f (A) = 4(1, A) +

(1, A).

VI-13. For the scattering data {r({) = 0, (1,2,3), (1,1,1)}, find the potential u(x)

and the Jost solutions ft(x,(). VI-14. Calculate the scattering data for each of u(x + 1) and u(-x), assuming that {r({), (271, ... ,T)N), (cl, ... , cN)} are the scattering data for u(x) and that u(x) satisfies a condition Eu(x)I < Ae-k1=I for some positive numbers A and k.

Hint. Let f f(x, () be the Jost solutions for u(x). Two quantities a(() and are given by

a(S) =

2C

I

f+(x,C) f_(x,<)

and

1

(X,0 f-' (x:0

I

respectively.

The Jost solutions f o r u(x + 1 ) are e ' ft (x + 1, (). Therefore, the scattering data for u(x + 1) are

(171,... ,nN), (e-2"'c1,... ,e-2"r'cN)}. The Jost solutions for u(-x) are ff(-x, (). Hence, the scattering data for u(-x) are

b(l)' a(C)

(171,...,17N)I

(

1

cia<(iv71)2

,...,-

1

l}

CNa<(LT,N)21 J

Note that

f-(x,iil3) = i a<(iul,)f+(x,iu,)

EXERCISES VI

195

VI-15. Let u(x) be real-valued, continuous, and periodic of period t > 0. Also, for every real , let O(x, {, A) be the solution of the differential equation 2 + (A - u(x))y = 0

(Eq)

satisfying the initial conditions 0({, {,A) = 0 and {G'(£, t, A) = 1. Let

A = Al (0 < A2W < µ3(f) < ... be all roots of ii({ + t, , A) = 0 with respect to A. Show that

( + [', , A) = Q

Am(t) - A +00

2

m=1

e

Hint. Step 1. Let us construct O(x, , A) for negative A. To do this, change (Eq) to the integral equation sinh(p(x

- )) + 1 r p p f

sinh(µ(x -

where it = v > 0. Since e-;`(=-() sinh(µ(x - )) is bounded as p - +oo, it can be shown that e-v(x-E)+G(x,

, A) =

e-al=-E)

such{µ x- f ))

+ O(

on the interval 0 < x - < t as A -e -oo. Thus, we derive lim

t'(4+f,.,A) = 1. (sinh(e))

Step 2. There exists an entire function R(A) of A such that

sin(Pf) R( A ) =

J

a

sinh(CX)

for

A > 0,

for

A <0.

This function R(A) has the following factorization:

-_,

R(A) = t m=1

\1

A

1

VI. BOUNDARY-VALUE PROBLEMS

196

where Cm =

mr

t

Step 3. If the function V)((+1, 1, A) is entire in .\ and ''(+t, 1,,\) = O(exp(1 JaJ)) as A -+ oo, we can write 1'(( + 1, t,,\) in the following form:

[i(A]

lp(f + t, (, a) = c

cn,J is bounded

where c is independent of A. Here, we used the fact that +oc (cf. Theorem VI-3-14). as m

Step 4. Note that

- 'c' + [0-W _- a

_((+t,t,X) (1)

R(A)

(

)H

cam, - A

]

Note also that

c -a

cm -A

This implies the uniform convergence of the infinite product on the right-hand side Z'((+ 1, t, _ = 1 = c

of (1) for -oo < A < 0. Thus, lim

a-.-OO

l

R(a)

Remark. For expressions of analytic functions in infinite products, see, for example, (Pa, pp. 490-5041.

VI-16. The functions

are defined in §VI-9. Calculate G4(u) and G5(u).

VI-17. Use the same notations as in Theorem VI-10-6. Show that if J

I

u(t)dt = 0,

0 ft

then Ao < 0. Also show that if / u(t)dt = 0 and

= 0, then u(x) = 0 identically

Jo

on-oc
Hint. Since j3o(z) # 0 on -oc < x < +oo, set w(z) = !Lx). Then, w'(x) + w(x)Z + A - u(x) = 0 (cf. 1A'IW, Theorem 4.4, p. 621).

V1-18. Set F(() = log f (() for `a( > 0, ( # 0, where (i) f (() is continuous for QJ( > 0, ( 0, (ii) f (() is analytic for `£( > 0, (iii) ((f (() - 1) is bounded for !a( > 0,

(iv) f (() # 0 for !'( > 0, (0 0. Denote by SR the semicircle {(: 1(I = R,0 < arg(< r) which is oriented counterclockwise, where R is a positive number. Show that 1

1

(a) (b)

2ri IR

F(() = tai

F(z) dz =

2Ri

j

F'(()

f F(() 0

d{ + 7 f- g

where fl() is the complex conjugate of F().

(`1( > 0,1(1 < R),

(`'(<0,1(1
for a(> 0,

CHAPTER VII

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF LINEAR SYSTEMS

In this chapter, we explain the behavior of solutions of a homogeneous linear +oo in the case when the coefficient matrix A(t) = A(t)y as t system has a limit A0 as t -+ +oo. The purpose is to show how much information we can glean from the limit matrix Ao. We are interested in the exponential growth of solutions and the asymptotic behavior of solutions. In order to measure the exponential growth of a function, we use Liapounoff's type numbers which was originally introduced by A. Liapounoff in [Lia]. Liapounoff's type numbers are explained in §VII-1. Also, we explain Liapounoff's type numbers of solutions of a homogeneous linear system in §VII-2. (Liapounoff's type numbers are also found lim A(t) = Ao, we in L. Cesari [Ce, pp. 50-551.) In §VII-3, assuming that t-too calculate Liapotimoff's type numbers of solutions in terms of the eigenvalues of A0 (cf. [Huk3j, [Hart, Chapter X1, and [Si9]). In §VII-4. we explain how to derive the asymptotic behavior of solutions by diagonalizing the given system. A theorem of M. Hukuhara and N. Nagumo gives the original motivation (cf. Theorem VII-4-1). The main result is Theorem of N. Levinson (cf. Theorem VII-4-2). (Generalizations and refinements of Theorem of Levinson are found, for example, in [Bell], [Bel2], [CK], [Cop1]. [Dev], [DK], [El], [E2], [Gi], [GHS], [HarL1], [HarL2], [HarL3], [HW1],

[HW2], [HX1], [HX2], and [HX3].) In §VII-5, the Theorem of Levinson is applied

to a system whose matrix has a limit as t -+ +oc and its derivative is absolutely integrable on the interval 0 _< t < oo. The topics of §§VII-4 and VII-5 are also found in [CL, §8 of Chapter 3, pp. 91-97]. In §VII-6, we explain how we can reduce

some problems such as the differential equation dt2 + {X + h(t) sin(at)}rj = 0 to

the Theorem of Levinson, even if the derivative of h(t)sin(at) is small but not absolutely integrable on the interval 0 < t < oo. The main idea is to apply the Floquet theorem (Theorem IV-4-1) to z + {1 + e sin(ot)}rt = 0 to eliminate the periodic parts of coefficients so that we can use the Theorem of Levinson (cf. [HaS3]; see also [HarLl], [HarL2], [HarL3]).

VII-1. Liapounoff's type numbers In order to measure the exponential growth of a function, let us introduce Liapounof''s type numbers.

Definition VII-1-1. Let f (t) be a C"-valued function whose entries are continuous on an interval Z = {t : to < t < +oo}. Let us denote by A the set of all real 197

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

198

numbers a such that exp[-at] f (t) is bounded on the interval Z. Set

1 (f) =

(VII.1.1)

+oo inf{o : a E A}

ifA =0,

-00

ifA=R.

i f A j4R and A 3 40 ,

The quantity A (f) is called Liapounoff's type number off at t = +oo.

Note that if a E A and a <,6, then a E A.

Example VII-1-2. J+ (0) = -00,

A (exp[-t2j) _ -00, A (tm) = 0 (for all constants m),

{ A (exp[ort]) = a,

A (exp[t2j) _ +00.

Here, it was assumed implicitly that to > 0 if m < 0. The following lemma can be proved easily.

Lemma VII-1-3. Let A (f) be Liapounoff's type number of a C' -valued function

f (t) whose entries are continuous on an interval Z = It: to < t < +oo}. Then,

(i) exp I - (. (f) + e) t] f(t) is bounded on I if e > 0, and unbounded if e < 0, whenever A (f) # -oo, (ii) A

(tii) A

fi

< max{A (f,)

f,

= A(ft),

:

j = 1,2,... m

tfA(f1) > A(fi)forj=2....,m,

1
(iv) f1, f2, ... , f, are linearly independent on the interval if A if 1) , ... , A (fm) are mutually distinct, m

(V) A(f1f2...fm) <- FA(f,) in the case when f1i... , fm are C-valued junctions.

(vi) A (P(t)f) = A (f), if the entries of an n x n matrix P(t) and the entries of its inverse P(t)-1 are bounded on the interval Z. The following lemma characterizes Liapounoff's type number.

Lemma VII-1-4. If A(f) is Liapounoff 's type number of a function f(t) at t = +oo, then a(f)

=

log If (s) I lim sup s t-.+oo lt
2. LIAPOUNOFF'S TYPE NUMBERS OF A LINEAR SYSTEM

199

Proof.

Since there exists a positive constant K such that I f (s)I < Keia(f)+`i' for e > 0 and large values of s, it follows that

logIf(s)I


for

t < s.

Hence, log If(3)1

< a(f) sup s It<5<+m Also, for a fixed positive number a and any positive integer in, there exists a large lim t-+m

value of sm such that

If (sm)1 and

lim sm = +oo. Hence,

log m

+

sm

e < log If (sm)1 Therefore, we obtain Sm

lim sup t-+oo
log If(s) s

I

> (f )

Thus, Lemma VII-1-4 is proved.

VII-2. Liapounoff's type numbers of a homogeneous linear system In this section, we explain Liapounoff's type numbers of solutions of a homogeneous linear system (VII.2.1)

dy

dt

= A(t)9

under the assumption that the entries of the n x n matrix A(t) are continuous and

bounded on an interval T = It : to < t < +oc}. Let us start with the following fundamental result.

Theorem VII-2-1. If y" = fi(t) is a nontrivial solution of system (VII. 2. 1) and if f A(t)e < K on the interval I = It : to < t < +oo} for some non-negative number K, then (VII.2.2) Proof.

Let the n x n invertible matrix 4P(t) be the unique solution of the initial-value dY problem 7 = A(t)Y, Y(to) = In, where In is the n x n identity matrix. Then,

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

200

dZ 4i(t)-1 is the unique solution of the initial-value problem dt 1, we obtain I (cf. Lemma IV-2-4). Since

= -ZA(t), Z(to) _

1 + KJ I4(t)Idt, to

I

+K/

1

for

t ,I

t r- T.

dt

Therefore,

I < exp[K(t - to)] for t E T

)4i(t)I < exp[K(t - to)) and

(cf. Lemma 1-1-5). Observe that fi(t) = 4D(t)j(to) and b(to) the definition of the norm of a matrix, it follows that

From

exp[-K(t - to)] < 1 (t)I < 1 (to)j exp[K(t - to)] for t E Z. Therefore, (VII.2.2) follows, since I¢(to)I 0 0.

Set A = {a

Q is a nontrivial solution of (VII.2.1)}. Then, (iv) of Lemma VII-1-3 implies that A is a nonempty subset of R which contains at most n numbers. Let Al > A2 > ... > Am (1 < m < n) be all of the distinct numbers in A. Set (VI1.2.3)

Vj = {j :

is a solution of (VII.2.1) such that A (p) < A-}

for j = 1,2.... Tn. Then, (ii) and (vi) of Lemma VII-1-3 imply that V, is a vector space over C. Set (VII.2.4) 1i = dim Vj (j = 1,2,... ,m). The following lemma states that there exists a particular basis for each space V, which consists of y, solutions whose type numbers are equal to A,. Lemma VII-2-2. For system (VI1.2.1) and y, given by (VII. 2-4), it holds that

(_) yi = n,

(ii) ym < -Ym-1 < ... < y, (ii:) for each j, there exists a basis for V, that consists of y, linearly independent solutions d,,- (v = 1,2,... , y,) of (VIL2.1) such that A (p,,°) = A3. Proof It is easy to derive (i) and (ii) from the definition of V, and the definition of the numbers at, ... , am. To prove (iii), let y,,,, (v = 1, 2, ... . y,) be a basis for Vj. Assume that A, for v = 1,...t, ( for

v=f+1...... ).

It follows from (ii) of Lemma VII-1-3 and definitions of V, and A, that t > 1. Set for

+

for

v = 1.... , t, v = t + l.... , y2.

Then, , , (v = 1, ... , y,) satisfy all the requirements of (iii).

2. LIAPOUNOFF'S TYPE NUMBERS OF A LINEAR SYSTEM

201

Observation VII-2-3. The maximum number of linearly independent solutions of (VII.2.1) having Liapounoff's type number A,, is rye.

Definition VII-2-4. The numbers Al, A2, ... , Am are called Liapounoff's type numbers of system (VIL2.1) at t = +oc. For every j = 1, 2,..., m, the multiplicity of Liapounoff's type number A, is defined by

hj _

(VII.2.5)

for j=1,2,... in- 1, for j = in.

7m

The structure of solutions of (VI1.2.1) according with their type numbers is given in the following result.

Theorem VII-2-5. Let {(1, ¢2r ... , iin} be a fundamental set of n linearly independent solutions of system (VII.2.1). Then, the following four conditions are mutually equivalent:

(1) for every j, the total number of those 4t such that A (y5t) = A, is h, (cf. (VII. 2.5)), (2) for every j, the subset {¢t : A (Qt) < A, } is a basis for V,,

ctt

(3) A

> A) if the constants ct are not all zero,

= max {A (dt)

(4) A

:

ct

0} for every nontrwzal linear combi-

n

nation

FC, it of

e

Wn}

t=1

Proof.

Assume that (1) is satisfied. Then, the total number of those ¢t such that m

A (3t) < A, is Fht = ryr = dims Vj. Hence, (2) is also satisfied. Conversely, t=j assume that (2) is satisfied. Then, the total number of those t such that A (fit) < Aj is equal to dime V) = -,. Hence, (1) is also satisfied (cf. (VII.2.5)).

Assume that (2) is satisfied. Then, if A follows that

ctc

ct4

E V,, and hence

clot =

tt

A, for some ct, it

202

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

for some c e. Since {¢j : t = 1,2,... , n} is linearly independent, all constants cc

must be zero. Thus, (3) is satisfied.

n

Assume that (3) is satisfied. Write a linear combination I:ct4t in the form t=1

m

cj¢t. Then, A

CWI

ci4r 96 0. Hence, (4)

= A), if

j=1 a(ft)=A, a(mt)=a1 follows from (iii) of Lemma VII-1-3.

a(Ot}=a,

Finally, assume that (4) is satisfied. Then, every solution d of (VII.2.1) with A (j) < A, must be a linear combination of the subset {¢t : A ((rt) S Aj }. Hence,

(2) is satisfied. 0 Definition VII-2-6. A fundamental set (01,02, ... , On } of n linearly independent solutions of system (VIL2.1) is said to be normal if one of four conditions (1) - (4) of Theorem VII-2-5 is satisfied.

Since V. C Vm-1 c . C V2 C V1, it is easy to construct a fundamental set of (VII.2.1) that satisfies condition (1) of Theorem VII-2-5. Thus, we obtain the following theorem.

Theorem VII-2-7. If the entries of the matrix A(t) are continuous and bounded on an interval Z = It : to < t < +oo}, system (VIL2.1) has a normal fundamental set of n linearly independent solutions on the interval Z. = Ay" with a constant matrix A, Lia Example VII-2-8. For a system pounoff's type numbers A1, A2, ... , Am at t = +oo and their respective multiplicities

h1, h2, ... , hm are determined in the following way. Let IL I , p2, ... , Pk be the distinct eigenvalues of A and M1, m2, ... , Mk be their

respective multiplicities. Set v, = R(pj) (j = 1, 2,... , k). Let Al > A2 > be the distinct real numbers in the set {v1i v2, ... , vk }. Set h, _

> Am

mr for

t=at dy

j = 1, 2, ... , m. Then, A1, A2, .... Am are Liapounoff's type numbers of = Ay" at t = +oc and h1, h2, ... , hm are their respective multiplicities. The prom of this result is left to the reader as an exercise.

Example VII-2-9. For a system (VII.2.6)

dy dt = A(t)y

with a matrix A(t) whose entries are continuous and periodic of a positive period c..' on the entire real line R, Liapounoff's type numbers )k1, A2, ... , Am at t = +oo and their respective multiplicities h1, h2, ... , hm are determined in the following way:

There exists an n x n matrix P(t) such that (i) the entries of P(t) are continuous and periodic of period w, (ii) P(t) is invertible for all t E R.

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

203

(iii) the transformation y = P(t)z changes system (VII.2.6) to (VII.2.7)

with a constant matrix B. Therefore, (vi) of Lemma VII-1-3 implies that systems (VII.2.6) and (VII.2.7) have the same Liapounoff's type numbers at t = +oo with the same respective multiplicities. Liapounoff's type numbers of system (VII.2.7) can be determined by using Example VII-2-8. Note that if pl, p2,. .. , p" are the multipliers of system (VII.2.6),

then p,

(j = 1, 2,... , n) are the characteristic exponents of system

log[p,]

(VII.2.6), i.e., the eigenvalues of B, if we choose log[p3] in a suitable way. Hence, R(µ1)

(j = 1, 2,... , n).

log[[p2I]

Those numbers are independent of the choice of branches of log[p,].

Example VHI-2-10. For the system

the fundamental set {et

=I J

eu [0] I is normal, but the fundamental set I

[b],

{[] [J} -e is not normal. 2t

,

VII-3. Calculation of Liapunoff's type numbers of solutions The main concern of this section is to show that Liapounoff's type numbers of

a system J = B(t)y" at t = +oc and their respective multiplicities are exactly the = Ay" with a constant matrix A if iimoB(t) = A. same as those of the system t-+0 dt It is known that any constant matrix A is similar to a block-diagonal form

diag[pllm, + W.421,., + Rf2i ... , µkl,,, + Mk],

where µl, ... , Pk are distinct eigenvalues of A whose respective multiplicities are is the mj x mj identity matrix, and M1 is an mj x m, nilpotent m1, ... , mk, matrix (cf. (IV.1.10)). Consider a system of the form (VII.3.1)

10 = A2y1 + EBjt(t)Ut

(j = 1,2,...

t=1

where y2 E C'-,, A. is an n, x n) constant matrix, and B2t(t) is an n, x nt matrix whose entries are continuous on the interval Za = {t : 0 < t < +oo}, under the following assumption.

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

204

Assumption 1. (i) For each j, the matrix A. has the form

Aj = )1In, + E. + N.,

(VII-3.2)

(j = 1,2,... ,m),

where Al is a real number, In, is the nJ x nl identity matrix, E. is an n. x nj constant diagonal matrix whose entries on the main diagonal are all purely imaginary, and Nj is an nj x n, nilpotent matrix, (ii) Am < )1m-1 < ... < 1\2 < Al,

(iii) N,E, = E?NJ (j = 1,2,... ,m), (iv)

t

limp B,t(t) = 0 (j,t = 1,2,... ,m).

The following result is a basic block-diagonalization theorem.

Theorem VII-3-1. Under Assumption 1, there exist a non-negative number and a linear transformation

y , = z ' , + I: T , (t)zt

(VII-3.3)

tj

to

(j = 1,2,... ,m)

with ni x nt matrices T,t(t) such that

(1) for every pair (j, t) such that j

f, the derivative dt tt (t) exists and the

entries of Tat and dj tt are continuous on the intervalI = (t : to < t < +oo)

lim T;t(t) = O (j # e), t-+00 (3) transformation (VII.3.3) changes system (VI1.3.1) to (2)

(VII.3.4) t Lzj

(j=1,2,...,m).

A,+B,,(t)+EBjh(t)Th,(t)I

Proof.

We prove this theorem in eight steps.

Step 1. Differentiating both sides of (VII.3.3), we obtain dzl dt

+

dTit5t

T,tdz"t dt

t#?

df

m

(VII.3.5)

= A) % + E Tj tat t0r

Aj 4' Bjj +

BjhThj

h*

Bjt t + t=1#t i, +

t

Tt V

AjTjt + Bjt +

BjhTht zt,

hit

205

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

where j = 1, 2,... , m. Note that r

in

e=1

n+

2, = F v=1

Vol

1

BjnTnv zv h#v

Rewrite (VII.3.5) in the form

{

-[At+Fj }+ET,t{ Lz' -[At+Fe]zt} t#i

JJJJJJ

_

[AJTJt + Bit +

t#,

1

l

BlhThe - Tlt (At + Ft)

dtt

h#t

Ztf

where j = 1, 2,... , m, and

F t = B» +

B)hTh)

M)

h#j

Define the Tit by the following system of differential equations: (VII.3.6)

T

dtt

(.l r e)

h#t

Then,

-[A,+F,]x",}+Tj dt -[At+Ft]zt}=Q (j=1,2,...,m).

{

This implies that we can derive (VII.3.4) on the interval I if the Tit satisfy (VII.3.6) and condition (2) of Theorem V1l-3-1, and to is sufficiently large.

Step 2. Let us find a solution T of system (VII.3.6) that satisfies condition (2) of Theorem VII-3-1. To do this, change (VII.3.6) to a system of nonlinear integral equations (VII.3.7)

T,t(t) =

J

exp[(t - s)A,]U,t(s,T(s))exp[-(t - s)A1]ds,

for j 0 e, where the initial points rat are to be specified and

U,t(t,T) = B11(t) + E Bjh(t)Tht - Tjt Btt(t) + E Bth(t)Tht h#t

h961

Using conditions given in Assumption 1, rewrite (V11.3.7) in the form

rt

(VII.3.7')

Tit(t) = J exp 2(a1 -

1

t)(t - s)]

s,s,T(s})ds,

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

206

t and Wjt(r,s,T)

where j (VII.3.8)

[(A,

= exp

- At)r] exp[rEj j exp[rNj[UJt(s, T) exp[-rEtj exp[-rN,].

On the right-hand side of (VII.3.7'), choose the initial points rat in the following way

ra t

(VII.3.9)

1t..,.

( +00

j`

to

if Aj > At

(i.e., j < t),

if A < A

(ie j > t)

Step 3. Let us prove the following lemma. Lemma VII-3-2. Let a be a positive number and let f (t, s) be continuous in (t, s) on the region Z x Z, where Z = {t : to < t < +oo}. Then, (VII.3.10)

t III

ft"

exp[-a(t - s)] f (t, s)dsl <

- max l

f (t, s)to

-s
y J

for each fixed t E T. Also, if I f (t, s)' is bounded for to < t < s < +oo, then

rt

(VII.3.11) I

J+00

exp[a(t - s)j f (t, s)dsl

<_

1 sup { I f (t, $)I : to < t < s < +oo} a

for each fixed t E Z. Proof In fact, estimates (VII.3.10) and (VII.3.11) follow respectively from

rt

Jto

exp[-a(t - s)]ds = a {1 - exp[-a(t - to)]} <

t

+oo

exp[a(t - s)]ds =

- 1a

O

.

Step 4. Let us estimate s, T) fort < s < +oo if A j > at, and for to < s < t if A_, < at. If A, > At, r < 0, and s > t, it follows that Irlm'+mi-2)

i1',t(r,s,T)I < Ko(l +

exp 12(A, - A,) 7-1 IVit(s,T)I

for some positive number 1Co. It can be shown

`easily that there exist a positive number X and a non-negative valued function p(t) such that

I + Ir1m'*m`_2 exp [(A1 - At)r < X, ]

B, (s) lim

Q(t)

for

for r < 0,

t < s < +oo and all pairs (p, q),

p(t) = 0.

t-+00 Hence, if Aj > At, r < 0, and s > t, we obtain IW,t(r,a,T)I < /C2,3(t) {1 + ITI2} for some positive number 1C2, where ITI = maxIT,t[. Similarly, the estimate t#j I Wjt(r, s, T) I < 1C219(to) { 1 + ITI2) is obtained by choosing a positive number 1C2

sufficiently large, if A., < at, r > 0, and to < s < t.

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

207

Step 5. In a manner similar to Step 4, we can derive the following estimates: IWj1(T,s,T)

- Wj1(r,s,T)I

K3$(t) {1 +ITI + ITI} IT - TI if A, > A1, T < 0, s > t,

1 K3$(to){1+IT{+ITI}IT-T{

if A0, to
for some positive number K. Step 6. Define successive approximations on the interval I as follows:

TO ye(t) = 0,

TP1t;)t(t) =

f

[ exp

[(A,

i..

- A,)(t - s)j 9,1(t - s, s, Tp(s))ds,

where j 54 a and p = 1, 2, .... Suppose that

on the interval I = ft: to < t < +oo}

I Tp;j,(t) j < C

(VU.3.12)

for some positive number C. Then, Lemma VII-3-2 and Step 4 imply that K2i3(t){1 + C2}

if A) > A,,

K21300){I +C2}

if A, < '\1

2

I TP .iat(t)

I

<_

A'2- Al Al

Aj

on the interval Z. Hence, choosing to so large that 2

IA, - A,I

K2/3(t){1 + C2} < C

on Z,

we obtain ITp+1,je(t)j < C

on the interval I

from (VII.3.12).

Step 7. Suppose that (VII.3.12) holds for p = 0, 1, 2,.... Then, lTp+i.t(t)

where ITpI =

i

- Tp,1(t)I S 2

aup ITP(s)

- TP-1(s)I <_

WP

C on Z,

ITpael if to is so large that K30(t){1 + 2C} <

on Z. Since

P

Tp.e(t) _

{TQae(t) - TQ_lae(t)}, it can be shown easily that

lim Tpe(t) _

Tje(t) exist for all (j, 1) such that j # f uniformly on the interval T. The limit Tj1(t) satisfies integral equation (VII.3.7).

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

208

Step 8. In this final step, we prove that the bounded solution Tjt of (VII.3.7) satisfies condition (2) of Theorem VII-3-1. It easily follows from Steps 6 and 4 that 1

lim o

t

exp

[(A3 - at)(t - s)1 ir4jt(t - s, s,T(s))ds = 0

1+ 00

> X1.

if

For (j, f) such that ), < at, write the right-hand side of (VII.3.7') in the following form:

s)] IV1t(t - s, s,T(s))ds

t

Jto exp 12 (A Q

11

fexp

L(A

- a)(t - s)]

- s, s,T(s))ds

or

r

+

Jo

t

exp 12 (At

l

1

- at)(t - s)] bt%tt(t - s, s, T(s) )ds

for any a such that to < or < t. Observe that

fexp

- At)(t - s)]

Wjt(t - s,s,T(s))dso

< exp L2 (.1j - '\t)(t - a)]

12 (a) - at)(a - s)J 14 t(t -- s, s, T(s))ds

I 1'. a

<

2KO(t°) At - A)

exp `

{l + C2} exp 1(aI - AE)(t - a) J , 1:5

if JT(t)[ < C on I (cf. (VII.3.10)). Observe also that

f

exp

L2(,\j

j - s,s,T(s))dsl s)]ji)t(t - at)(t -

A i3(a) {1 +C2}

if T(t)j < C on I (cf. (VII.3.10)). Hence,

jexp 1(A3 - a)(t - s)] t < if

2K1\1 211-

- s, s, T(s))dso

C2} {(to)exP {(A, - At)(t -

)1

+

J

T(t) I < C on T. Therefore, letting a and t tend to +oo, we obtain rt

lim

t-+3o

I exp 12 (A, - At)(t - s)] Yt,jt(t - s, s, T(s))ds = 0. to

L

This completes the proof of Theorem VII-3-1. In order to find Liapounoff's type numbers of system (VII.3.4), it is necessary to establish the following lemma.

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

209

Lemma VII-3-3. If N is a nilpotent matrix, then for any positive number e such that 0 < e < 1, there exists an invertible matrix P(e) such that

P(e)-INP(e)I < Xe for some positive number IC independent of e. Proof.

Assume without any loss of generality that N is in an upper-triangular form with

zeros on the main diagonal (cf. Lemma IV-1-8). Set A(e) = diag[1, e, ... , e"-1]. Then, A(e)-'NA(e) = [ ek-Jz,k ] (a shearing transformation). Hence, as 0 < e < 1 and j < k, we obtain IA(e)-1NA(e)I < e(N(. 0 Let us find Liapounoff's type numbers of system (VII.3.4), i.e., dzj

a. = IA + B,,(t) +

z,

(7 = 1,2,... ,m).

h*j

Theorem VII-3-4. A system of the form dz

dt

= (AIn+E+N+B(t)Ja

has only one Liapounoff's type number ,\ at t = +oo if (i) A is a real number, (ii) In is the n x n identity matrix, E is an n x n constant diagonal matrix whose entries on the main diagonal are all purely imaginary, N is an n x n constant nilpotent matrix, and EN = NE, (iii) the entries of the n x n matrix B(t) are continuous on the interval I = {t : to < t < +oo}, (iv)

t

lim B(t) = 0. +00

Proof.

Set i = ea`e`EU. Then, (VII.3.13)

dt = [N + C(t)) u,

C(t) = e-`EB(t)etE

Let p be any Liapounoff's type number of (VII.3.13) at t = +oo. Then, (µ( < sup(N + C(t)( (cf. Theorem VII-2-1). This is true even if to -+ +oo. Since tez

lim C(t) = 0, we conclude that 1µ1 < (N(. Using Lemma VII-3-3, it can be t+00

shown that any Liapounoff's type number p of (VII.3.13) at t = +oo is zero. This, in turn, completes the proof of Theorem VII-3-4. 0

Remark VII-3-5. For a nilpotent matrix N, the two matrices N and eN are similar to each other for any nonzero number e. To prove this, use the Jordan canonical form of N. From the argument given above, we conclude that Liapounoff's type numbers of system (VII.3.4) are at, ... ,.1,,, and their respective multiplicities are n1, ... , r4,,. Thus, the following theorem was proved.

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

210

Theorem VII-3-6. Liapounof's type numbers of a system

= B(t)y at t =

+oo and their respective multiplicities are exactly the same as those of the system dy

dt= Ay with a constant matrix A if the entries of the matrix B(t) are continuous

on the interval Io = {t : 0 < t < +oo} and lim B(t) = A. t-.+ao

= B(t)y', we obtain

Applying Lemma VII-1-4 to the solutions of the system the following corollary of Theorem VII-3-6.

Corollary VII-3-7. If an n x n matrix B(t) satisfies the conditions (i) the entries of B(t) are continuous on the interval To = {t : 0 < t < +oo}, (ii) lim B(t) = A exists, t-.

dy

= B(t)y, lim then, for every nontrivial solution ¢(t) of the system t--+oo dt p exists and p is the real part of an eigenvalue of the matrix A.

log 1-0(t)I

t

Remark VII-3-8. The conclusion of Theorem VII-3-1 still holds even if condition (iv) of Assumption 1 of this section is replaced by IB)k(t)j < f (t) (j, k = 1, 2,... , m) on the interval Zo, where f (t) satisfies rP

(VII.3.14)

sup(1 +p- t)-1 J f(s)ds - 0 p>t

as t

+oo.

t

To see this, let us assume that a positive-valued function f (t) satisfies condition tP

(VII.3.14). Set h(t) = sup ((1 + p - t)-1 ( f (s)ds) for a fixed positive number

ft

p>t \\

t. Also, set E(t) = suph(r). Then, lim E(t) = 0, and E(t2) < E(t1) if t1 < t2. t--+oo

r> t

Now, it is sufficient to prove the following lemma.

Lemma VII-3-9. For any positive numbers t, to, and c, it holds that (a)

rt e-c(t-a) f(s)ds < (I + -) E(to), to

f

+

ec(t-s) f(s)ds <

lim t-.+oo

1+

f t e-`(t-`) f (s)ds = 0. o

cE(t).

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

211

Proof of (a).

Set p(t) = f e-`(t-°)f t (s)ds. Then, 0'(t) = -cb(t) + f (t) and, hence,

I.

0(r) - 0(o) = -c /

T

th(s)ds + / T f (s)ds

0

0

for to < a < r. Suppose that there exists a positive number 6 such that

b(r) =

C1

+

and

- +6

0(a) = C- + 6 E(to),

E(to),

(!± b E(to)

0(s) >

for

a < s < r.

Then,

E(to) + c l! + 6 E(to)(r - a) < E(a)(1 + (r - o)).

\

This is a contradiction. Proof of (b).

Set ,L(t) =

f

et-f (s)ds for 0

T for a fixed T > 0. Then, "(t) _

t

cO(t) - f (t) and, hence, w(r) - ?P(t) = cJ ;(s)ds r

r

J f (s)ds for t < r < T.

Suppose that there exists a positive number 6 such that t

y(t) = (1 + 1 +6) E(t),

>
1l'(s) >

(!+o)E(t)

for

t < s < r.

Then,

E(t) + c (1+ 6) E(t)(r - t) < E(t)(1 + (r - t)). This is a contradiction. This, in turn, proves that T

f (s)ds <

J

C1

+ c j E(t)

for

T > t.

+430

Since lim E(t) = 0, the integral

t-+o

j+00

e

ft

+

e'('-")f (s)ds exists and

ec(t-e)f(s)ds <

\

1+ 1 f E(t). c

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

212

The proof of (c) is left to the reader as an exercise. 0 For the argument given above, see [Harl).

VII-4. A diagonalization theorem Liapounoff's type number of a solution is useful information since it provides some idea about the behavior of the solution as t --. +oo. However, it is not quite enough when we look for a more specific information. For example, Liapounoff's type number of the second-order differential equation x

dt2 + {1 + R(t)}r, = 0 is 0 and its multiplicity is 2 if the function R(t) is continuous on the interval 0 < t < lim R(t) = 0 (cf. Theorem VII-3-6). However, this does not imply the +oc and t-+"0 boundedness of solutions as t --+ oc. 0. Perron [Per3[ constructed a function R(t) satisfying the two conditions given above in such a way that solutions of (VII.4.1) are not bounded as t +oc. Looking for better information concerning equation (VII.4.1), M. Hukuhara and M. Nagumo [HNI[ proved the following theorem.

Theorem VII-4-1. Every solution of differentud equation (VII.4.1) is bounded as t

+x, if

1

+C0

+

jR(t)ldt < +oc.

0

Proof.

First, fix to > 0 in such a way that a

r+x

jR(t)ldt < 1. Write a solution 0(t)

SU

of (VII.4.1) in the form

b(t) = q(to) cost - to) + 0'(to) sin(t -

to) - J t R(s)¢(s) sin(t - s)ds. to

Choose a positive number K so that 4(to)j + kd'(to)j < K and choose another KQ positive number At so that M > 1 > K. Then,

K + At J t JR(s)jds < At

for

to < t < tt

to

if jb(t)j < At for to < t < t1. Hence, 14(t)I < M for to < t < +oc. 0 In this section, we explain the behavior of solutions of a system of linear differential equations under a condition similar to the Hukuhara-Nagumo condition. Precisely speaking, we consider a system of the form (VII.9.2)

d

under the following assumptions.

[A(t) + R(t))1i

4. A DIAGONALIZATION THEOREM

213

Assumption 2. Assume that A(t) is an n x n diagonal matrix A(t) = diag(A1(t), A2(t), ... , An (t)], R(t) is an n x n matrix whose entries are continuous on the interval Zo = {t : 0 < t < +00)1 and (VII.4.3)

(VII.4.4)

J

IR(t)Idt < +oo.

Set

A,k(t) = al(t) -)tk(t) and D-,k(t) = R(AJk(t))

(VII.4.5)

(j,k = 1,2,... ,n).

Concerning the functions A. (t) (j = 1, 2,... , n), the following is the main assumption. Assumption 3. The functions Al (t), A2(t), ... , A,, (t) are continuous on the inter-

val 4. Furthermore, for each fixed j, the set of all positive integers not greater than n is the union of two disjoint subsets P.t and P32, where

(i) kEP) I if

lim I Djk(r)dr = -oo and

t-+oo 0

Dik(r)dr < K

for 0 < s < t

Li

for some positive number K,

(ii) k E Pj2 if

`t f/a

Dik(r)dr < K

for

s > t > 0

for some positive number K.

Remark VII-4-2. Assume that the functions A1(t), ... , An(t) are continuous on the interval Z o and that lim A, (t) = pj (j = 1, 2, ... , n) exist. Then, the functions t +oo A1(t),... ,An(t) satisfy Assumption 3 if the real parts of pl,... p, are mutually distinct. The proof of this fact is left to the reader as an exercise. The main concern in this section is to prove the following theorem due to N. Levinson.

Theorem VII-4-3 ([Levil]). Under Assumptions 2 and 3, there exists an n x n matrix Q(t) such that

(1) the derivative dQ(t) exists and the entries of Q and Q are continuous on d

the interval Z0,

(2) t1 1 Q(t) (3) the transformation (VII.4.6)

y' = V. + Q(t)] E

changes system (VII.4.2) to (VII.4.7)

d"

dt= A(t)i

on the interval Z0, where I is the n x n identity matrix. Prool We prove this theorem in six steps.

214

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

Step 1. Differentiating both sides of (VII.4.6), derive

a + (In + Qj dr = [A(t) + R(t)J (I + Q) z. Then, it follows from (VII.4.7) that Q should satisfy the linear differential equation

dt _ (A(t) + R(t)1[I,, + Q] - (In + Q] A(t)

(VII.4.8)

or, equivalently, (VII.4.9)

dQ = A(t)Q - Q A(t) + R(t) (In + Qf .

The general solution Q(t) of (VII.4.9) can be written in the form (VII.4.9')

Q(t)

='Nt)C41(t)-1 +

f0(t)O(s)-1R(s)`I,(s),I,(t)-1ds,

where C is an arbitrary constant matrix, 44(t) is an n x n fundamental matrix of

Al

dt = [A(t) + R(t)]4>, and 'F(t) is an n x n fundamental matrix of = (cf. Exercise IV-13). Thus, the general solution Q(t) of (VII.4.9) exists and satisfies condition (1) of Theorem VII-4-3 on 10. Therefore, the proof of Theorem VII-4-3 will be completed if we prove the existence of a solution of (V1I.4.9) which satisfies

condition (2) of Theorem VII-4-3 on an interval I = {t

:

to < t < +oo} for a

large to.

Step 2. Now, construct Q(t) by using equation (VII.4.9) and condition (2) of Theorem VII-4-3. To do this, let 4?(t. s) be the unique solution of the initial-value problem

dY = A(t)Y, dt

Y(s) = In.

Then, (VII.4.9) is equivalent to the following linear integral equation: (VII.4.10)

Q(t) = J

1

(t, s)R(s) f 1 + Q(s)145(t, s)'1ds,

where 4?(t, s) = diag(Fl (t, s), F2(t,1s), ... , Fn(t, s)],

= exp [ I A,(r)d; {

F, (t, s)

(j = 1,2,. .. ,n).

JJJ

Step 3. Letting q,k(t) and rik(t) be the entries on the j-th row and the k-th column of Q(t) and R(t), respectively, write the integral equation (VII.4.10) in the form

(VII.4.10')

ggk(t) =

J

exp

[ft

Aik(r)drr)k(s) + F 11

A=1

ds,

215

4. A DIAGONALIZATION THEOREM

where j, k = 1, 2,... , n and the .1jk(t) are defined by (VII.4.5). Note that lexp

t

t ( J(Ajk(r)dr} I

I

= exp I J Djk(T)dr] <

eK

a

ll

if 0< s< t

t< s< +oo

and

k E Pj1

for

k E Pj2,

for

where j, k = 1, 2,... , n and the Djk(t) are defined by (VII.4.5). The initial points rjk are chosen as follows: if

to

rjk = { +00

k E P1I,

if k E Pj 2

for some to > 0.

Step 4. Define successive approximations by

gojk(t) = 0,

gPJk(t) = j

t

l

exp LJ

rk(S) +

t Ajk(r)dTJ

h=1

rja(S)gp-I;hk(s) ds,

where p = 1, 2, .... Then, we obtain I gpuk(t) I < eK (1 + nC]J

r(s)ds on the

to

interval I={t:to
on the interval 2,

Igp_I,jk(t)I < C

where r(t) = max(j,k) Irjk(t)I. Using assumption (VII.4.4), choose to so large that

e'11 + nCJ

r+oo

Jto

r(s)ds < C.

Then, from (VII.4.11), it follows that Igpjk(t)I < C on the interval Z.

Step 5. Similarly, if (VII.4.11) holds for p = 1, 2, ... , we obtain P

Igp+Iuk(t) - gpjk(t)l < (2} C on the interval I = {t : to < t < +oo} if to is chosen sufficiently large. Hence, plim gpUk(t) = gjk(t) exists uniformly on the interval Z, and the limit satisfies integral equation (VII.4.10').

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

216

Step 6. Let us prove that the bounded solution qjk(t) satisfies condition (2) of Theorem VII-4-3. If k E Pj2, then Tjk = +oo. Hence, lim q)k(t)

1im f exp f f

AJk(T)d-rl

[rik(s)+rih(s)chk(s)](is = 0.

LJ

h=1

In the case when k E PJ1, note that t

[f>tik(r)dr]

q)k(t) = f exp

[r)k(s)+r)h(s)hk(s)} ds h=1

and IrJ// exp IJ \Jk(r)dr] L s

ft v

=f

[ft

o

r)h(S)hk(S)

rik(S) +

A=1

l1

ll

.\)k(T)dr] [r)k(S) +> rJh(S)hk(s)J ds

exp

h=1

\.k(T)dr] {r)k(S) +

+ s

n h-1

for any a such that to
exp [ft ak(T)dr] fIrk(s)+

rJh(s)hk(s)

1

1

= eXp Lf t Dk(r)dTJ (

f

exP

dsl rJh(S)ghk(S)ds

[fC.Jk(r)dr] [rJk(s) +

r)h(s)hk(s)l1 ds h=1

t

JI

+oo

< exp J f D)k(r)dr] eK (1 + nCj ft) l o if (q)k(t)(< C on Z. Observe also that

f

t

exp I

f

r(s)ds

t

AJk(T)dr] [r)k(s) +

rJh(s)hk(s) ds h=t

I

< eK(1 + tnC}

f

+co

r(s)ds

0

fix a positive number a so large

if lq)k(t)E < C on I. For a given positive number

ft-

that eK (1 + nCj

r(s)ds <

Since

= -oo

if

k E P)

exp f DJk(r)dr] eK [1 + nC]

f+

r(s)ds <

lim f t D)k(r)dT

t-»+0 for a fixed a, we obtain

t

L

o

1

e

to

for large t. Therfore, for any positive number e, there exists e for t > t(e). This complete the proof of Theorem IV-4-3.

such that Iq,k(t)( <

4. A DIAGONALIZATION THEOREM

217

Remark VII-4-4. The n x n matrix W(t, s) = [In + Q(t)j$(t, s) is the unique solution of the initial-value problem dY = [A(t) + R(t)]Y,

Y(s) = In + Q(s),

dt

where di(t, s) is the diagonal matrix defined in Step 2 of the proof of Theorem VII-4-3 (cf. (VII.4.10)). Since det[%P(t,s)] i4 0 for large t, the matrix WY(t,s) is invertible for all (t, s) in Yo x Yo (cf. Exercises IV-8). This, in turn, implies that the matrix In + Q(t) is invertible for all t in Io.

Remark VII-4-5. Theorem VII-4-3 has been shown to be the basis for many results concerning asymptotic integration (cf. [El, E2] and [HarLl, HarL2, HarL3]).

Remark VII-4-6. Using the results of globally analytic simplifications of matrix functions in [GH[, H. Gingold, et al. [GHS] shows some results similar to Theorem VII-4-3 with Q(t) analytic on the entire interval Io under suitable conditions.

Remark VII-4-7. Instead of (VII.4.4), [HX1] and [HX2] assume only the integrability at t = oc for above (or below) the diagonal entries of R(t) and obtain the results similar to Theorem VII-4-3. More results were obtained in [HX3] applying a result of [Si9j. The following example illustrates applications of Theorem VII-4-3.

Example VII-4-8. Let us look at a second-order linear differential equation

d 22 + p(t)rt = 0.

(VII.4.12)

If we set

1=

A(t) =

,

do

1l 0J -P(t) 0

dt

equation (VII.4.12) becomes the system dy

(VII.4.13)

dt

= A(t)y

The two eigenvalues and corresponding eigenvectors of the matrix A(t) are '\t(t)

911 _ [i p(t)1/2]

= i p(t)112,

1

pct) _ [-i pt)1/2J

A2(t) = -t p(t)'/

Set Po(t) _ [i P(t)'/2 dPo(t) dt

-

ip'(t) 2p(t)'/2

-ip(t)'/2J.

[0 1

01 1

Then,

POW-1 = '

-i 2p(t) '/2

[ip(t)'/2

11 1

jP(t)1/2

,

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

218 and

Po(t)-'A(t)Po(t) _

P(to

[i

_i

1/2

pt)ti21 0

The transformation y = Po (t) i changes system (VII.4.13) to

-

P0(t)-1 {A(t)Po(t)

dP t)j z.

-

Using the computations given above, we can write this system in the form (VII.4.14)

.ii =

{t P(t) 1/ 2 [0

- 49t)

01]

] [ 11

11

Suppose that (1) a function p(t) is continuous on the interval Zo = {t : 0 < t < +oo}, (2) there exists a positive number c such that p(t) c > 0 on the interval Zo, (3) the derivative p'(t) of p(t) is absolutely integrable on Io,

Then, Theorem VII-4-3 applies to system (VII.4.14) and yields the following theorem (cf. IHN2]).

Theorem VII-4-9. If a function p(t) satisfies the conditions (1), (2), and (3) given above, every solution of equation (IV.4.12) and its derivative are bounded on the interval Zo.

The proof of this theorem is left to the reader as an exercise. Note that condition (3) implies the boundedness of p(t) on the interval 1 . If p(t) is not absolutely integrable, set

z = [I2 + q(t)E]t,

(VII.4.15)

where I2 is the 2 x 2 identity matrix, q(t) is a unknown complex-valued function, and E is a constant 2 x 2 unknown matrix. Then, the transformation (VII.4.15) changes system (VI I.4.14) to (VII.4.16)

dt = [Iz + q(t)E]-1 { [ip(t)h12A0

- 4p(t) A11 [12 + q(t)E] J

dtt) E} u,

where

Ao =

Al =

Anticipating that (i) q(t)I and Ip'(t)I are of the same size, (ii) Jq'(t) I and Ip"(t)I are of the same size, choose q(t) and E so that two off-diagonal entries of the matrix on the right-hand side of (VII.4.16) become as small as Ip'(t)I2 + [p"(t)I. In fact, choosing

p'(t) , q(t) = 8-ip(t)312

E = [01 -11 0J

219

S. SYSTEMS WITH ALMOST CONSTANT COEFFICIENTS

we obtain

(12 + q(t)E]'

= 1 + I9(t)2 (12

-

q(t)E(

and q(t)E]-1 { [1(t)1/2

(12 +

Ao - 4p(t) A11 (12 + q(t)E1 jip(t)1/2A0 _ p'(t) EAo - p'(t)

1

1 + q(t) 2

8p(t)

4p(t)

dt) E}

Al

+g(t}AoE+E(p,p',p") - q(t) dtt)12} I 1 -r q(t) 2

I

lp(t)1/2Ao- p,(t)12+E(p,p,p')-q(t)dq(t)12 dt

4p(t)

where

EAo = -AoE =

E2 = -12,

0 11

,

Il 0

and E(p, p', p") is the sum of a finite number of terms of the form

a (P (t)

12

+ $ p"(t)

p(t)h/2

with some rational numbers or and /3 and some positive integers h. Applying Theorem VII-4-3 to system (VII.4.16), we can prove the following theorem.

Theorem VII-410. Suppose that (1) a function p(t) is continuous on the interval Zo = {t : 0 < t < +oo}, (2) there exists a positive number c such that p(t) > c > 0 on the interval ID, (3) +00

(Ip(t)12 + Ip'(t)I } dt < +oo. 0

(4)

limop'(t) = 0.

Then, every solution of equation (VII.4.12) is bounded on the interval 7o.

The proof of this theorem is left to the reader. Condition (3) of Theorem VII-4-10 does not imply the boundedness of p(t) on the interval Io. For example, Theorem VII-4-10 applies to equation (VII.4.12) with p(t) = log(2 + t).

VII-5. Systems with asymptotically constant coefficients In this section, we apply Theorem VII-4-3 to a system of the form (VII-5.1)

d9

where A is a constant n x n matrix and V(t) is an n x n matrix whose entries and their derivatives are continuous in t on the interval Zo = It : 0 < t < +oo} under the following assumption.

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

220

Assumption 4. The matrix A has n mutually distinct eigenvalues p 1, u2, ... , An, and the matrix V(t) satisfies the conditions

lim V(t) = 0 t

+00

and

+ Cc

+

[V'(t)[dt < +oo.

o

Let A1(t), 1\2 (t) .... , an(t) be the eigenvalues of the matrix A+V(t). Then, these are continuous on the interval I. Furthermore, it can be assumed that

lim ),(t) = µj t-+,

(j = 1,2,... ,n).

Choose to > 0 so that A) (t),.\2(t), ... , an(t) are mutually distinct on the interval

I={t:to
F(t, A) = det[AII - A - V(t)].

Then, F(t, \j(t)) = 0 on II for 1 = 1,2,... ,n, and

Also,

j

(t, A, (t)) + as (t, \,(t)) A (t) = 0 (j =1,2,... n) OF 8A

(t, \j (t)) # 0

on I.

(j = 1,2,... ,n),

8F (t, since III, µ2i ... , p, are mutually distinct. Observe that .X (t)

aF

A, (t ))

is

8a (t, af(t)) linear homogeneous in the entries of the matrix V'(t). In this way, we obtain the following lemma.

Lemma VII-5-1. Under Assumption 4, the derivatives of the eigenvalues of the matrix A + V (t) are absolutely integrable over the interval I = {t : to < t < +oo}, i. e.,

JA(t)dtl w

< +oo

(j = 1,2,... ,n).

An eigenvector p, (t) of the matrix A + V(t) associated with the eigenvalue as(t) can be constructed in the following manner. Observe that the characteristic polynomial of A + V(t) can be factored as

F(t, \) = (A - \1(t))(A -1\2(t)) ... (A - \.(t)) Hence, by virtue of Theorem IV-1-5 (Cayley-Hamilton), we obtain

(VII.5.2) (A+V(t)-.11(t)In)(A+V(t)-.\2(t)In)...(A+V(t)-an(t)IJ) = 0

5. SYSTEMS WITH ALMOST CONSTANT COEFFICIENTS

221

on Zo. Furthermore, if t > to,

J1 (A+V(t) - Ah(t)In) # 0, h 0j

lim fl(A + V(t) - Ah(t)In) = H(A - phln) 36

t-+oo

h#l

0,

h#j

for j = 1, 2,... , n. From (VII.5.2), it follows that

[A + V(t)] 1(A+ V(t) - Ah(t)In) = A,(t) [I(A + V(t) - Ah(t)In) h#J

h#) H(A+V(t)-Ah(t)In), we obtain

Hence, choosing a suitable column vector j,(t) of h#)

p3(t)

1-1

0 and [A+V(t)]p,(t) = \j(t)jYj(t)

on the interval I = It : to < t < +oo} if to > 0 is sufficiently large. Furthermore, lim j5, (t) = 4, t+oo

(.7 = 1,2,... n)

are eigenvectors of the matrix A associated with the eigenvalues p. (j = 1, 2,... , n), respectively. Observe that the entries of the vectors pF,(t) (j = 1,2,... , n) are polynomials in the entries of V(t) and A1(t),... ,An(t) with constant coefficients. Hence, +00

Ip"j'(t)I dt < +oc

(j = 1,2,... ,n).

Jta

Thus, we proved the following lemma.

Lemma VII-5-2. Under Assumption 4, them exists a non-negative number to such that (1) the matrix A + V (t) has n mutually distinct eigenvalues A1(t)...... n (t) on the interval I = It : to < t < +oo}, (2) the eigenvalues Al(t), ... , An(t) are continuously differentiable on I and

lim Aj(t) = p, (j = 1,2,... ,n), (3) the matrix A + V(t) has n eigenvectors p1(t),... ,#n(t) associated with the eigenvalues \I(t) .... ,An(t), respectively, such that lim 73 (t) = 4j (j = t-+oo 1, 2, ... , n) are eigenvectors of the matrix A associated with the eigenvalues

p) (j = 1,2,... ,n),

(4) the derivatives of the eigenvalues A,(t) (j = 1,2,... , n) and the derivatives o f the eigenvectors p ' , (t) (j = 1, 2, ... , n) with respect to t are absolutely integrable on the interval Z. Set

Po(t) _ [ 91 (t) #2 (t) ...

( t ) [[ ,,

A(t) = diag[A1(t), A2(t), ...

, An (t)[,

Q = [ 91 92 ... 4n [ , M = diag[pl, p2, ... , pn]-

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

222

Then,

lim Po(t) = Q,

lim A(t) = M,

t-+oo

Po(t)-'[A + V(t)]Po(t) = A(t),

f

+oo I Po(t)_ 1 dP°(t) I

dt

to

Q-'AQ = M,

dt < +oo.

Observe that the transformation y"= P°(t)z changes system (V11.5. 1) to

dt =

POW-1 [[A + V(t)]P0(t) -

dl t)1 i=

[A(t)

- Po(t)-'d t)]

Y.

Applying Theorem VII-4-3 to this system, we obtain the following theorem.

Theorem V11-5-3. Under Assumption 4, if the eigenvalues A1(t)...... n (t) of the matrix A + V (t) satisfy all requirements given in Assumption 3 (cf. § VII--4) on the interval .7 = {t : to < t < +oo}, a fundamental matrix solution of (VII.5.1) will be given by

1(t) = P°(t)[In + H(t)]exp I / ( A(s)ds], where H(t) is an n x n matrix whose entries are continuously differentiable on the

interval I andt+00 lim H(t) = O. Remark VII-5-4. 1 r: (a) As t -. +oo, the matrix exp [ - J A(s)ds] 4i(t) approaches the matrix Q, (b) we can prove a result similar to Theorem VII-5-3 even if system (VII.5.1) is replaced by

d = (A+V(t)+R(t)]y, where the matrix A + V (t) satisfies all the requirements of Assumption 4 and the entries of the matrix R(t) are absolutely integrable for t > 0, i.e., f+00 0

JR(t)jdt < +oo.

Observation VII-5-5. Let us look into the case when the matrix A has multiple eigenvalues. To do this, consider system (VII.3.1) under Assumption 1 given in §VII-3. By virtue of Theorem VII-3-1, system (VII.3.1) is changed to system (VII.3.4) by transformation (VII.3.3). Furthermore, (VII.3.4) is changed to (VII.5.3)

dt

[ N 3 + R 3 (t)) u 3

(j = 1, 2, ... m),

where

R3(t) = e-'E, B33(t)+E B3h(t)Ttj(t)

etE,

(j=1,2,...,m)

h #)

by the transformation (.) = 1, 2, ... , m). z3 = explt(A3I., + E, )]u3 Therefore, we look into the following two cases.

5. SYSTEMS WITH ALMOST CONSTANT COEFFICIENTS

223

Case VII-5-5-1. If N is an n x n nilpotent matrix such that N' = 0 and R(t) is an n x n matrix whose entries are continuous on the interval 0 < t < +oo and satisfy the condition (VII.5.4)

+

jt2(T_1)IR(t)Idt < +oo,

we can construct a fundamental matrix solution 45(t) of the system dy dt

(VII.5.5)

such that lim

t -.+00

= (N + R(t))y'

e-' N4(t) = I, , where I is the n x n identity matrix. In fact, the

transformation y" = etNii changes system (VII.5.5) to

dii

= e-tNR(t)e1NU. Since

leftNI < Kjtjr-1 for some positive number K, the integral equation X(t) = In

j

e-R(r)erNX(r)dT can be solved easily.

Case VII-5-5-2. Let N be an n x n nilpotent matrix such that Nr = 0 and let R(t) be an n x n matrix whose entries are continuous on the interval 0 < t < +oo and satisfy the condition +00

tr-1IR(t)jdt < +oo.

(VII.5.6) 10

Exercise V-4 shows that this case is different from Case VII-5-5-1. As a matter of fact, in this case, we can construct a fundamental matrix solution 'P(t) of system (VII.5.5) such that (VII.5.7)

lira t-(k-t) (41(t)

t-+00

-

etN) c" = 0',

whenever

Nkc_ = 0,

where k < r. We prove this result in three steps. Step 1. First of all, if an n x n matrix T(t) satisfies condition (VII-5.7), then We N'5 c' = 0 if 41(t)c = 0 for large t. In fact, noice first that lim e tr if f!

etNt: Also, note that lim t-+oo tktkl

NPe = 0 for p = F + 1, ... , r - 1. = 0 if a(t)e = 0 and Nkcc = 0. Therefore, N'lE = 0 if k = r. Hence, Nr-2c' = 0. In this way, we obtain c" = 0. Thus, we showed that if the matrix %P(t) satisfies the differential equation (N + R(t))* and condition (VII.5.7), then %P(t) is a fundamental matrix solution of (VII.5.5).

Step 2. To prove the existence of such a matrix W(t), we estimate integrals of ske(t-")NR(s) with respect to s, where 0 < k < r - 1. Observe that ske(t_e)N =

r1

Sk(t - S)hNh h1

h=0

Let us look at the quantity sk(t - s)h. Note that 0 5 k 5 r - 1 and 0 5 h < r -

1.

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

224

Case 1. k + h < r - 1. In this case, since sk(t - s)h = sk+h C t _ 11, define

-

s

Sk(th'

Jt

8)h N°R(s)ds = / t

8k(th! S)h NhR(s)ds.

Case 2. k + h > r. In this case, look at sk(t - 8)h = D-1)p

(hp)

Sk+pth-p

p_Q

Subcase 2(i). k + p < r - 1. In this case,

sk+pth-p = Sr-I

\s

r-1-u=k+p,u+v=h-p. Since

1 t", where

v = h - p - p = (h-p) + (k+p) - (r-1) = (k+h) - (r-1) = k - (r-1-h), we obtain 0 < v < k. Now, in this case, define h

t

Irt

Sk+pth-PNhR(S)dS

j

= J+ao p) sk+Pth-PNhR(s)ds.

Subcase 2(ii). k + p > r. In this case,

Sk+pth-p = Sr-1(tlµt",

r-l+p=k+p, v -p=h -p.

where

Since

v = h - p + p = (h-p) + (k+p) - (r-1) = (k+h) - (r-1) = k - (r-i-h), we obtain 0 < v < k. Also, note that h-p<(r-1)+(k-r)=k-1. Now, in this case, define =

; k+Pt h- PNhR{s)d s = -3 h

k

)

r

t

o

-

kh $

Pt P Nh R(s)d$, k+h-

where

From the definition given above, it follows that t

Ske(t-s)NR(S)d8 = etN / t Ske sNR(s)ds,

f',

v rt

lim I / ske(`'")NR(s)ds = O (k = 0,... , r t+00 F. q

Setting

r U(t) = f t ske(`-e)NR(s)ds,

1),

t > to.

6. AN APPLICATION OF THE FLOQUET THEOREM

225

we obtain

dU(t) = NU(t) + tkR(t). dt Step 3. Let us construct n x n matrices Uk (t) (k = 0, 1, 2, ... , r - 1) by the integral equations

Uk(t) = Nk +

ft

1

ske(t-")NR(s)Ui(s)ds

Then, lira

t-++00

Uk(t) = Nk

Observe that tkUk(t)

tkNk

k!

k!

rt e(t_s)NR(s)skUk(s)ds. k!

n

Hence, setting tkUk(t) k!

k=0

we obtain

1(t) = eIN +

r-1 t

k r e(t-,)NR(s) s

Y' r

k!!

k=o n

This implies that

do(t) dt

_ (N + R(t))W(t).

It can be easily shown that lim

t-+oo

L(t)c = tk

ds.

etNc'

Urn

t-+00

-

Nk"

tic

if Nk+16= 0. Thus, the construction of W(t) is completed. Case VII-5-5-2 was given as Exercise 35 in [CL, p. 1061.

VII-6. An application of the Floquet theorem The method discussed in §VII-5 does not apply directly to the scalar differential equation

d-t2 + {1 + h(t)sin(at)} ri = 0 when h(t) is a small function such as t112, i , l01 t, and

sin(tt/2) since the log g t derivative of h(t) sin(at) is not absolutely integrable over the interval 0 < t < +oo. In this section, using the Floquet theorem (cf. Theorem IV-4-1), we eliminate g

periodic parts of coefficients so that Theorem VII-4-3 applies. Keeping this scheme in mind, consider a system of the form djj (VII-6-1)

under the following assumption.

= A(t,h(t))y

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

226

Assumption VII-6-1. (1) The entries of an n x n matrix A(t, ej are continuous in (t, e) E R x A(r) and analytic in f E A(r) for each fixed t E LR, where FE C'" unth the entries et, ... , A(r) = IF: Ie < r}, and r is a positive number. (2) The entries of A(t, e) are periodic in t of a positive period w. (3) The entrees hl,... , h,,, of a C"t -valued function h(t) are continuous on the

interval T(to) = {t lim h(t) = 6.

:

to < t < +oo} for some non-negative number to and

t+00

Let us consider the following two cases.

Case 1. The function h(t) is supposed to be continuously differentiable on T(to) and satisfy the condition +00

(VII.6.2)

Ih(t)Idt = +oo

J

f + Ih'(t)Idt < +oo.

and

I.

Case 2. The function h(t) is supposed to be twice continuously differentiable on T(to) and satisfy the condition lim

t -+00 +00

+00

r= +00,

(VII.6.3)

= +oo,

0

L00I;(t)ldt

J0

/

Ih"(t)Idt < +oo,

J + Ih'(t)I2dt < +oo 0

For example, the function h(t) _

I

satisfies (VI1.6.2), whereas h(t) =

satisfies (VII.6.3).

sin(f)

f

Observation in Case 1. In Case 1, we use the following lemma. Lemma VII-6-2. If a matrix A(x, ej satisfies conditions (1) and (2) of Assumption VII-6-1, them exist n x n matrices P(t,e) and H(t") such that

(i) the entries of P(t, ) are continuous in (t, e) E R x A(f) and analytic in FE A(f) for each fixed t E R, where f is a suitable positive number. (ii) P(t + w, ej = P(t, t-) for (t, F) E R x ©(f),

(iii) P(t, fj is invertible for every (t, t) E R x A(f), (iv) the entries of H(t) are analytic in FE A(f), 2ai (v) any two distinct eigenvalues of H(0) do not differ by integral multiples of -, w

(vi) a P(t, a exists for (t, e) E R x L(f) and given by (VII.6.4)

for (t, e) E R x L(f).

P(t, eM = A(t, t)P(t, e) - P(t, e)H(e)

6. AN APPLICATION OF THE FLOQUET THEOREM

227

Proof

In order to prove this lemma, construct a fundamental matrix solution $(t, e)

of the differential equation d = A(t, e)y by solving the initial-value problem dX = A(t, e-) X, X (O) = I, where In is the n x n identity matrix. The endt

bog((w

, and tries of 4!(w, ej are analytic in A(r). Define H(e) by H(ej = P(t,') = 4i(t, cl exp(-tH(E)]. Then, (VII.6.4) follows. The most delicate part of this proof is the definition of H(). Details are left for the reader as an exercise (cf. [Sill).

Changing system (VI I.6.1) by the transformation

W = P(t, i(t))u

(VII-6-5)

we obtain the following theorem.

Theorem VII-6-3. Transformation (VII.6.5) changes system (VII.6.1) to (VII.6.6)

di =H(i(t)) -

hj(t)aP(t,h(t)) }u.

P(t,h(t))-1

1
Proof In fact,

dil iiT

(VII.6.7)

= P(t, h(t))-1 {A(t(t))P(t&(t)) = P(t, i(t))-1( A(t, h(t))P(t, h(t))

-

dt [P(t, h(t))] } u" (t, K (t))

1 <
Since H(e) is given by (VII.6.4), equation (VII.6.6) follows from (VII.6.7). Observe that r+oo

(IV.6.8) fro

P(t, h(t))-' 1
j

h(t)) dt < +oo

under assumption (VII-6.2). Also, observe that H(i(t)) does not contain any periodic quantities and

dH(i(t)) dt is absolutely integrable. Therefore, if eigenvalues

of H(6) satisfy suitable conditions, the argument given in §VII-5 applies to system (VII.6.6).

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

228

Observation in Case 2. Set Bo (f) = (VII-6-9)

8P

B1(t, E, iZ) _ -P(t, E)`1 I
u 8e

(t, E1,

where µ E C' and the entries of µ are u1, P2, ... , µm. Then, differential equation (VII.6.6) can be written in the form

dt = {Bo(h(t)) + B,

(VII.6.10)

Notice that the entries of the matrix B, (t, E, µ) are periodic in t of period tv and that B1 (t, E, 6) = 0. Furthermore, any two distinct eigenvalues of Bo (6) do not 27n

differ by integral multiples of -. Set D(r) =

IE + lµI < r}.

In Case 2, the following lemma is used.

Lemma VII-6-4. There exist n x n matrices P, (t, e, {'t) and HI (E, µ) such that (:) the entrees of P1 (t, E, ui) are continuous in (t, E, u') E R x D(r) and analytic in (E, )i) E D(r) for each fixed t E Ii£, where r is a suitable positive number, (i:) P, (t + w, F,)!) = P, (t, e, it) for (t, F, tt+) E 1 x D{r (iii) P1(t, E, 6) = O for (t, i) E R x 0(r), (iv) the entries of H1 (E, u') are analytic in (E, Et) E D(r) and H1 (E, 6) = 0 for E E O(r), (v) at P, (t, e, µ') exists for (t, E, p) E R x D(r) and is given by 19

(VII.6.11)

Pi(t,e,u) ={Bo(e) + B1(t,e,µ)) (In + Pi(t,E,ls)} + P1 (t, i,#)) { Bo(E) + Hi (E, µ) }

for (t, E, µ) E IR x D(r-), where I, is the n x it identity matrix.

Remark VII-6-5. Equation (V1I.6.11) can be simplified as (VII.6.12)

a P1(t,Ej) = Bo(E)P1(t,E,i) - Pi(t,E., )Bo(Ej

+ {B1(t,E,N)P:(t,E,Ji) -

+B1(t,,rA-)} -Hl(E,u-)-

Proof of Lemma VII-6-4. M

Given p = (pi,... ,p,,,), where the p, are non-negative integers, denote EIpjI =1

and 4'

um by Ipl and 91, respectively. Set gP1,p(t, E),

Pi (t, E, u') _ ipl>1

$pl?1

y Ip1?1

B, (t, e,

6. AN APPLICATION OF THE FLOQUET THEOREM

229

Then, equation (VII.6.12) is equivalent to (VII.6.13)

8

p = B0(E)P1,p - P1,pBo(E) + Q1,p(t, E) -

where

E

(VII.6.14) Q1,p(t,E') =

(B1,ptP1,p2 - Pi,p2Hl,ps) + B1,p.

P1+p2=p, 0p1IJ{r21>_1)

Hence,

{c(

Pi,p(t,e-) =

+ 1 exp[-sBo(0

(VII.6.15)

x 1 Q1,p(s, ) - H1,$,(e)) exP[sBo(E))ds} exp[-t B0(Z)1,

where C(e) and H1,p(e) are n x n matrices to be determined by the condition that P1,p(t,e) is periodic in t of period w, i.e., exp[wBoQ-)1C(E) - C(i) exp[wBo(E)1

(VII.6.16)

- exp[wBo(E)1

_ - exp[wBo(E)1

J0

exp[-sBo(e)1H1,p(i) exp[sBo(i)1ds

exp[-sBo(e)1Ql.p(s, e) exp[sBo(t1ds. 0

It is not difficult to see that condition (VII.6.16) determines the matrices C(ep) and Hl,p(ej. Then, the matrix P1,p(t, E) is determined by (VII.6.15). The convergence of power series P1 and H1 can be shown by using suitable majorant series. 0 In the same way as the proof of Theorem VII-6-3, the following theorem can be proven.

Theorem VII-6-6. The transformation

it = {In + Pi (t,h(t),h'(t)))v"

(VII.6.17)

changes differential equation (VII.6.10) to

dv = f H(i(t)) + H1(h(t), h'(t)) dt l (VII.6.18)

- [In

+ P1(t, h(t), hh'(t)),'1

+ h (t) a_ (t, h{t), h'(t))J j v

I hj(t) aP1(t, h(t), h'(t})

l ,

230

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

where P1(t, E, µ) and H1(E, µ) are those two matrices given in Lemma VII-6-4.

Observe that under assumption (VII.6.3), we obtain

In + P1(t,h(t),h'(t))j_1

f0+00

8Ej

(VII.6.19)

(t, h(t), h'(t))J dt < +oo.

+ hJ (t) a.

Observe also that the matrix H(h(t))+H1(h(t), h'(t)) does not contain any periodic

terms. Furthermore, H(6) + H1(6, 6) = H(0). It is clear that the derivative of H(h(t)) is not necessarily absolutely integrable over to < t < +oo. Therefore, in order to apply the argument of MI-5, the matrix H(h(t)) must be examined more closely in each application. Details are left to the reader for further observation.

EXERCISES VII

VII-l. Find the Liapounoff's type number of each of the following four functions

f (t) at t = +00: (1) exp [t2sin

(1L1)

]'

sint(dt](3)

(2) exp Lfo

inin(exp[3t],exp[5t]sin67rt]),

(4) the solution of the initial-value problem y"-y'-6y = eat, y(O) = 1, y'(0) = 4.

VII-2. Find a normal fundamental set of four linearly independent solutions of the system dy = Ay" on the interval 0 < t < +oo, where

A=

252

498

4134

698

-234

-465

-3885

-656

15

30

252

42

-10

-20

-166

-25

Hint. See Example IV-1-18.

VII-3. Let 4i(x) be a fundamental matrix solution of the system

=

log(l+x)

e=

expx2

x3

1+ 1+ x

sin x

exp(e')

cosx

arctanx

on the interval 0 < x < +oo. Find Liapunoff's type number of det 4i(x) at x = +oo.

EXERCISES VII

231

VII-4. Assuming that the entries of an n x n matrix A(t) are convergent power series in t-1, calculate lim p log I (t) for a nontrivial solution ¢(t) of the differential equation t

= A(t)9 at t = +oo.

Hint. Set t = e".

VII-5. Let

= xP-'A(x)i be a system of linear differential equations such that y E C" is an unknown quantity, p is a positive integer, the entries of an n x n matrix A(x) are convergent power series in x-1, and lim A(x) is not nilpotent. Show that dY

there exists a solution g(x) of this system such that lim r-.+oo number for some real number 6.

ln(l y-(Teie)j)

rp

is a positive

Hint. Set x = re!e for a fixed 0 and t = rp. Then, the given system becomes d#

e'

e+'0 PO = -A(re's)y'. Note that lim e'-A(re'B) _ -A(oo). The eigenvalues

r-.+oo P

P

ei>'

P

A(oo) are ei_)Iof p ' where the a, are the eigenvalues of A(oo). Now, apply ,

P

Corollary VII-3-7.

VII-6. For each of two matrices A(t) given below, find a unitary matrix U(t) analytic on (-oo, oo) for each matrix such that U-' (t)A(t)U(t) is diagonal or uppertriangular. 1 t 0 r (a) A(t) = Ot 0J, (b) A(t) = t 1 + 2t 0 11

i

L

0

0

1+t2

Hint. See [GH]. VII-7. Show that if a function p(t) is continuous on the interval 0 < t < +oo and lim t-Pp(t) = I for some positive integer p, the differential equation d2 t +p(t)y = +oo 0 has two linearly independent solutions rlf(t) such that t

77.k(t) =

P(t)-114(1 +o(1))exp [±ij t

)ds(, JJ

77, (t)

= P(t)114(+1 +o(1))exp

[f

iJ t

d4]

,

to

as t +oo, where to is a sufficiently large positive number and o(1) denotes a quantity which tends to zero as t - +oo. Hint. Use an argument similar to Example VII-4-8. m-1

VII-8. Let Q(x) = x' + E ahxh and P(x, e) _ ,Tn+l + Q(x), where in is a h=o

positive integer, x is an independent variable, and

am-1) are complex

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

232

parameters. Set A(x, f) =

{(0) 11.

Also, let Ro, po, and ao be arbitrary but

fixed positive numbers. Suppose that 0 < po < 2. Show that the system d:V

= A(x,f)y,

y = 17121

has two solutions y, (x, f) (j = 1, 2) satisfying the following conditions: (a) the entries of y, (x, f) are holomorphic with respect to (x, f, ao,... , am_ 1) in the domain

D={(x,f,ao,...,am_,): rEC, 0
(b) y', (x, f) (j = 1, 2) possess the asymptotic representations 111 +o(1)JP(x,f)

P(t,f)1/2dtl J L it - o(1)]P(x,f)1/4

exp L fro

exp

- Jro P(t,E) 1/2dt , l

11 +o(1)] P(x,E)-1/4

f -l + o(1)]P(x, f)'/4, respectively as x - +oc on the positive real line in the x-plane, where xo is a positive number depending on (Ra.po.ao) and o(1) denotes a quantity which tends to 0 as x ---. +oo on the positive real line uniformly with respect to (f, ao,... , am-I) for IEI < ao, I arg EI < po, and Iaol + ... + lam-1l < Ro. Hint. Use a method similar to that for Exercise V1I-7. See, also, [Nful and [Si13, Chapter 3].

VII-9. Let A1(t)...... (t) be n continuous functions of t on the interval Zo =

{tER:01(j=1,...,n-1) on Also, let A(t) be an n x n matrix such that the entries are continuous in t on Zo and that

sup(1 + p - t)

1 1 p I A(s) l ds - 0

p>t

as

t -+ +oo.

1

Show that there exist a non-negative number to and an n x n matrix T(t) such that

(t) exists and the entries of T(x) and . (1) the derivative in t on the interval 2 = It : to < t < +oo}, (2) t limo T(t) = 0, (I + T(x)) z changes the system (3) transformation

(x) are continuous

E

dt =

(diag[A1(t),A2(t),...

+T(t))y

to

d

dt

= diag(A1(t) + bl (t), A2(t) + b2(t), ... , An(t) +

where y E C, i E C, n functions b1(t),... ,b,,(t) are complex-valued and continuous in t on T, and lim b,(t) = 0 (j = 1, ... , n). t

+00

233

EXERCISES VII

VII-10. Show that if R(t) is a real-valued and continuous function on Zo = {t E It : +00

0 < t < +oc} satisfying the condition J

JR(t)j < +oo, the differential equation

0

d2y + (1 + R(t))y = 0 has a solution q5(t) such that lim (O (t) - sint) = 0. Also, c-+OQ dt2 = 7r. show that 0(t) has infinitely many positive zeros an such that lim

n+00 n n

tz + R(t)y = 0 has at most a finite number of zeros on the interval Zo = {t E Lea : 0 <_ t < +oo} VII-11. Show that every solution of a differetial equation

if R(t) is a real-valued and continuous function on Zo satisfying the condition +a,

L

tIR(t)j < +oc. +00

tjR(t)j = +oo,

Remark. See (CL, Problem 28 on p. 1031. For the case when f o

f+a but J IR(t)l < +oo, see Example VI-1-i. 0

VII-12. Suppose that u(t) is a real-valued, continuous, and bounded function of t r+00

on the interval 0 < t < +oo. Also, assume that J

ju(t)jdt < +oc. Show that if

0

X is an eigenvalue of the eigenvalue problem

f+00

d'y

7t2 + u(t)y = Ay,

y(t)2dt < +oo,

y(O) = 0,

J0

then 0 < A <

sup

o<:<+c*

Ja(i)l.

VII-13. Let A(t) be an n x it matrix whose entries are real-valued, continuous, and periodic of period 1 on R. Show that there exist two n x n matrices P(t, c) and B(c) such that (a) the entries of P(t, c) and B(c) are power series in c which are uniformly convergent for -oc < t < +oo and small JEJ,

(b) P(t, 0) = I, (-oo < t < +oo), where In is then x n identity matrix, (c) the entries of P(t, c) is periodic in t of period 1, (d) P(t, c) and B(E) satisfy the equation

eP

8t =

cIA(t)P(t, e) - P(t, c)B(c)j.

VII-14. Apply Lemma VII-6-2 to the following system: dy

dt

= [nL + (a + 8cos(2t))(K - L)Jg,

g_

[Y2J

where n is a positive integer, the two quantities a and,3 are real parameters, and f0 K= and L = [ 01 0] U]

234

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

VII-15. Using Theorem VII-6-3, find the asymptotic behavior of solutions of the differential equation

2 + {1 + h(t) sin(at)} rl = 0, as t -+ +oo, where a is a real parameter.

h(t) = ln(2 + t)

CHAPTER VIII

STABILITY

In the previous chapter, we explained the behavior of solutions of linear systems as t -. +oo. In this chapter, we look into similar problems for nonlinear systems. To start with, in §VIII-1, we introduce the concepts of stability and asymptotic +oo. We illustrate those concepts stability of a given particular solution as t with simple examples. Reducing the given solution to the trivial solution by a simple transformation, we concentrate our explanation on the stability property of the trivial solution. It is well known that the trivial solution is asymptotically +oo if real parts of eigenvalues of the leading matrix of the given stable as t system are all negative. This basic result is given as Theorem VIII-2-1 in §VIII-2. The case when some of those real parts are not negative is treated in §VIII-3. In particular, we discuss the stable and unstable manifolds. In §VIII-4, we look into the structure of stable manifolds more closely for analytic differential equations. First we change a given system by an analytic transformation to a simple standard form. By virtue of such a simplification, we can construct the stable manifold in a simple analytic form. This idea is applied to analytic systems in IR2 in §VIII-6. In §§VIII-7-VIII-10, using the polar coordinates, we explain continuous perturbations of linear systems in J2. In §VIII-5, we summarize some known facts concerning linear systems with constant coefficients in JR2. The topics discussed in this chapter

are also found in [CL, pp. 371-388], [Har2, pp. 160-161, 220-227], and [SC, pp. 49-96]. The materials in §§VIII-4 and VIII-6 are also found in [Du], [Huk5], and [Si2].

VIII-1. Basic definitions Let us consider a system of differential equations (VIII.1.1)

dy d

t=

where y E lR" and the R'-valued function At, M is continuous on a region

A(ro) = Zo x D (ro) = {(t, y1 E J"+I : 0 < t < +oo, 1171 < ro}. Also, assume that a solution mo(t) of (VIII.1.1) is defined on the entire interval 10 and that (t, mo(t)) E A(ro) on Yo. The main topic in this chapter is the behavior of solutions of the initial-value problem (VIII.1.2)

as t

dt = f (t, o'

+oo. To start, we introduce the concept of stability. 235

VIII. STABILITY

236

Definition VIII-1-1. The solution do(t) is said to be stable as t - +oo if, for any given positive number c, there exists another positive number b(c) such that whenever I&(0) - 7 < 5(c), every solution 4(t) of initial-value problem (VIII.1.2) exists on the entire interval Zo and satisfies the condition (VIII.1.3)

I$o(t) - (t)I < E

on

10.

Remark VIII-1-2. Suppose that the initial-value problem (VIII.1.2.r)

dt

= At' y),

y(r) = it

has the unique solution y = (t, r, i) if (r, rl) E 0(ro). Then, qs(t, r, 771 is continuous with respect to (t, r, '). Therefore, for any r E Zo and any given positive numbers T and f, there exists a positive number p(r,T,e) such that whenever 100(r) - )1 <

p(r, T, c), the solution 0(t, r, 71) exists on the interval 0 < t < T and I0o(t) ¢(t, r, Y-7)1 < E on the interval 0 < t < T (cf. §11-1). This implies that if the solution

+oo, then for any r E Zo and any positive number c, there exists another positive number b(r, c) such that whenever loo (,r) - n7 < b(r, E), the solution (t, r, rt) exists on the entire interval 10 and (t, r, J) satisfies the condition

0o(t) is stable as t

IV'0(t) - (¢(t, r, T-D I < E on I.

We also introduce the concept of asymptotic stability.

Definition VIII-1-3. The solution do(t) is said to be asymptotically stable as

t-4 +00 if (i) the solution do(t) is stable as t - +oo, (ii) there exists a positive number r such that whenever I o(0) - i7 < r, every solution fi(t) of initial-value problem (VIII. 1.2) satisfies the condition

lim I&(t) - (t)I = 0.

(VIII.1.4)

Remark VIII-1-4. Set g= E+ ¢0(t). Then, system (VIII.1.1) is changed to

dt = At' z + do(t)) - f'(t, do(t)).

(VIII.1.5)

Hence, the study of the solution o(t) of (VIII.1.1) is reduced to that of the trivial solution z1(t) = 0 of (VIII.1.5). Thus, the solution .o(t) of (VIII.1.1) is stable (respectively asymptotically stable) as t -+ +oo if and only if the trivial solution of (VIII.1.5) is stable (respectively asymptotically stable) as t -- +co. In the following sections, we shall study stability and asymptotic stability of the trivial solution. The following three examples illustrate stability and asymptotic stability. Example VIII-1-5. The first example is the system given by (VIII.1.6)

ddtl

= -yi,

dt = (yl - y2)y2

1. BASIC DEFINITIONS

237

To find the general solution of (VIII.1.6), solve the first-order equation (VIII.1.7)

= -y2 +

dye

dyi

.

y1

The transformation u = b2 changes this equation to a first-order linear differential du

2

equation7Y1 - = 2u - -. Thus, we obtain u(yl) = ezvi c - 2'I

vt a-2n

yl

dq . Since

rl

u = yz > 0, the quantity c must be positive. It is evident that y1(t) = y e-t satisfies the first equation of system (VIII.1.6) with the initial value yt(0) = y. Therefore, (VIII.1.8)

1

yl(t) = -( e-t,

112(t) = ±

u(yi(t))

is a solution of (VIII.1.6). Observe that

y0 ry, (t)-2

for t > 0. Therefore, e2171

c

J J.

dt) < 0 for t > 0. Thus, y2(t)2 <

e-2Yt(t) C

77

= e21',1e2'ry2(0)2 for t > 0. This proves that the trivial solution of system

(VIII.1.6) is stable as t

+oc. Furthermore, since fvt(t) a-2q

lim J t-+o0

-dt7 = lim Y1 -0

fYi

a-2n

r)

di = -oo,

11

the trivial solution of system (VIII.1.6) is asymptotically stable as t -- +oo. This result is shown also by Figure 1. Y2

FIGURE 1.

Example VIII-1-6. The second example is a second-order differential equation (VIII.1.9)

d-t2

+ g(t7) = 0,

VIII. STABILITY

238

where 9(17) is a real-valued and continuously differentiable function of g on the real line R. Equation (VIII.1.9) is equivalent to the system (VIII.1.10)

dti = Y2,

= -9(yi)

d

If g(a) = 0, then system (VIII.1.10) has a constant solution yt = a, Y2 = 0. Set yi = zi + a and y2 = z2. Then, system (VIII.1.10) becomes

di = Az + #(z-), iii

(VIII.1.11) where

z = lz2J

,

A = j-9'(a) 0]

[-[g(21

+ a) -9 ()z1)]

The eigenvalues of A are ±(-g'(a))I/2. Note also that lim z,-0 g(zl + a)zi- g'(a)zt = 0. Set 17

G(i7) = / g(s)ds.

(VIII.1.12)

0

Then, G(n) takes a local minimum (respectively a local maximum) at n = a if g(a) = 0 and g'(a) > 0 (respectively g'(a) < 0). Also, set H(yi , y2) = G(yi) +

(VIII.1.13)

Then,

dH(yi(t),y2(t)) dt

=

9 ( Y1 (t))

dyl(t) dt

y2 2

+

(t) dy2(t) Y2

dt

Therefore, dH(yi(t),y2(t)) dt

= 9(YI(t))y2(t) - y2(t)9(yI(t)) = 0

if (yi(t),y2(t)) is a solution of (VIII.1.10). This means that

(t) + a) + z2(t)2 = (VIII.1.14)

G(zi

2

(0) + a) + z2 0)2 G(zi

2

for all t

if z(t) is a solution of (VIII.1.11). Case when g'(a) < 0: If g'(a) < 0, there exists a positive number PO such that G(rl) < G(a) for 0 < 177 - al < Po Let t;(t) be the unique solution of (VIII.1.11)

satisfying the initial condition z1(0) = 0, z2(0) = Co > 0. Since t;2(t) = (1(t),

239

1. BASIC DEFINITIONS

2(G(a) - G((1(t) + a)) + co >- Co as long as 0 < (1(t) < po. Hence, there must be a positive number to depending on (o such that (I (to) = Pa no matter how small to may be. This implies that the trivial solution of (VIII.1.11) is not stable. Case when g'(a) > 0: For a given positive number p, let V (p) be the connected component containing 0 of the set {[; : G(( +a) < G(a)+p}. Then, V (pl) C V (P2)

we obtain ('(t) =

if p1 < p2. Furthermore, if g(a) = 0, g(a) > 0, and if a positive number p is sufficiently small, there exist two positive numbers e1(p) and e2(p) such that (1) e2(P) < e1(P),

(2) Plmoe1(P) = 0,

< el(p)} (cf. Figure 2).

(3) {( : ISO < C2 W} C 7) (P) C

S=o G= G(S+a)

G=G(a)+p

G=G(a) 1 P)

FIGURE 2.

Observe also that G(a) < G(( + a) < G(a) + p for [; E V (p) if p is a sufficiently small positive number. For a given positive number e, choose another positive number p so that ei(p) < e and p:5 e. Choose also the initial value z(0) so that Jz1(0)j < e2 (2) and z2(0)2 < p.

Then, G(z1(0)+a) < G(a)+ since z1(0) EP (2). Hence, (VIII.1.14) implies that G(zi(t) + a) < G(a) + p for all 2 t. Observe that the set {z1(t) : all t} is connected and zi(0) E D (2) C V (p). Therefore, zi(t) E D(p) for all t. Hence, Iz1(t) < ei (p) < e. On the other hand, G(a) < G(z1(t) + a) < G(a) + p since z1(t) E D (p) for all t. Hence, (VIII.1.14) implies that2(tz2< G(z1(0) +a) - G(a) + < p < e. This proves that the trivial solution of system (VIII.1.11) is stable as t -+ 2 +oo. The analysis given in Example VIII-1-6 is an example of an application of the Liapounoff functions to which we will return in Chapters IX and X. The third example is the following result.

Theorem VIII-1-7. If f (x, y) and g(x, y) are real-valued continuously differentiable functions such that (i) f (0, 0) = 0 and g(0, 0) = 0, (ii) (f (x, y), g(x, y)) 0 (0, 0) if (x, Y) # (0, 0),

(iii) 8x (x, y) + 49Y (x, y) = 0, then, the trivial solution (x, y) = (0, 0) of the system (S)

dt = f(x,y),

dt =

g(x,y)

VIII. STABILITY

240

is not asymptotically stable as t -- +oo. Proof.

A contradiction will be derived from the assumption that the trivial solution is asymptotically

stable. Let

(,t CI,t,C2)2)

x(t, ct, c2)

be the unique solution of

y(t,cl,c2)

system (S) satisfying the initial condition 0(0.cl, c2) =

[ci].

If the trivial solution

is asymptotically stable as t -. +oo, the trivial solution is also stable as t -. +oo. Therefore, for every positive number c, there exists another positive number 8(e) such that it (t,cl,c2)1 < e for 0 < t < +oo whenever max{jcj1,lc21} < b(e). Since f and g are independent of t, O(t - r, cl, c2) is also a solution of (S) and satisfies the initial condition (x(r), y(r)) = (Cl, c2). This implies that I¢(t, cl, c2)1 < e for r < t < +oo if I¢(r,c1,c2)I < 8(e). Denote by 0(r) the disk {(c1,c2) E 1R2 : Ic112 + 1c212 < r}. Also, for a fixed value r of t, let us denote by D(r, r) the set {0r,C1,c2) : (cl,c2) E A(r)}. Then, the mapping (C1, C2) --+ (r,Cl,C2) is a homeomorphism of 0(r) onto D(r, r) (cf. Exercise 11-4). Fix a sufficiently small positive number ro. Since (0, 0) is asymptotically stable as t +00 and the disk A(ro) is compact, there exists a positive number ro such that D(ro,ro) C 0 ro l 2 ). This implies that the area of V(ro, ro) is definitely smaller than the area of A(ro). Observe that (VIII.1.15)

area of D(ro, ro) =

fA(r0)

det

tO(roc1c2) a$(ro, C1, C2)

It is known that the matrix 1L(t,cl,c2) =

8c2

8c1

dc 1 de,.

aj(ro,c1,c2) L

ac,

0Ce

is the

unique solution of the initial-value problem dX dt

1of of ax

8y

a9

09

ax

Dy

X (0) = I2.

X. tr,y)=Oit.c,,cz)

where 12 is the 2 x 2 identity matrix (cf. Theorem 11-2-1). Therefore, r

/'zJ

det '(t, cl, c2) = expLI

( o

8f (x, y) + ag (x, y)1 I

l ax

Oly

dtJ = 1

ft=,v>=ecl.c,,cz1

(4) of Remark IV-2-7). Now, it follows from (VIII.1.15) that the area of D(ro, ro) is equal to the area of 0(ro). This is a contradiction. 0 (cf.

2. A SUFFICIENT CONDITION FOR ASYMPTOTIC STABILITY

241

VIII-2. A sufficient condition for asymptotic stability In this section, we prove a basic sufficient condition for asymptotic stability. Let us consider a system of differential equations

= Ag + g(t,yj

(VIII.2.1)

under the assumptions that (i) A is a constant n x n matrix, (ii) the entries of the 1R"-valued function §(t, y-) are continuous and satisfy the estimate for (t, y)) E A(ro), (VIII.2.2) I9(t,y)l < k(t)lyy + colylt+w where, vo > O, co > O, ro > O, k(t) > O and bounded fort > 0, lim k(t) = 0,

t+x

and A(ro) = {(t, yj : 0 < t < +oo, Iy1 < ro). The following theorem is the main result in this section. Theorem VIII-2-1. If the real part of every eigenvalue of A is negative, the tnvial solution g = 0 of (VIII. 2.1) is asymptotically stable as t -i +oo. Proof.

We prove this theorem in four steps. Step 1. Let A, (j = 1, 2,... , n) be the eigenvalues of A. Then, RIA,I (j = 1, 2, ... ,

n) are Liapounoff's type numbers of the system d= Ay (cf. Example VII-2-8). t Therefore, choosing a positive number p so that 0 < p < -9RIA,I f o r j = 1, 2, ... , n, we obtain IeXpIAtll < Ke-"t on the interval To = {t : 0 < t < +oc} for some positive constant K. Step 2. Fixing a non-negative number T, change system (VIII.2.1) to the integral equation

g(t) =

e(t-T)Ag(T)

+ fTte(t-')Ag(s,g(s))ds.

If Ig(t)I < b for some positive number b on an interval T < t < T1, then ly(t)l

<-

Ke-"(t-T)Ig(T)I +

Kff e-"(t-d) {k(s) + cabv0} Ig(s)Ids

on the same interval T < t < Ti. Setting p(T) = sup k(s), change this inequality a>T

to the form e"(t-T)Ig(t)I <- Klg(T)I + K {p(T) + coa"o}

e"('-T)Iy(s)Ids

T

for T:5 t < Tt. Then, e"(t-T)lg(t)I <- Klg(T)I exp[K(p(T) + cob&°)(t -T)I and, hence, (VIII.2.3) lg(t)I <- KI g(T)I exp I- (u - K(p(T) + cotVO))(t - T)J for T:5 t < Tl (cf. Lemma 1-1-5).

VIII. STABILITY

242

Step 3. Fix a non-negative number T and two positive numbers 6 and 61 in such a way that

0<61 <6,

p-K(p(T)+co6'o) > 0,

K61 <6.

Assume that 19(T)J < 61. Then, inequality (VIII.2.3) holds for T < t < T1 as long

as Iy(t)I < bon the interval T < t < T1. This, in turn, implies that

Jg(t)J < K61 < 6

for T < t < T1.

This is true for all T1 not less than T. Hence, inequality (VIII.2.3) holds for t > T.

Step 4. If Jy(t)J < b < 1 for 0 < t < T, there exists a positive number is such that

Id

t)

I < n19(t)j, for 0 < t < T. This implies that Jy-(t)J < Jy-(0)JeK' as long

as Jy(t)I < 6 < I for 0 < t < T. Therefore, Jy(T)J is small if Jy(0)j is small. Thus, it was proven that (VIII.2.3) holds for t > T if Iy(0)J is small. This completes the proof of Theorem VIII-2-1. O

Example VIII-2-2. For the following system of differential equations d = A#+ where

y=

[y Y

j

A=

r -0.4

-2

-

(Y2

1

.2

( 1 + y2) I 11 J

the trivial solution y = 0 is asymptotically stable as t -' +oc. In f act, the characteristic polynomial of the matrix A is PA(a) _ (A + 0.3)2 + 1.99. Therefore, the real part of two eigenvalues are negative. Remark VIII-2-3. The same conclusion as Theorem VIII-2-1 can be proven, even if (VIII.2.2) is replaced by (VIII.2.4)

19(t, y-)J < (k(t) + h(t)Jyj")Jy7

for

(t, y) E A(ro),

where p is a positive number, and two functions h(t) and k(t) are continuous for

t > 0 such that k(t) > 0 for t > 0, lim k(t) = 0, h(t) > 0 fort > 0, and Liapounoff's type number of h(t) at t = +oo is not positive (cf. [CL, Theorem 1.3 on pp. 318-319]). Remark VIII-2-4. The converse of Theorem VIII-2-1 is not true, as clearly shown in Example VIII-1-5. In that example, the eigenvalues of the matrix A are -1 and 0, but the trivial solution is asymptotically stable as t -+ +oo. Also, in that example, solutions starting in a neighborhood of 0 do not tend to 0 exponentially as t -+ +oo.

Remark VIII-2-5. Even if the matrix A is diagonalizable and its eigenvalues are all purely imaginary, the trivial solution is not necessarily stable as t -+ +oo. In such a case, we must frequently go through tedious analysis to decide if the trivial solution is stable as t -+ +oo (cf. the case when g'(a) > 0 in Example VIII-1-6). We shall return to such cases in IR2 later in §§VIII-6 and VIII-10 .

243

3. STABLE MANIFOLDS

Remark VIII-2-6. If the real part of an eigenvalue of A is positive, then the trivial solution is not stable as it is claimed in the following theorem.

Theorem VIII-2-7. Assume that (i) A is a constant n x n matrix, (ii) the entries of y(t, yam) are continuous and satisfy the estimate

for (t,y-) E o(ro) _ {(t,y-) : 0 5 t < +oo, Iy-I <_ ro},

I9(t,y)l 5 e(t,yyly7

i(ro),

lim e(t, y) = 0, t-1+19{o (iii) the real part of an eigenvalue of the matrix A is positive.

where ro >

0, e(t, y-)

> 0 for (t, y) E

and

dy

= Ay + §(t, y-) is not stable as t -. +oo. dt A proof of this theorem is given in (CL, Theorem 1.2, pp. 317-3181. We shall prove this theorem for a particular case in the next section (cf. (vi) of Remark VIII-3-2). Then, the trivial solution of the system

An example of instability covered by this theorem is the case when g'(a) < 0 of Example VIII-1-6. The converse of Theorem VIII-2-7 is not true. In fact, Figure 3 shows that the trivial solution of the system I

= -yI,

dY2

= (yi + yi)112

is not stable as t - +oo. Note that, in this case, eigenvalues of the matrix A are -1 and 0. Y2

Yt

FIGURE 3.

VIII-3. Stable manifolds A stable manifold of the trivial solution is a set of points such that solutions starting from them approach the trivial solution as t --. +oo. In order to illustrate such a manifold, consider a system of the form (VIII-3-1)

dx

= AIi +

LY = Alb + 92(40,

where Y E iR", 11 E R n, entries of R"-valued function gl and RI-valued function g2 are continuous in (1, y-) for max(IiI, Iyj) 5 po and satisfy the Lipschitz condition 19't(_,y)

- 9,(f,nil 5 L(P)max(Ii-.1, Iv-ffl)

(1=1,2)

VIII. STABILITY

244

for max(jil, Iy1) < po and max(11 i, Ii1) < po, where po is a positive number and u oL(p) = 0. Furthermore, assume that gf(0, 0) = 0 (j = 1, 2). Two matrices AI and A2 are respectively constant n x n and m x m matrices satisfying the following condition: le(t-9)A' I < Je(t-s)A21 < K2e-02(t-8)

for

Kie-o1(t-a)

(VIII.3.2)

t

for

t > s, t < s,

where K, and of (j = 1,2) are positive constants. Condition (VIII.3.2) implies that the real parts of eigenvalues of AI are not greater than -01, whereas the real parts of eigenvalues of A2 are not less than -02. Assume that

a1 > 02.

(VIII.3.3)

Let us change (VIII.3.1) to the following system of integral equations: I

x(t, c) = e1AI C + J e(1-a)A' gi (Y(s, c-), y(s, c))ds, 0

(VIII.3.4)

At, c ) =

Jt to0 e(t

8)A2 2(x(s, c), y(s, c))ds.

The main result in this section is the following theorem.

Theorem VIII-3-1. Fix a positive number c so that a1 > 02 + E. Then, there exists another positive number p(() such that if an arbitrary constant vector c' in R' satisfies the condition KI I c1 < be constructed so that

1(0,61=6 and

(VIII.3.5)

2 , a solution (i(t, c), y"(t, cam)) of (VIII.3.1) can

max(Ii(t, c)I, I y(t, c)I) < p(e)e-(Ol -t)t

for 0 < t < +oo. Furthermore, this solution (i(t, cl, #(t, c)) is uniquely determined by condition (VIII.3.5). Proof Observe that if

max(Ix(t,c)I,Iy(t,c)I) <

pe-(°'-`)t

0 < t < +00,

for

then t

e(t-s)A'91(Y(s, c), y(s, c))ds1 < K1 L(P)P f t e-`(t-8)e-(" -`)sds

I fo

0

KiL(P)Pe-0,t

(e" - I) <

Ki L(P)Pe-(o, -c)t

and +00

t

IJ

+ 00

e(t-a)A2 2(j(s, c), y(s, c))dsl < K2L(p)p f t

K2L(P)P

al -o2-E

e

e-02(t-8)e-(0 $ -')8ds

245

3. STABLE MANIFOLDS

This implies that if a positive number p is chosen so small that

<

and K1L(p)

K2L(p)

<

1

01-o2-e - 2

and if the arbitrary vector c' in R" satisfies the condition

2 successive approximations, a solution (i(t, cl , y(t, c)) of , then, using 2 (VIII.3.4) can be constructed so that

Kl I i <

x(O, c-)= c and

max(Ii(t, c1I, Iy(t, c)I) < pe-(c

for

0 < t < +oo.

Details are left to the reader as an exercise.

Remark VIII-3-2. (i) The positive number a is given to start with and the choice of p depends on e. However, since this solution approaches the trivial solution, the constant e may be eventually replaced by any smaller number, since the right-hand side of (VIII.3.1) is independent of t. This implies that the curve (i(t, c), y'(t, c-)) is independent of a as t -. +oo. More precisely, if a solution (x"(t), y"(t)) of (VIII.3.1) satisfies a condition

max(Ii(t)I, ly(t)I) <

Ke-(°'-'0)t

for

0 < t < +00

with some positive constants K and eo such that of -o2-eo > 0, then for every positive a smaller than co, there exists to > 0 such that (i(to + t), y(to + t)) _ (i(t, ), y(t, cam)), where c = ?(to). (ii) The initial value of y(t, c-) is given by 0

00, cl = f e-sA292(i(s, c_), y(s, c_))ds. +oo

(iii) If

and

exist and are continuous in a neighborhood of (6,6) and if

ay 09 8 (6, U = 0' and

(0, 0) = 0', then i(t, c) and y"(t, cl are continuously differentiable with respect to c. (iv) If the real parts of all eigenvalues of the matrix Al of (VIII.3.1) are negative and the real parts of all eigenvalues of the matrix A2 are positive, then the stable manifold of the trivial solution of system (VIII.3.1) is given by S =

89

((6, 9(0, c-)) : 161 < p}, where p is a sufficiently small positive number.

(v) Consider a system dy-

(VIII.3.6)

dt

Ay + 9(y

in the following situation:

(a) A is a constant n x n matrix, (b) the entries of §(y1 are continuous for Iy1 < po and satisfy the Lipschitz

-

L(p)ly - r11 for ly-I where PO is a positive number and lim L(p) = 0, condition 19(y-)

p-0

< po and i31 < po,

VIII. STABILITY

246 (c) 9(0,6) = 6.

Suppose further that A has an eigenvalue with positive real part. Then, applying

= -Ay' - g(y-), we can construct the stable

Theorem VIII-3-1 to the system

manifold U of the trivial solution. This means that if a solution fi(t) of (VIII.3.6) starts from a point on U, then urn Q(t) = 6. This shows that the trivial solution of t - +oo, and Theorem VIII-2-7 is proved for (VIII.3.6). The set U is called the unstable manifold of the trivial solution of (VIII.3.6). The materials in this section are also found in [CL, §§4 and 5 of Chapter 81 and [Hart, Chapter IX; in particular Theorem 6.1 on p. 2421.

VIII-4. Analytic structure of stable manifolds In order to look closely into the structure of the stable manifold of the trivial solution of a system of analytic differential equations (VIII.4.1)

dy

dt

= Ay + f (y),

let us construct a formal simplification of system (VIII.4.1). To do this, consider system (VIII.4.1) under the following assumption.

Assumption VIII-4-1. The unknown quantity g is a vector in C" with entries {y1, ... , y" }, A is a constant n x n matrix, and (VIII.4.2)

f (Y-) = E #vfa jpI>2

is a formal power series with coefficients fp E C', where p = (pl,... , p") with n

non-negative integers pl, ... , p",

> Ph, and yam' = yl'

yP,,-.

h=1

The following theorem is a basic result concerning formal simplifications of system (VIII.4.1).

Theorem VIII-4-2. Under Assumption VIII-4-1, there exists a formal power series

(VIII.4.3)

P'(u) = Pou" +

EPP,, IpI>2

in a vector u E C" with entries {ul,... ,u"} such that (i) Po is an invertible constant n x n matrix and PP, E C", (ii) the formal transformation y' = P(u") reduces system (VIII 4. 1) to (VIII.4.4)

j = Boii + g"(u)

4. ANALYTIC STRUCTURE OF STABLE MANIFOLDS

247

with a constant n x n matrix Bo and the formal power series g(ui) _ E upgp Ipi>2

with coeficients gp in Cn such that (iia) the matrix B0 is lower triangular with the diagonal entries Al.... , An,

and the entry bo(j, k) on the j-th row and k-th column of B0 is zero whenever A, j4 Ak,

(iib) for p with IpI > 2, the j-th entry gp, of the vector gp is zero whenever n

A1

(VIII.4.5)

1: PhAh h=1

Proof Observe that if y" = P(d), then d9 =

Po +

dt

1: [ L

p

p

Pp P22 Pp

Pp

un Pp

...

W>2

Boi + F 4Lpgp Ipl>2

and

Ay + f(y) = A Pod + F u-'pPp + E P(u)p fp. IpI>2

IPI>_2

Furthermore, [Plup,5, P2upP-p

Po + Ipl>2

poop P_p

Bo u'

...

u2

ul

Un

J

n

= PoBou" +

phAh lpl>2

u"PPp + 1
h_1

uh plp),Aul

where 03,k is the entry of B0 on the 7-th row and k-th column. Let us introduce a linear order pi -< p2 for p. = (p,',... , pin) (j = 1, 2) by the relation and Plh = p2h for (h < he) P1ho < P2ho Now, calculating the coefficients of iip obtain

1) on both sides of (VIII.4.1), we

PoBo = APo

(VIII.4.6) and

(VIII.4.7)

(PhAh)P + Po9p - APp h=1

=j (I5,' p -
:

10
VIII. STABILITY

248

for Jpj > 2. From (VIII.4.6), it is concluded that the diagonal entries A1, ... , A,, of Bo are eigenvalues of A and that this allows us to set A = B0 and P0 = In, where I,, is the n x n identity matrix. Then, (VIII.4.7) becomes

(PhAh)PP + 9p - Bo,6,

(VIII.4.8)

h=1

= ff(PP,

p) + 9$'05P" 9P, : 10 < 1p1)

Solve (VIII.4.8) by solving equations of the form (VIII.4.9)

PP,1 + 9P.J = F'p,J

(PhAh - A7

h-

J

successively, where PP,, and gp,, are the j-th entries of the vector P. and respectively, and FP,, are known quantities. If >phAh - Aj 36 0, set 9,j = 0 and h=1

solve (VIII.4.9). If >phAh - A? = 0, then set 9P,, = Fr,, and choose PPJ in any h=1

way. This completes the proof of Theorem VIII.4.2.

Observation VIII-4-3. Assume that R(AE) < 0 for j = 1, ... , r and R(A3) > 0 1,... , r. In this case, if uh = 0 for h I, ... , r, the a-th entry of the for j vector Bou + g"(u) is zero for t ¢ I, ... , r. In fact, look at g"(u"). Then, the fn

th entry of the coefficient g"p is zero if Al 96 >phAh. Note that, if t > r and h=1

n

At = > Ph Ah, then (Pr+ 1,

.

Pn) 0 (0, ... , 0). Hence, in such a case, uP = 0 if

h=1

(ur+1, ... , u,) = (0, ... , 0). Therefore, the 1-th entry of the vector Boti + 9"(uis

zero forI>rif(ur+1,...,u,,)=(0,....0).

Observation VIII-4-4. Under the same assumption on the Aj as in Observation VIII-4-3, set (ur+1, ... , un) = (0,... , 0). Then, the system of differential equations has the form on (u1,... dud

dt

=

A3u3

+

QJ,huh

(VIII.4.10)

+

9(P,.

.P04ul1

... u p ,

( i=1 , -...- - , r)-

A,=p,a,+ +p.a.,lai>2

r

r

Observe that since A. -

ph is sufficiently large, the right-hand

ph Ah # 0 if h=1

h=l

members of (VIII.4.10) are polynomials in (ul,... ,ur).

249

4. ANALYTIC STRUCTURE OF STABLE MANIFOLDS

Observation VIII-4-5. Assume that R[Ah+1J <_ R[Ah]. Then, (V111.4.10) can be written in the form dul

dt

= Aiu1 and

du

= A)u) +

u_,-i)

for

j=2,... ,r.

Hence, system (VIII.4.10) can be solved with an elementary method. To see the structure of solutions of (VIII.4.10) more clearly, change (ul, ... , ur) by u2 _ ea'ivi (j = 1,... , r). This transformation changes (VIII.4.10) to dv)

dt

Q),hvh an

...vf

g(P,..

(.i

A, =P, a,+...+prAr,Ipl>2

This, in turn, shows that the general solution of (VIII.4.10) has the form u) _ ea'ii) (t, cl, ... , cr) (j = 1.... , r), where (cl,... , c,.) are arbitrary constants and

0,(t,c1,....c,-) are polynomials in (t.cl,... ,G) An analytic justification of the formal series 15(ii) is given by the following theorem.

Theorem VIII-4-6. In the case when the entries of the Z"-valued function f'(y-) on the right-hand side of (VI11.4.1) are analytic in a neighborhood of 0, under the same assumption on the A) as in Observation VIII-4-3, the power series P(d) is convergent if (ur+i, ... , u") = (0,... , 0). The proof of this theorem is straight forward but lengthy (cf. [Si2J). The key fact is the inequality

A) - >PhAh h=1

h=1

r

for some positive number a if E ph is large. In this case, a iajorant series for P h=1 can be constructed.

The construction of such a majorant series is illustrated for a simple case of a system dy = Ay + f (y), dt

where A is a nonzero complex number and AM is a convergent power series in y given by (VIII.4.2). According to Theorem VIII-4-2, in this case there exists a dd = Ad, where formal transfomation ff = d + Q(u) such that dt

This implies that E AIpJi1PQg, = AQ(u) + f (u" + IeI>2

IpI?2

Set f (ii + Q(u)) _

VIII. STABILITY

250

ui"A,. Then, lal>2

Ap

a= AOPI-1) for all p (IpJ > 2). Set 1

' (y1= > IfP11F IpI>2

1

Then, F(y-) is a majorant series for ft y-). Determine a power series V (u-) =

uupii

IpI>2

by the equation v =

ICI 0(u"

iBa. Then,

+ v'). Set F(u + V(U)) _ Ipl>2

vp =

B

Ii

for all p (JpJ > 2).

It can be shown easily that Y(ul) is a convergent majorant series of Q(ur). This proves the convergence of Q(ii). Putting P(u") and the general solution of (VIII.4.10) together, we obtain a particular solution y" P(#(t, cl) of (VIII.4.1), where

= (t,cj = (e'\It7Pi(t,cl,... ,cr),... ,e*\1 'Wr(t,cl,... ,cr),0,... ,0).

This particular solution is depending on r arbitrary constants c = (cl,... , c,.). Furthermore, this solution represents the stable manifold of the trivial solution of

(VIII.4.1)if W(Aj)>0for j#1,...,r. Remark VIII-4.7. In the case when y, A, and AM are real, but A has some eigenvalues which are not real, then P(yl must be constructed carefully so that the

particular solution P(i(t, c)) is also real-valued. For example, if A =

a b

b

a

,

the eigenvalues of A are a ± ib. If IV = ['] is changed by ul = 1/i + iy2i and

=

- iy, system (VIII.4.1) becomes dul OFF

(VIII.4.11)

due dt

= (a + ib)ul +

9P,,p2u 'uZ Pl+P2>2

_ (a - ib)u2 +

,

where gP,,- is the complex conjugate of g,,,p,. If a 54 0, using Theorem VIII-4-2, simplify (VIII.4.11) to (VIII.4.12)

= (a - ib)v2

dtl = (a + ib)vl, ddt

S. TWO-DIMENSIONAL LINEAR SYSTEMS

251

by the transformation PP,.P2'UP11t?221

VI + p, +p2>_2

(VIII.4.13)

7P,.p2V2P1VP13.

V2 +

P,+p2>2

Now, system (VIII.4.12), in turn, is changed back to - = At by wI = and w2 =

VI - V2

2i

,

V 2 V2

where (w1, w2) are the entries of the vector tu. Observe that U l + u2

w1 + E

2

p,+p2>2

uI - u2

U'2 +

2i

g2.P,,pzu'P1 u" p, +p3 22

where ql,p,,p2 and g2,p,,p2 are real numbers. Similar arguments can be used in general cases to construct real-valued solutions. (For complexification, see, for example, [HirS, pp. 64-65].) For classical works related with the materials in this section as well as more general problems, see, for example, [Du).

VIII-5. Two-dimensional linear systems with constant coefficients Throughout the rest of this chapter, we shall study the behavior of solutions of nonlinear systems in 1R2. The R2-plane is called the phase plane and a solution curve projected to the phase plane is called an orbit of the system of equations. A diagram that shows the orbits in the phase plane is called a phase portrait of the orbits of the system of equations. As a preparation, in this section, we summarize the basic facts concerning linear systems with constant coefficients in R2. Consider a linear system dy =

(VIII.5.1)

dt

Ay,

where

E R2 and A is a real, constant, and invertible 2 x 2 matrix. Set p = trace(A) and q = det(A), where q ¢ 0. Then, the characteristic equation of the matrix A is 1\2 - pA + q = 0. Hence, two eigenvalues of A are given by

AI =

2

+

4

q

and

A2 = 2 -

4

- q.

It is known that p = AI +A2,

q = AIA2,

and

Al - A2 =2

P2 - q. 4

VIII. STABILITY

252

Also, let t and ij be two eigenvectors of A associated with the eigenvalues Al and A2, respectively, i.e., At = All;, l 0 0, and Au = A2ij, ij 0 0. Observe that, if y(t) is a solution of (VIII.5.1), then cy(t +r) is also a solution of (VIII.5.1) for any constants c and T. This fact is useful in order to find orbits of equation (VIII.5.1) in the phase plane.

Case 1. Assume that two eigenvalues Al and A2 are real and distinct. In this case, two eigenvectors { and ij are linearly independent and the general solution of differential equation (VIII.5.1) is given by 1!(t) = cl eAI t

+ c2ea'tti = e)lt (cl +

c2e(1\2-,,,)t11= ea2t[cie(AI -a2)t (+ c2i17,

where cl and c2 are arbitrary constants and -oo < t < +oo. 2

la: In the case when Al > A2 > 0 (i.e., p > 0, q > 0 and 4 > q), the phase portrait of orbits of (VIII.5.1) is shown by Figure 4. The arrow indicates the direction in +oo. Note that as which t increases. The trivial solution 0 is unstable as t and -oo , the solutions y(t) tends to 0 in one of the four directions of t -ij. The point (0, 0) is called an unstable improper node. 2

lb: In the case when 0 > Al > A2 (i.e., p < 0, q > 0 and 4 > q), the phase portrait of orbits of (VIII.5.1) is shown by Figure 5. The trivial solution 0 is stable as t --i +oo. The point (0, 0) is called a stable improper node. 4

FIGURE 5.

FIGURE 4.

1c: In the case when Al > 0 > A2 (i.e., q < 0), the phase portrait of orbits of (VIII.5.1) is shown by Figure 6. The trivial solution 0 is unstable as Itl +oo. Note that as t -+ -oo (or +oo), only two orbits of (VIII.5.1) tend to 0. The point (0, 0) is called a saddle point. 2

Case 2. Assume that two eigenvalues Al and A2 are equal. Then, q = 4 and

Al=A2=29& 0. 2a: Assume furhter that A is diagonalizable; i.e., A = 212i where 12 is the 2 x 2 identity matrix. Then, every nonzero vector c is an eigenvector of A, and the general

solution of (VIII.5.1) is given by y"(t) = exp

[t]e.

In this case, the phase portrait

of orbits of (VIII.5.1) is shown by Figures 7-1 and 7-2. As t -+ +oo, the trivial solution 6 is unstable (respectively stable) if p > 0 (respectively p < 0). Note that,

5. TWO-DIMENSIONAL LINEAR SYSTEMS

253

for every direction n, there exists an orbit which tends to 6 in the direction n" as t tends to -oo (respectively +oo). The point (0, 0) is called an unstable (respectively stable) proper node if p > 0 (respectively p < 0).

p<0

p>0 FIGURE 6.

FIGURE 7-1. 2b: Assume that A is not diagonalizable; i.e., A = p

FIGURE 7-2.

212 + N, where 12 is the 2 x 2

identity matrix and Nris a 2 x 2 nilpotent matrix. Note that N 0 and N2 = O. Hence, exp[tA] = exp 12t] {I2 + tN}. Observe also that a nonzero vector c' is an

eigenvector of A if and only if NcE = 0. Since N(Nc) = 0, the vector NcE is either 6 or an eigenvector of A. Hence, NE = ct(-) , where is the eigenvector of A which was given at the beginning of this section and a(c) is a real-valued linear homogeneous function of F. Observe also that at(c) = 0 if and only if c' is a constant multiple of the eigenvector . The general solution of (VIII.5.1) is given

by y(t) = exp

[t] {c + ta(c7}, where c" is an arbitrary constant vector. In this

case, the phase portrait of orbits of (VIII.5.1) is shown by Figures 8-1 and 8-2. The trivial solution 6 is unstable (respectively stable) as t - +oo if p > 0 (respectively p < 0). The point (0, 0) is called an unstable (respectively stable) improper node if p > 0 (respectively p < 0).

p>0

p<0

FIGURE 8-1.

FIGURE 8-2.

Case 3. If two eigenvalues \I and .12 are not real, then q >

and

4 Al =a+ib,

A2=a-ib,

a=

2,

4

Note that b > 0. Set A = a12 + B. Then, B2 = -b212, since two eigenvalues of B

VIII. STABILITY

254

are ib and -ib. Therefore, exp[tB] = (cos(bt))I2 +

Smbbt)

B.

3a: Assume that a = 0 (i.e., p = 0). Then, the general solution of (VIII.5.1) is given by '(t) = exp[tB]c = (cos(bt))c"+

slnbbt)

Bc, which is periodic of period 2b

in t. The phase portrait of orbits of (VIII.5.1) is shown by Figures 9-1 and 9-2. The trivial solution 0 is stable as It 4 +oo. The point (0, 0) is called a center. It is important to notice that every orbit y(t) is invariant by the operator . In fact, r,

B j(t)

T

=

cos(bt) b

b

sin bt +

Bc-(sin(bt))c = (cos (bt + ))e+

b

2 Be= 17 (t +

In other words, dy(t) = by (t + Note that "" is 1 of the period dt 2b) 2b 4

j 2b

)

2r

.

FIGURE 9-1.

b

FIGURE 9-2.

3b: Assume that a 0 0 (i.e., p 0 0). Then, the general solution of (VIII.5.1) is given by y(t) = exp[at]il(t), where u(t) is the general solution of dt = Bu. The solution 0 is unstable (respectively stable) as t - +oo if p > 0 (respectively p < 0). The orbits, as shown by Figures 10-1 and 10-2, go around the point (0, 0) infinitely many times as y(t) -- 0. The point (0, 0) is called an unstable (respectively stable) spiral point if p > 0 (respectively p < 0).

(0.0)

p>0 FIGURE 10-1.

p
Let us summarize the results given above by using Figure 11: (1) (0, 0) is an improper node. (2) (0, 0) is a saddle point.

255

6. ANALYTIC SYSTEMS IN R2

(3) (0, 0) is a proper or improper node. (4) (0, 0) is a center.

(5) (0, 0) is a spiral point. Stability of the trivial solution 6 is summarized as follows: (1) If q < 0, the trivial solution 0' is unstable as Itl -+ +oo.

(II) If q > 0 and p > 0, the trivial solution 0 is unstable as t +oo. (III) If q > 0 and p < 0, the trivial solution 0 is stable as t -+ +00. (IV) If p = 0 and q > 0, the trivial solution 0 is stable as It( - +oo. (Cf. Figure 12).

q(P0)

q(P=0) P2

(5)

4

(5)

(IV)

(4) (3)

(3)

--------

-------

(Ill

(111)

FIGURE 11.

FIGURE 12.

VIII-6. Analytic systems in R2 In this section, we apply Theorems VIII-4-2 and VIII-4-6 to an analytic system in R2. Consider a system (VIII.6.1)

dt

Ay" + E ypfp, tp!>2

where y' E LR2 with the entries yj and y2, A is a real, invertible, and constant 2 x 2 matrix. p = (pi, p2) with two non-negative integers pi and p2i Ipl = pi + p2, yam' = y'' y2 , the entries of vectors fp E R2 are real constants independent of t, and the series on the right-hand side of (VIII.6.1) is uniformly convergent in a domain

A(po) = (y E R2 : fyj < po) for some positive number po. Let us look into the structure of solutions of (VIII.6.1) in the following five cases. Case 1. If the point (0, 0) is a stable proper node of the linear system = Ay, then A = )J2i where A is a negative number and 12 is the 2 x 2 identity matrix. To apply Theorem VIII-4-2 to this case, look at the equation A = pi.1 + p2A on non-negative integers pi and p2 such that pl + p2 > 2. Since no such (pi, p2) exists, there exists an R2-valued function P(u) whose entries are convergent power series in a vector

it E R2 with real coefficients such that 5 () = 12 and that the transformation = P( u reduces system (VIII.6.1) to dt` = Ail. This, in turn, implies that the

256

VIII. STABILITY

point (0, 0) is also a stable proper node of (VIII.6.1) and that the general solution of (VIII.6.1) is y = P(eJ1°cl, where c is an arbitrary constant vector in R2.

Case 2. If the point (0, 0) is a stable improper node of the linear system dt = Ay, then we may assume that either (1) A = AI2+N, where 1A is a negative number and

N 4 0 is a 2 x 2 nilpotent matrix, or (2) A = I 1 a2 J , where Al and A2 are real negative numbers such that Al > A. In case (1), the same conclusion is obtained concerning the existence of an 12-valued function P(i7). Therefore, the point (0,0) is a stable improper node of (VIII.6.1), and the general solution of (VIII.6.I) is y" = P(eAt(12 + tN)c), where c is an arbitrary constant vector in 1R2. In case (2), looking at the equations Al = p1 A 1 + p2A2 and A2 = p1 At + p2A2 on non-negative integers pi and p2 such that pl + p2 ? 2, it is concluded that there exists an 1R2valued function P(ui) whose entries are convergent power series in a vector ii E R2 with real coefficients such that

dP

reduces system (VIII.6.1) to dt =

(d) = 12 and that the transformation y" = P(u) ['Ju.&Ajul

2u21, where (u1, u2) are the entries of

1

i , M is a positive integer such that A2 = MA1, and -y is a real constant which must be 0 if A2 54 MA1 for any positive integer M. Therefore, in case (2), the general A,:

([(''

solution of (VIII.1.6) is Cty"7t=+ C2)e PAlt ) , where cl and c2 are arbitrary constants. This, in turn, implies that the point (0, 0) is a stable improper node of (VIII.6.1).

Case 3. If the point (0, 0) is a stable spiral point of the linear system

dy

= Ay",

at

then we may assume that A = I b

ab

1,

where a and b are real numbers such that

a < 0 and b # 0. The eigenvalues of A are At = a ± ib. This implies that there are no non-negative integers pi and p2 satisfying the condition A = p1 A+ +P2A_ and pl + > 2. Therefore, there exists an R 2-valued function P(g) whose entries are convergent power series in a vector u" E R2 with real coefficients such that LP (0) 8u

_

12, and the transformation y = P(u) reduces system (VIII.6.1) to dt = Au (cf. Remark VIII.4.7). This, in turn, implies that the point (0, 0) is also a stable spiral point of (VIII.6.1) and that the general solution of (VIII.6.1) is y = P(eAC1, where c" is an arbitrary constant vector in 1R2.

Case 4. If the point (0, 0) is a saddle point of the linear system

= Ay, the

eigenvalues At and A2 of A are real and At < 0 < A2. Construct two nontrivial and real-valued convergent power series fi(x) and rG(x) in a variable x so that ¢(eA'tcl) (t > 0) and ,G(eA2tc2) (t < 0) are solutions of (VIII.6.1), where cl and c2 are arbitrary constants (cf. Exercise V-7). The solution y' = *(eAt tct) represents the stable manifold of the trivial solution of (VIII.6.1), while the solution y" =1 (eA2tc2) represents the unstable manifold of the trivial solution of (VIII.6.1). The point (0, 0)

6. ANALYTIC SYSTEMS IN R2

257

is a saddle point of (VIII.6.1). In the next section, we shall explain the behavior of solutions in a neighborhood of a saddle point in a more general case. dy Case 5. If the point (0, 0) is a center of the linear system - = Ay, both eigenvalues at

of A are purely imaginary. Assume that they are ±i and A = 1i

y" = 2 L

01

Set

J.

[t1]. Then, the given system (VIII.6.1) is changed

v, where v =

1

[0

J

2

to

du dt

(VIII.6.2)

-

Ig(vl,v2)l 9(27)

012, v1)

where +00

g(tvl, v2) = ivl +

9P'9111

012,V0 = -iv2 +

2,

p+9=2

p.9v2v1 p+9=2

Here, a denotes the complex conjugate of a complex number a. Let us apply Theorem VIII-4-2 to system (VIII.6.2).

Observation VIII-6-1. First, setting Al = i and A2 = -i, look at Al = p1A1 + p2A2 and A2 = Q1A1 +g2A2, where p1, p2, q1, and q2 are non-negative integers such that p1 + P2 > 2 and ql + q2 > 2. Then, p1 - 1 = p2 and q2 - 1 = q1. This implies

that system (VIII.6.2) can be changed to du1

(VIII . 6 . 3)

dt

=

i W ( ulu2 ) u1,

d = -iC due

0 ( u1U2 ) U2,

+oo

where w(z) = 1 + E Wmxm, by a formal transformation m=1

v = f(17) =

(T)

ul + h(ul,u2) u2 + h(u2,u1)

Here,

+0

+00

h(ul,u2) _

hp.qupU2,

h(u2,u1) _

_ hP,9u'lU2

p+9=2

p+q=2

In particular, h(ul, u2) can be construced so that (VIII.6.4)

the quantities hp+1,p are real for all positive integers p.

Observation VIII-6-2. We can show that if one of the Wm is not real, then y = 0 is a spiral point (cf. Exercises VIII-14). Hence, let us look into the case when the +00

Wm are all real. Furthermore, if a formal power series a(t) = 1 + real coefficients am is chosen in a suitable way, the transformation (VIII.6.5)

u1 = a(B11j2)131,

u2 = W10002

with m=1

VIII. STABILITY

258

changes (VIII.6.3) to another system d,31

= in(AiQ2)Qi,

2 _ -0#10002,

where f2(() is a polynomial in C with real coefficients which has one of the following two forms: 1, Case I Q(o 1 + coS"'°, Case II

-

where co is a nonzero real number and tno is a positive integer.

Observation VIII-6-3. Let us look into Case I. Assume that system (VIII.6.3) is

(VIII.6.3.1)

due

dul

dt

= iu1, dt

=

-iu2.

Note that transformation (VIII.6.5) does not change (VIII.6.3.1). Using a transformation of form (VIII.6.5), change h(ul, u2) so that (VIII.6.6)

1,P = 0

for all positive integers p.

Since (u1,u2) = (ce`t,de-'t) is a solution of system (VIII.6.3.1) with a complex arbitrary constant c, the formal series

v = f(-) =

(FS-1)

cell + h(eett,

cue-,t)

ce-u + h(ce-tt cett)

is a formal solution of system (VIII.6.2) which depends on two real arbitrary constants. Let

u(t, , ) _

(S)

Vt + H(t, fe-'t

be the solution of system (VIII.6.2) satisfying the initial condition

t'(0, f, 6 = III].

(C)

Note that +«0

H(t, , ) _

HP,q(t)ef°, D+9=2

H(tto = E

P+v=2

are power series in { and which are convergent uniformly on any fixed bounded interval on the real t line.

6. ANALYTIC SYSTEMS IN R2

259

Using (VIII.6.6), we obtain

I 0

Zx

r 2* h(Ce-'t, cett)e'Ldt

h(ceic, ee ")e-'tdt = 0,

= 0.

0

0

Fix c and c by the equations

c = £ +T7r-

/

2x

H(s, t:, )e-"ds and e = { + 2 J2w H(s, {, {)e'sds. 0

0

Then,

= c + -(c, c)

t; = e + ?(c, c),

and

where =(c, e) and c(e, c) are convergent power series in (c, e). Now, we can prove that two formal solutions +h(ce",Ce-'t)

ce`t

v'(t,c+F(c,e),e+ =2(c' i!))

and ee-tt +

h(&-'t, cent )

j

of system (VIII.6.2) are identical. The first of these two is convergent; hence, the second is also convergent. This finishes Case I.

Observation VIII-6-4. Let us look into Case II. Assume that system (VIII.6.3) has the form (VIII.6.3.2)

dtl = iul(l +

dt2 = -iu2(1 +

co(u1u2)'0),

co(uiu2)"'°)

Note that we have (VIII.6.4). Since (ul,u2) = (ce't(1+co(ce)'"°) ee-u(1+c0(ct)'"0)) is a solution of system (VIII.6.3.2) with a complex arbitrary constant c, the formal series ce't(I+co(ce)'"0) + h(ceit(l+co(ct)"'O),

(FS-2)

ee-'t(1+c0(ct)_0))

v = f (u') = h(ce-tt(1+co(ce)'"° ),

ee-'t(1+c0(ce)'"0) +

celt(1+co(ct)"O) )

is a formal solution of system (VIII.6.2) which depends on two real arbitrary constants. Again, as in Observation VIII-6-3, let (S) be the solution of system (VIII.6.2) satisfying the initial condition (C). Set (VIII.6.7)

t; = ?(c, cJ = c + h(c, c)

and

= =(c, c) = e + h(e, c).

Then,

I

ce1t(1+co(ct)"°)

+

h(ceit(1+co(ce)'"°),ee-'t(l+co(ct)"'0)

ee-tt(1+c0(ct)'"0) + h(ee-it(t+co(ct)'"O),ceut(l+co(et)"O))

VIII. STABILITY

260

as formal power series in (c, e). This series has a formal period

T(cc) =

27r

1 + co(ce)m0

with respect to t, i.e., E(c, c)

(VIII.6.8)

v(T(cc), =[c, e], E[c, c]) E(e, C)

Solving (VIII.6.7) with respect to (c, e), we obtain two power series in

c=

and

e = e(

,

£).

t) = T(c(., t)e(, £)). Then, (VIII.6.8) becomes

Set

(VIII.6.9)

Using (VIII.6.9) as equations for P({,£), it can be shown that the formal power series is convergent. This implies that 1/7+o

C(S,S)C(CS)

[(CO)

1/

J

is convergent. (Here, some details are left to the reader as an exercise.) On the other hand,

T(ct)

{{ese

0 T(cz)

H(s,e-.(1+c0(c4),"O)ds

+ 1Cesa(1+co(ct)'"°)

L

h(ceps(1+Co(cr)m0),ce-1e(1+co(Ce)m°))J

+ +00

x e-fs(1+G)(cz)m°)ds = c 1 + Ehp+l,p(cz)P P=1

and `T(cC) J/

{e

0

+ H(s, , ) }

+oo e'8(1+co(cc)"° )ds

1 + t he+1,P(cz)p P=1

since the quantity hP+1,P are real (cf. (VIII.6.4)). This proves that

C(Cr/

) is conver-

C(l, 7)

gent. Hence, c(l;, ) and 4) are convergent. Thus, the proof of the convergence of A g) in Case II is completed. Thus, it was proved that if all of the coefficients w,,, on the right-hand side of system (VIII.6.3) are real, the point (0,0) is a center of system (VIII.6.1). The general solution of (VIII.6.1) can be constructed by using various transformations of (VII.6.1) which bring the system to either (VIII.6.3.1) or (VIII.6.3.2). Periods of solutions in t are independent of each solution in Case I, but depend on each solution in Case H.

7. PERTURBATIONS OF AN IMPROPER NODE AND A SADDLE PT. 261

Remark VIII-6-5. The argument given above does not apply to the case when the right-hand side of (VIII.6.1) is of C(O°) but not analytic. A counterexample is given by _

2

[h(1)

dt

where h(r2) = e-1jr2 sin I

r

h(r)

r2 = Y1 + y2+

y'

I. Using the polar coordinates (r,O), this system is

changed to dt = rh(r2)

and

and periodic solutions are given by r2 =

1 MR

d with any integers nt. This is an

example of centers in the sense of Bendixson (cf. [Ben2, p. 26] and §VIII-10). For more information concerning analytic systems in JR2, see [Huk5].

VIII-7. Perturbations of an improper node and a saddle point In the previous section, the structure of solutions of an analytic system in JR2 was showen by means of the method with power series. Hereafter, in this chapter, we consider a system dy _

(VIII.7.1)

dt

Ay + §(y-),

under the assumption that (i) A is a real constant 2 x 2 matrix, (ii) the entries of the R2-valued function §(y) are continuous in g E JR2 near 0' and satisfies the condition lim 9(y)

(VIII.7.2)

y-o Iy1

= 0.

In this section, it is also assumed that the initial-value problem (VIII.7.3)

df = Ay + §(y-),

g(0) = g

has one and only one solution as long as g belongs to a sufficiently small neighborhood of 0.

Remark VIII-7-1. In the case, when the entries of an 1R2-valued function f (Z-) are continuously differentiable with respect to the entries z1 and z2 of i in a neigh-

borhood of i = 0 and f (0) = 0, write the differential equation dt = f(E) in form (VIII.7.1) by setting A

=

of (o) i9

aft 0z' Lof2 8x1

(o)

eft 19x2

(o) and

aft az2

( )

g(y)=f(y)-Ay.

VIII. STABILITY

262

Using thepolar coordinates (r, 0) in the g-plane, write the vector g in the form

y=r

[c?}. Then, system (VIII.7.1) is written in terms of (r, B) as follows: d=r[a, i cos2(B) + a22 sin2(9) + (a12 + a21) sin(g) cos(9)] + 91(-1 COs(0) + 92() sin(g),

(VIII.7.4)

r dte

=r[-a12 sin2(0) + a21 COS2(0) + (a22 - all) sin(e) cos(0)] - 91(y-) sin(g) + 92 M cos(e),

all a12 J and g"(y) = [iW)l Here, use was made of the formula

where A =

92 M

22

a21

dr =

dy1

dt

dt

r de

cos (9) + dye sin(6),

dt

= -dy1 sin(g) + dye cos(9). dt

dt

dt

In this section, we consider the case when the two eigenvalues Ai and A2 of the

matrix A are real, distinct, and at least one of them is negative, i.e., A2 > Al and Ai < 0. Throughout this section, assume that all = Al, a22 = A2, and a12 = a21 = 0. Then, in terms of the polar coordinates (r, 0), system (VIII.7.1) can be writen in the form (VIII.7.5)

dt = d9

1

r [Ai(cos(8))2 + A2(sin (e))2 +

(e) + g2(g)sin (e))]

= (A2 - Ai )sin (0) cos (0) + r

(e) + 92(y-)cos (e))

Observation VIII-7-2. The point (0,0) is not a proper node of (VIII.7.1). In fact, if (r(t), 9(t)) is a solution of (VIII.7.5) such that r(t) 0 and e(t) --+ w as

t - +oo or -oo, then

d8 --* (A2 - AI) sin w cos w as t -+ +oo or -oo. Hence,

sin w cos w = 0. This implies that w = -7r, -

7r,

0, or

2

Observation VIII-7-3. For any given positive number e > 0, there exists another positive number p(e) > 0 such that

de dt

>0

for

- 7r + e < 0 < - 2 - e,

<0 >0

for

-2

for

e
I<0

for

-+e<0
+ e << 0 < - e,

-

IT

2

if 0 < r < p(e) (cf. Figure 13). Observation VIII-7-4. If 0 > A2 > Al, there exists a positive number ro > 0 such that dt < Air for 0 < r < ro.

7. PERTURBATIONS OF AN IMPROPER NODE AND A SADDLE PT. 263

Observation VIII-7-5. In the case when 1\2 > 0 > AI, find a real number wo such that 0 < wo < 2 and tanwo =

-1. Then, for any given positive number 2

E > 0, there exists another positive number p(E) > 0 such that

>0 dr dt

<0 >0 <0

for for

(wo - 7r) + E < 0 < -"lo - e, - wo + f < 0 < wo - E,

for

wo + E <0< (7r - wo) - E,

for

(7r - wo) + f < 8 < (7r + wo) E,

if 0 < r < p(E) (cf. Figure 14).

FIGURE 14.

FIGURE 13.

Observation VIII-7-6. If two positive numbers f and p are sufficiently small, Observations VIII-7-2 and VIII-7-3 show that the behavior of orbits of (VIII.7.1) in the sectorial region S = {(r cos 0, r sin 0) : $6 < E, 0 < r < p} looks like Figure 15.

Denote by fl the set of all real numbers w such that

(i) IwI 5 E,

(ii) the orbit (r(t), 8(t)) of (VIII.7.1) such that r(0) = p and 0(0) = w leaves the sectorial region S through the boundary 0 = -E. Set a = sup(w : w E 0). Then, ja$ < E. Furthermore, the orbit (r(t), B(t)) of (VIII.7.1) such that r(0) = p and 0(0) = a satisfies the condition lim r(t) = 0 and Iim 6(t) = 0. This can be shown by using the continuity of orbits with +00 respect c-.+oo to initial data (cf. Figures 16-1 and 16-2). Here, use was made of the uniqueness of solutions of initial-value problem (VIII.7.3).

FIGURE 15.

FIGURE 16-1.

FIGURE 16-2.

VIII. STABILITY

264

Observation VIII-7-7. In the case when 0 > A2 > A1, the point (0, 0) is a stable improper node of (VIII.7.1) as t -+ +oo (cf. Figure 17).

Observation VIII-7-8. In the case when A2 > 0 > A1, the point (0,0) is a saddle point of (VIII.7.1) (cf. Figure 18).

FIGURE 17.

FIGURE 18.

Observation VIII-7-9. Figures 17 and 18 indicate that there are many orbits tend to the point (0,0) in the direction 0 = 0 as t +oc. This is the general situation, as the following example shows. Fixing two positive numbers c and v, define two regions Do and DI in the y"-plane as follows (cf. Figure 19):

U < yI}, DI = -2 - 19: Iy2I 5 (1 + E)yi+U, 0 < yl}. 'D0 = (!7: Iy2I <

yi+f.

Choose three real numbers A1, A2, and A3 so that Al < 0, A2 > Al, and A3

(VIII.7.6)

51

>1+

v.

Let µ(y) be a real-valued function such that (a) u is bounded on tit, (b) p is continuouslv differentiable in y" on e - {(0.0)}. (c)U(y) = A3 on Do. and

(d) µ(y') = A2 on DI. Set 9(y) =

0

(l4(y) - A2) Y21J

Furthermore, 0 < yl. 11121 5 (1 + ()y1+" if y""

Then, 9(y) = 0 if y" E Dl.

D1, and

I9'(y)I = I (µ(y) - A2)y2I 5 KIy2I < K(1 + E)yi"" 5 K(1 + e)Iyj1+

where K is a suitable positive constant. Consider the system (VIII.7.7)

ddll

= -\01-

dt

= AM + (µ(y) - A21y2

7. PERTURBATIONS OF AN IMPROPER NODE AND A SADDLE PT. 265 If y"(0) E Do, then

db2

=

(.!)

2

,\I

dy1

Hence, y2 = y1

' . Now, condition (VIII.7.6)

1/i

implies that g(t) E Do for t > 0, lim r(t) = 0, and lim 9(t) = 0. (cf. Figure 20). t-+oo t-+oo (I +E)Yj

.V

(I +e)Yi

FIGURE 19.

FIGURE 20.

Observation VIII-7-10. If we assume some condition on smoothness of §(y-), there is only one orbit of (VIII.7. 1) that tends to the point (0, 0) in the direction

9 = 0 as t - +oc. Theorem VIII-7-11. Assume that all = A1, a22 = A2, and a12 = a2I = 0, and that

(1) A2 > \1 and Al < 0, (2) g(j) is continuous in a neighborhood of

(3)y-.o lim " =

0,

lye

(4)

a9(y

exists and is continuous in a neighborhood of 6.

2

Then, there exists a unique orbit y(t) _ [01(1)1 such that 02(t)

01(t) > 0

for

urn 01(t) = 0,

i t-.+oo

t > 0,

lim 02(t) = 0,

b2(t)

= 0.

lim t--+co 01(t)

Proof.

The existence of such orbit is seen in previous observations. To show the uniqueness of such orbit, rewrite (VIII.7.1) in the form I

(VIII.7.8)

dye = 1\2Y2 + 92(Y) _ dy1 AIyI+91(91

All yz + a y192(y - (Aty1)291(Y) 1+ 1 91()J)

\Y1

AIYI

Next, introduce a new unknown u by y2 = y1u 1or u =

y2 yI.

Then, (VIII.7.8)

becomes JI2u

1

( VIII.7.9 )

y1du + u dy1

= (L2) u+

1y192(YIU) - \I-91(y1u) 1

1 +

Iy1gI(y1u) '\

VIII. STABILITY

266

where u =

[:1].

Observe that (a) 161 = 1 if Jul < 1, (b) -L (d) = U

g(0, y2)

-0 Y2

= d, and (c)

- 1 < 0. i

Write system (VIII.7.9) in the form

l

- 1 I u + G(yi.u).

Q 1

G(yi, u) =

-

A2u

92(yiu) - a

91(yl U

1

1 + -9i(y1iZ) aiyi

ac

It is easy to show that lim 5:u (y1, u) = 0 uniformly on Jul < 1. Hence, there exists

v --o

a non-negative valued function K(yj) such that

!G(yi, u) - G(y1, v)! < K(yi )lu - vl whenever Jul < 1, !v! < 1 and !y1! is sufficiently small. Furthermore, slim K(yt)

=0. Let u = 01(yi) and u = ip2(yi) be two solutions of (VIII.7.9) such that (1) ti i (yi) and .'2(y1) are defined for 0 < yi < r) for some sufficiently small

positive number 'I, (2)

lim i'1(yi) = 0 and

vl

o+

lim i2(yl) = 0.

v,

Set '(yi) = 01 (y1) - s (yi ). By virtue of uniqueness, assume without any loss of generality that y(yi) > 0 for 0 < yi < ri. Then, for a sufficiently small positive r?, d y1 -)tt'(yi) for 0 < y1 < i , where 7 = 2 1 I < 0. This implies that J

d [y1 "V,(yi)j < 0 for 0 < yi < rj. Hence, 0 < y1 lim yi 'W(yi) = 0 for Y:- 0 dyi 0 < y1 < 17. Therefore, ;1(yi) = 0 for 0 < y1 < r). The materials of this section are also found in (CL, §§5 and 6 of Chapter 151.

VIII-8. Perturbations of a proper node In this section, we consider a system of the form (VIII.8.1)

dt

= All +

where y" E R2,

under the assumptions that (i) A is a negative number, (ii) the entries of the R2-valued function g(y} are continuous in a neighborhood of 0,

(iii) 1§(y -)l < K!y"l1+i in a neighborhood of 6, where K is a nor-negative number and v is a positive number. The main result is the following theorem.

267

8. PERTURBATIONS OF A PROPER NODE

Theorem VIII-8-1. Under the assumption given above, the point (0, 0) is a stable proper node of system (VIII.8.1) as t --, +oo. Before we prove this theorem, we illustrate the general situation by an example.

Example VIII-8-2. Look at the system

dt = -y + 9(yj,

(VIII.8.2)

9(y _ (yi + y2)

[

y

where y E R2 with the entries yl and y2 It is known that the point (0,0) is a stable proper node of the linear system dt = -y. To find the general solution of (VIII.8.2), change (VIII.8.2) to (VIII.8.3)

by the transformation y' = e-'Z. Next, introduce a new independent variable s by s = 2e-2t. Then, system (VIII.8.3) becomes (VIII.8.4)

ds

-g(z)

Observe that

If a solution y(t) of system (VIII.8.2) satisfies an initial condition

9(o) = n =

(VIII.8.5)

171 ?12

'

the corresponding solution z(s) = et y'(t) of system (VIII.8.4) is the solution of the l initial-value problem ds = r(7)2 f zz2

z" C 2

r(i) =

(VIII.8.6)

/

= rj, where

(n,)2 + (m)2.

Hence,

zl(s) = r(ir)sin(r(i)2s + 6(rl)),

z2(s) = r(i)7)cos(r(1-7)2s + 0(r1)),

where

z, (0) = r(rl sin

{ Z2(0) = r(n7) cos

2

111 = r(i) sin(

in = r( n

oos

(

0(rl 2

VIII. STABILITY

268

(cf. Figure 21). The general solution of (VIII.8.2) is 2

yI(t) = e-tr(7f)sin (L(7 a-2t y2(0 = e-tr(771 cos

(!2e_2t

where y(0) and r(fl) are given by (VIII.8.5) and (VIII.8.6), respectively. Every orbit of (VIII.8.2) goes around the point (0, 0) only a finite number of times. Figure 22 shows that the point (0, 0) is a stable proper node as t -+ +oo.

FIGURE 22.

FIGURE 21.

Remark VIII-8-3. If condition (iii) of the assumption given above is relaxed, the point (0, 0) may become a stable spiral point (cf. Exercise VIII-10). Proof of Theorem VIII-8-1.

We prove Theorem VIII-8-1 in two steps. To start with, using the polar coordinates, write system (VIII.8.1) in the form

dr (VIII.8.7)

J

dt

= Ar + gI (y) cos(9) + g2(y) sin(g),

dO

r t = -g1(y) sin(g) + g2(y COs(9) (VIII.7.4)). By virtue of the assumption given above, there exists a positive number ro such that (cf.

(VIII.8.8)

2Ar < Ar + gi (y-)cos(9) + g2(y) sin(g) <

2r < 0

for0
r(0) = p, 9(0) = w,

where p is a positive number and w is a real number. This solution exists for t > 0. Furthermore, estimate (VIII.8.8) implies that the function r(t) satisfies

8. PERTURBATIONS OF A PROPER NODE

269

the following two conditions: (a) r(t) is strictly decreasing as t -+ +oo and (b) lim r(t) = 0. Therefore, we obtain (c) 0 < r(t) < _ p for t >_ 0. t-. +00

Since r(t) is strictly decreasing for t _> 0, the variable t can be regarded as a function of r, i.e., t = t(r), where t(p) = 0 and lim t(r) = +oo. Use r as the independent variable. Then, (VIII.8.7) becomes ( V1II . 8 . 10 )

d9

Tr

F (r, 0 ) _

-gi (y-)sin(g) + g2(Y)cas(e) r[Ar + g1(y) cos(0) + g2(y) sin(0)J

Let 0 = 4D(r) = 0(t(r)) (0 < r < ro) be the solution of (VIII.8.10) satisfying the initial condition (b(p) = w. Then,

fi(r) = w +

JP

(0 < r < ro).

r F(s, I(s))ds

Estimate (VIII.8.8) implies that 2Ar2

I - gt (y sin(0) + 92(Y) cos(0) I

0 < r < ro.

for

to

J

F(s, $(s))ds =

P

r(t)

ar2

(I g1(y)I + Ig2(y1I)

Now, by virtue of condition (iii), the improper integral lim r r-.0 JP F(s, 4(s))ds exists. Observe that 0(t) = 4(r(t)) = w +

F(s, (s))ds and that P

F(r, 0) <

lim r(t) t--+oo

= 0.

Therefore,

lim 0(t)

t-'+oo

=w+

0

Thus, we conclude that the point (0,0) is a stable node of

JP

(VIII.8.1) as t

+oc.

Step 2. In this step, we prove that the point (0,0) is a proper node. To do this, for a given real number c, find a solution (r(t), 0(t)) of (VIII.8.7) such that rim 0(t) = c and t -+oo lim r(t) = 0. Selecting a sequence {pm : m = 1, 2, ... } of t-+oo positive numbers such that 0 < p,,, < ro and lim pm = 0, define a sequence of functions {cm(r) : in = 1, 2.... } by the initial-value problems d0 WT

= F(r, 0),

0(Pm) = c

(m = 1, 2.... ),

respectively. Those functions are defined for 0 < r < ro, and

c+/r

F(s, 4m(s))ds

(m = 1, 2,

... ).

P

Set

Cm =

- +«; 4)m(r) = c + jP1.o lim

F(s, Om(s))ds.

VIII. STABILITY

270

Then, 4b,,,(0) = c,,,,

lira c,,, = c, and m=+oo

(m = 1,2.... )

(r) = cm + J r F(s, 4,,,(s))ds 0

for 0 < r < ro.

Since the sequence {4D m(r) : m = 1, 2.... } is bounded and

equicontinuous on the interval 0 < r < ro, assume without any loss of generality fi(r) exists uniformly on the interval 0 < r < ro (otherwise that lim M +00 choose a subsequence). It can be shown easily that

fi(r) = c +

fF(s,(s))ds

(0

r

ro).

Therefore, 7(r) is a solution of (VIII.8.10) such that rim 1(r) = c. Define r(t) by the initial-value problem dr dt

= )r +

where g(r) = r

92(17(r))sm(4'(r)),

r(0) = ro,

[c?1]. The function r(t) is defined for t > 0, is decreasing,

and tends to 0 as t -4 +oo. Set 0(t) = $(r(t)). Then, it is easily shown that (r(t), B(t)) is a solution of (VIII.8.7) such that limo 0(t) = 1i mt(r) = c. t

+0

0

The materials of this section are also found in (CL, §3 of Chapter 151.

VIII-9. Perturbation of a spiral point In this section, we show that (0,0) is a stable spiral point of (VIII.7.1) as t -, +oo

if (0, 0) is a stable spiral point of the linear system the following theorem.

= Ay". The main result is

Theorem VIII-9-1. If (i) two eigenvalues of A are not real and their real parts are negative, (ii) the entries of the L2-valued function ff(y-) is continuous in a neighborhood of the point (0,0),

(iii) lira MI = 6,

g-alyl

then the point (0, 0) is a stable spiral point of (VIII. 7.1) as t

+oo.

Proof.

Assume that (VIII.9.1)

all = a22 = a < 0

and

a12 = -a21 = b 0 0.

10. PERTURBATIONS OF A CENTER

271

Then, from system (VIII.7.4), it follows that dr

ar + 91(yjcos(9) + 92(ylsin(g),

dt

rt

= -br - 91(y-) sin(g) + g2(Y)cos(0). 1at

Therefore, a positive number ro can be fouind so that r(t) < r(0) exp

J for t > 0

if 0 < r(0) < ro. This, in turn, implies that lim r(t) = 0 and that dt < t -+oo

if

2 b > 0, while de > - 2 if b < 0, whenever 0 < r(0) < ro. Therefore, lim00(t) = -oo if b > 0 and lim 0(t) = +oo if b < 0, whenever 0 < r(0) < ro. Thus, we conclude t too that (0, 0) is a stable spiral point of system (VIII.7.1) as t -+ +oo. 0

The materials of this section are also found in fCL, §3 of Chapter 151-

VIII-10. Perturbation of a center We still consider a system (VIII.7.1) under the assumption that the entries of the RI-valued function §(y) is continuous in a neighborhood of 0 and satisfies condition (VIII.7.2). In this section, we show that if the 2 x 2 matrix A has two purely

imaginary eigenvalues Al = ib and A2 = -ib, where b is a nonzero real number,

then the point (0,0) is either a center or a spiral point of

(VIAssume

without any loss of generality that the matrix A has the form

[°b

(VIII.7.4) becomes

and system

b J

_ (0) , dt = 91Mcos (0) + 92(y-)sin

dr

(VIII . 10 . 1)

dB dt

= -b + r 1

(B) +

92(y')cos (B)).

Observe that system (VI I I.10.1) can be written in the form (VIII.10.2)

dr = d8

91(y')cos (0) + 92(y-)sin (0)

-b + r (-91(yjsin (0) + g2(y-')cos (0))

If a positive number p is sufficiently small, the solution r(0, p) of (VIII.10.2) satisfying the initial condition r(0, p) = p exists for 0 < 0 < 27r.

Case 1. If there exists a sequence (pm : rn = 1, 2, ...) of positive numbers such that lim p,,, m-.+00

=0

and r(2xr, Pm) = Pm

(m. = 1, 2, ... ),

then the point (0,0) is a center of (VIII.7.1) in the sense of Bendixson (Bent, p. 261 (cf. Figure 23).

VIII. STABILITY

272

Case 2. Assume that b < 0. If there exists a positive number po such that

r(2ir, p) > p

(respectively < p)

whenever 0 < p < po, then the point (0, 0) is an unstable (respectively stable) spiral point as t - +oo (cf. Figures 24-1 and 24-2).

FIGURE 24-1.

FIGURE 23.

FIGURE 24-2.

Example VIII-10-1. The point (0,0) is a center of the system d

[ yuj

dt [y2J

since

dr d8

+

yi +y2 I

yy1J

= 0.

Example VIII-10-2. The point (0, 0) is an unstable spiral point of the system d

dt

= [ Y2

l

Y2

l

-Yi J

+ ` y 2I + Y22 yI

Y2

as t - -oc, since dr = r2, which has the solution r = 1 r(0) = dt ro - t r

ro. (Note: t < ro.)

+oo when t

1

ro

and

Example VIII-10-3. For the system

dt

= - y,

d y = x + x2 - xy +

cry2,

the point (0, 0) is (a) a stable spiral point if a < -1, (b) a center if a = -1, and (c) an unstable spiral point if a > -1. Proof.

Use the polar coordinates (r, 0) for (x, y) to write the given system in the form

(1+rcos8(cos20-cos0sin8+asin28))

= r2sin8(cos28-cos0sin8+asin20).

273

10. PERTURBATIONS OF A CENTER

Setting

too

r(9, c) _ E r,,, (9)c"', where r(0, c) = c, m=1

and comparing r(2r, c) and c for sufficiently small positive c, it can be shown that

r1(t) = 1,

r3(2r) =

r2(2r) = 0,

(1 + Or 4

Thus, r(27r, c) < c if a < -1. Therefore, (a) follows. Similarly, r(2r, c) > c if a > -1. Therefore, (c) follows. x2 - xy - y2 = (x - +2 ) (x - (1-2 ) . Set In the case when a

W = - (1 X - (1 +

Then. a =

,/)y

2

and v = x - (1 du

Therefore, changing x and y by u =

(1

)y, the given system is changed to

2

dv

Wv(1 + u),

= ---u(1 + v).

dt

Hence, on any solution curve, the function W2(v

- ln(1 + v)) + (u - ln(1 + v)) =

u2

t2W2v2

+

is independent of t. This implies that in the neighborhood of (0, 0), orbits are closed

curves. This shows that (0, 0) is a center. 0

Example VIII-10-4. The point (0,0) is a center of the system dy dt

= bAy' +

9(y),

y= y'J y2

if (1) b is a nonzero real number, (2) A = 101

"

1J, (3) the entries of the R2-

valued function g"(yj is continuous and continuously differentiable in a neighborhood

of 0, (4) lim

g-6 1Y1

= 6, and (5) there exists a function M(y) such that M is positive

valued, continuous, and continuously differentiable in a neighborhood of 0 and that 8(Mf1)(y) + 8(M 2)(y..) = 0 in a neighborhood of 0, where f, (y-) and f2 (y-) are

the entries of the vector MY + §(y).

Proof The point (0, 0) is either a center or a spiral point. Look at the system dt

=bAy + 9(y)

and

7t- = M(y-

VIII. STABILITY

274

Using s as the independent variable, change the given system to ds = M(y)(bAy+ g"(y)). Upon applying Theorem VIII-1-7, we conclude that (0, 0) is not asymptotically stable as t - ±oo. Hence, (0, 0) is a center. The materials of this section are also found in [CL, §4 of Chapter 15] and [Sai, §21 of Chapter 3, pp. 89-1001.

EXERCISES VIII

VIII-I. For each of the following five matrices A, find a phase portrait of orbits of the system

d:i

= AY.

I 14

1]'

[ 11

[ 15 11]'

31 '

5

1

[1

5

VIII-2. Find all nontrivial solutions (x, y) = (d(t), P (t)). if any, of the system

-xy2, _ -x4y(1 + y) dt = satisfying the condition lim (th(t),tb(t)) = (0,0). c-.+00 VIII-3. Let f (x, y) and g(x, y) be continuously differentiable functions of (x, y) such that (a) f (0, 0) = 0 and g(0, 0) = 0,

(b) (f(x,y),g(x,y)) # (0,0) if (x,y) 0 (0,0), (c) f and g are homogeneous of degree m in (x, y), i.e., f (rx, ry) = rm f (x, y) and g(rx, ry) = rmg(x, y). Let (r, 8) be polar coordinates of (x, y), i.e., x = r cos 8 and y = r sin 0. Set F(8) = f (cos 0, sin 8) and G(O) = g(cos 8, sin 8).

(I) Show that

dr dt

(S)

= rm(F(8) cos 8 + G(8) sin 8),

d8

= rm-1(-F(8)sin8 + G(O)cosO).

dt

(II) Using system (S), discuss the stability property of the trivial solution of each of the following three systems: (i)

4ii

= x2

-

y2

= x3 (x2 + y2) - 2x(x2 + y2)2,

(iii)

d

= x4 -

6x2y2 + y4,

dt

= 2xy; = -y3(x2 + y2);

dy = 4x3y - 4xy3.

EXERCISES VIII

275

VIII-4. Let J be the 2n x 2n matrix defined by (IV.5.2) and let H be a real constant 2n x 2n symmetric matrix. Show that the trivial solution of the Hamiltonian system

a = JHy is not asymptotically stable as t

+oo.

V11I-5. Show that the trivial solution of the system dy'

d = Ay" + 9(t, y-) +oo if the following conditions are satisfied: is asymptotically stable as t (i) A is a real constant n x it matrix, (ii) the real part of every eigenvalue of A is negative, (iii) the entries of the IRI-valued function g(t, y-) are continuous in the region

0(ro) = To x D(ro) = {(t, yl : 0 < t < +oo, Iyj < ro) for some ro, (iv) g(t, yj satisfies the estimate 19(t, y)

1

eo(t, y')Iyj

_<

for

(t, y) E o(ro),

where Eo(t, y) is continuous, and positive, andfun +Ii7 -.o

Eo(t, y) = 0 in L(ro).

VIII-6. Show that the point (0, 0) is a stable improper node of the system

dy = Ay + Ng + 9(y),

y = Iyil,

where (1) A is a negative number,

(2) N is a real constant nilpotent 2 x 2 matrix and N # 0, (3) the 1R2-valued function y'(yj is continuous in a neighborhood of (4) 19(y-) 11-- cly-1I+" for some positive numbers c and v in a neighborhood of CA

Hint. Suppose N =

d

0 0

2

.

If we set yj = r cos 0 and y2 = r sin 6, we obtain

0

= r IA _

EA sin2 cos 0

+ 9i (y-)cos 9 + 92(y-)sin O ,

where 9"(y'i =

{9i]. 92(V

lim r(t) = 0 for t >- 0. t+0

Hence, if r(0) is small, r(t) is bounded by r(0) and This implies that if 1#(0)I is small, g(t) is bounded

and lim y(t) = 0'. Look at

t-+o

y(t) = eAtetN {(o) +

-sN9(y($))d$]

VIII. STABILITY

276

If we set y(t) = e' e'Nu, we obtain

r e-aae-aN9(g(s))ds.

d(t) = y-(0) +

0

Choose e > 0 so that 1 - e >

Then, condition (4) implies that

1

+ 6(+

+00

= 9(0) +

e-aae_a'

0

exists. Now,

(1) if d(+oo) = 0, then d(t) = 0 identically, since, in this case,

d(t) =

f

e-X$e-ajV9(y(s))ds;

+a

0 0, then t line-a`y"(t) = i"(+00);

(2) if t7(+oo) =

02

(3) if iZ(+oc) = RZJ with 2 # 0, then limo ( e _) L

Hence, the point (0, 0) is a stable improper node of the given system.

VIII-7. Determine whether the point (0,0) is a center or a spiral point of the system

dt

= -y,

dt = 2x + r3 - x2(2 - x)y.

VIII-8. Show that the point (0, 0) is the center of the system

_ -x + xy2 - yg.

+ xy3 - y7,

dt = y

Hint. This system does not change even if (t, y) is replaced by (-t, -y).

VIII-9. For the system dty = 27x + 5y(x2 + y2),

3y + 5x(x2 + y2), dt =

find an approximation for the orbit(s) approaching (0,0) as t - +oo. VIII-IO. Show that the point (0, 0) is a stable spiral point of the system

as t-++oc.

d

[y,]

dt

y2

[yj

+

1

- yIn yi + y2 --

[

y2

yi

277

EXERCISES VIII

Hint. If we set yl = r cos 0 and y2 = r sin 0, the given system becomes

dr _

dO

-r'

dt

1

dt = In r'

VIII-11. Suppose that a solution ¢(t) of a system dy _

dt

AJ + §(y-)

satisfies the condition

lim t-r 4(t) = 0 for a positive number r. Show that if (1) A is an n x n constant matrix, (2) the n-dimensional vector W(if) is continuous in a neighborhood of 0, (3) lien 9(Y) = 0, V-6 !y1

then ¢(t) = 6 for all values oft. (Note that the uniqueness of solutions of initial-value problems is not assumed.) VIII-12. Assume that (i) AI.... .. 1n are complex numbers that are in the interior of a half-plane in the complex A-plane whose boundary contains A = 0,

(ii) there are no relations \, = plat + P2 1\2 + + pnan for j = 1, ... , n and non-negative integers p,,... , pn such that pi + - - + pn >_ 2,

(iii) f(y' =

yam' ft, is a convergent power series in ff E Un with coefficients jp1>2

fpEC'. Show that there exists a convergent power series ((u) _

ul'Qp with coefficients Inl>2

_

dg

Qp E Cn such that the transformation if = u" + Q(ur) changes the system - _ A#+ f (y-) to

j

dt

= Au, where A = diag[.1t, A2,4, ... , A ].

VIII-13. Show that the trivial solution of the system dx

- -st(x +X22) + x2ezl+ 2,

is aymptotically stable as t

dr

dt

-22(22 + 22) _ Xlez'+ 2

+oo. Sketch a phase portrait of the orbits of this

system.

VIII-14. In Observation VIII-6-2, it is stated that if one of the then y" is a spiral point. Verify this statement.

VIII-15. Discuss the stability of the trivial solution of the system

j1 = x21 as t - +00.

722 = xi(I - x1)

is not real,

VIII. STABILITY

278

VIII-16. Find the general solution of the system

= 223 + 2j + X122 + 22,

= 3x4 + explicitly.

+ X122 t 2122 + 21X3 + x223,

CHAPTER IX

AUTONOMOUS SYSTEMS

In this chapter, we explain the behavior of solutions of an autonomous system dg = f (y). We look at solution curves in the y-space rather than the (t, y-)-space. dt Such curves are called orbits of the given system. In general, an orbit does not tend to a limit point as t -+ +oo. However, a bounded orbit accumulates to a set as t -4 +oo. Such a set is called a limit-invariant set. In §IX-1, we explain the basic properties of limit-invariant sets. In §IX-2, using the Liapounoff functions, we explain how to locate limit-invariant sets. The main tool is a theorem due to J. LaSalle and S. Lefschetz [LaL, Chapter 2, §13, pp. 56-71] (cf. Theorem IX-2-1). The topic of §IX-4 is the Poincare-Bendixson theorem which characterizes limitsets in the plane (cf. Theorem IX-4-1). In §IX-3, orbital stability and orbitally asymptotic stability are explained. In §IX-5, we explain how to use the indices of the Jordan curves to the study of autonomous systems in the plane. Most of topics discussed in this chapter are also in [CL, Chapters 13 and 16], [Har2, Chapter 7], and [SC, Chapter 4, pp. 159-171].

IX-1. Limit-invariant sets In this chapter, we explain behavior of solutions of a system of differential equations of the form

dg

(IX.1.1)

fW

where y" E Ill;" and the entries of the RI-valued function f (y) are continuous in the entire y-space II8". We also assume that every initial-value problem (IX.1.2)

dy

dt

= Ay),

y(0) = if

has a unique solution y = p(t, y7). System (IX.1.1) is called an autonomous sys-

tem since the right-hand side f (y) does not depend on the independent variable t.

Observe that p(t + r, r) is also a solution of (IX.1.1) for every real number T. Furthermore, At + r, rl) = p(r, il) at t = 0. Hence, uniqueness of the solution of initial-value problem (IX.1.2) implies that p(t +r, rt) = p(t, p(-r, rt)) whenever both sides are defined. For each il, let T(rl) be the maximal t-interval on which the solution p(t, r) is defined. Set C(rt) = {y" = p(t, rl) : t E Z(rl)}. The curve C(rl) is called the orbit passing through the point il. Two orbits C(iji) and C(rj2) do not intersect unless they are identical as a curve. In fact, (IX.1.3)

C(ijl) = C(i@

if and only if 7)2 E C(il, ). 279

IX. AUTONOMOUS SYSTEMS

280

If f (n) = 0, the point ij is called a stationary point. If , is a stationary point, the Generalizing property (IX.1.3) of consists of a point, i.e., C(17) = orbit orbits, we introduce the concept of invariant sets which play a central role in the study of autonomous system (IX.1.1).

Definition IX-1-1. A set M C R is said to be invariant if i E M implies C(n) C M. For example, every orbit is an invariant set.

Remark IX-1-2. If M1 and M2 are invariant sets, then M1 UM2 and M1 f1M2 are also invariant. For a given set f2 C 1R", let Ma (A E A, an index set) be all invariant subsets of S2, then U M>, is the largest invariant subset of Q. AEA

Hereafter, assume that every solution p(t, nJ) is defined for t > 0, i.e., (t : t > 0) C Z(n). In general, lim Pit, nom) may not exist. However, if p(t, n) is bounded for t > 0, the orbit C(171 accumulates to a set as t -+ +oo. This set is very important in the study of behavior of C(n) as t -+ +oo.

Definition IX-1-3. Let

C+(i7-7)

denote the set of all

y"

E

II8"

such that

lim p(tk, n ) = y f o r some increasing sequence {tk : k = 1, 2, ... } of real numbers

k-++oo

lim tk = +oo. The set C+(n) is called the limit-invariant set for the

such that

k-+oo initial point r7.

The basic properties of

are given in the following theorem.

Theorem IX-1-4. If p(t, n) is bounded fort > 0, then C+(n) is nonempty, bounded, closed, connected, and invariant. Proof.

(1) C+ (1-71 is nonempty: In fact, let {sk : k = 1,2,... } be an increasing sequence of real numbers such that lim sk = +oo. Since p(t, n) is bounded for t > 0, k-.+oo

there exists a subsequence {tk : k = 1, 2.... } such that lim tk = +oo and that

lim p(tk,

k-.+oo

exists. This limit belongs to f_+ (17).

(2) The boundedness of C+(n7) follows from the boundedness of p(t,i7) immediately. (3)

is closed: To prove this, suppose that

lim yk = y for 9k E L+(1-7).

k-+oo it follows that ilk = It must be shown that y" E L+(771. Since g k E lira p(tk,t, r 7 ) for some {tk,t : I = 1, 2, ... } such that lim tk,l = +00. Choose tk t-+oo t-.+oo

so that lim tk,t,, = +oo and 19k - P(tk,t,,,'Th !5 ! Then, since ly - P(tk,t4, n)I k-.+oo k

+ ly - ykl, we obtain lim oP(tk,l,,, n) = y E C+(1_7) k

(4) C+(777) is connected: Otherwise, there must be two nonempty, bounded, and closed sets Sl and S2 in R" such that

281

2. LIAPOUNOFF'S DIRECT METHOD

(1) S1 n S2 = 0, (2) S1 U S2

L+(rf)

yl E S1, y2 E S2}. Note that d > 1. Then, So is not empty, bounded, g: distance(y, SI) =

Set d = distance(S{{l, S2) = min{lyl - y21

:

0.

Set also So =

and closed. Furthermore, So n C+(i) =20. Choose two points y"I E SI and y ' 2 E S 2 and t w o sequences {tk : k = 1, 2, ... } and {sk : k = 1,2,...) of

lim tk = +oo, lim sk = +oo, real numbers so that tk < Sk (k = 1, 2, ... ), k-+oo k-+oo r), SO < y"1, and k lim G p1sk, r7) = 92. Assume that limp P(tk, k

Then, there exists a Tk for each k such that 2 and distance(p(sk, r), SI) > tk < Tk < sk and p-(Tk, rl) E So. Choose a subsequence (at : e = 1,2.... ) of {rk : k = 1, 2, ... } so that lim at = +oo and lim p(at, r) = # exist. Then, 2.

Y E So n C+ (Y-)) = 0. This is a contradiction. (5) C+ (1-7) is invariant: It must be shown that if ff E C+ (r), then p1t, y) E L+(1-7) f o r all t E 11(y). In fact, there exists a sequence {tk : k = 1, 2, ... } of real numbers such that Iii o tk = +oo and k lim o P-14, r) = 17. From (IX.1.1) and the continuity k

of P(t,y) of y", it follows that for each fixed t. k

li

+co

tk, n) =

k

UM P(t, Pptk, r1)) = Pit, y) E L+(7).

El

The materials of this section are also found in [CL, Chapter 16, §1, pp. 389-3911 and [Har2, Chapter VII, §1, pp. 144-1461.

IX-2. Liapounoff's direct method In order to find the behavior of pit, r)) as t +oo, it is important to locate L+(rl). As a matter of fact, if p(t, r)) is bounded for t _> 0 and if a set M contains ,C+ (1-7), then p(t. r)) tends to M as t -+ +oo, i.e., lim inf(1p(t, r)) - yj : y E M) = 0. t +a: Otherwise, there must be a positive number co and a sequence {tk . k = 1, 2.... } of real numbers such that lim tk = +00, limo P140 7) exists, and ipjtk, >)) -y"1 eo k k for all y E Jul. Hence, lim -1) V M. This is a contradiction. Keeping this k-.too fact in mind, let us prove the following theorem (cf. [LaL, Chapter 2, §13, pp. 56-71]).

Theorem IX-2-1. Let V(y) be a real-valued, continuous, and continuously differentiable function for Jyl < ro, where 0 < ro < +oo. Set

fDt = J#: V (y-) < e}, St = {y" E DI : V-(y) J(y) = 0}. Mt = the largest invariant set in St, yl fl( F'

a`,

where Vi(y) ' f(y) _ ayJ fJ(y), y J-1

, and f( y) _ yn

.

fn (y)

Suppose

IX. AUTONOMOUS SYSTEMS

282

that there exist a mat number t and a positive number r such that 0 < r < ro, Dt C {y : Iy1 < r}, and Vg(y-) f (y) < 0 on De. Then, L+ (1-7) C Me for all n E Vt. Proof.

Set u(t) = V(p(t,rl")). Then,

dot)

= V9(r(t,r7))'

dtp(t,i) =

du(t) < 0 as long as p(t, t) E Vt. This implies that u(t) < V(1-7) < e dt for r) E Vt as long as p(t, n-) E Vt. Hence, p(t, rl E Dt for t > 0 if ij E Vt. Consequently, Ip(t, rlI < r for t > 0. It is known that C+ (171 is nonempty, bounded, C Vt. closed, connected, and invariant (cf. Theorem IX-1-4). Furthermore, d t) < 0 if Let po be the minimum of V(rl') for Iy7 < r. Then, po < u(t) and it E 1)t. Hence, lim u(t) = uo exists and no > po. This implies that V(y) = uo Therefore,

for all y E C+(t)). Since C+(71 is invariant, V(p(t, y)) = uo for all y E L+(171 and

t > 0. Therefore, VV(p(t, y)) &I t, y)) = 0 for all y E C+ (q) and t > 0. Setting C St if i E Vt. Hence, t = 0, we obtain Vc(y) f (y) = 0 if y E C+(rl, i.e., C+() C Mt for if E Vt. The following theorem is useful in many situations and it can be proved in a way similar to the proof of Theorem IX-2-1.

Theorem IX-2-2. Let V (y) be a real-valued and continuously differentiable function for ally E lR' such that VV(y) f (y) < 0 for all y" E R". Assume also that the

orbit p(t, rte) is bounded fort > 0. Set S = 1g: Vi(y) f(y) = 0) and let M be the largest invariant set in S. Then, L+ (7-1) C M.

The proof of this theorem is left to the reader as an exercise. In order to use Theorem IX-2-2, the boundedness of p(t, advance. To do this, the following theorem is useful.

must be shown in

Theorem IX-2-3. If V (y) is a real-valued and continuously differentiable function for all y" E lR^ such that VV(y) f(y) < 0 for Iyi > ro, where ro is a positive number, and that lim V (y) = +oo, then all solutions p(t, r)7) are bounded for t > 0. 1Q1+00

Proof.

It suffices to consider the case when p(to,rl > ro for some to > 0. There are two possibilities: (1) IP-(t,r1)I > ro fort > to,

(2) IP(t,n')I > ro for to < t < t, and Ip1ti,17)1 = ro for some tl > to. Case (1). In this case, dt V (p( t, r)1) = Vj(p(t, rl) f (p(t, 4-7)) < 0 fort > 0. Therefore, V (r(t,

V (r(to, t))) for t > to. Hence, p(t, tt") is bounded for t > 0.

Case (2). In this case, it can be shown that V(p(t, 71)) <_ max IV(P(to,' )),

max(V(y) :

1yi <_ ro)1

283

3. ORBITAL STABILITY

in a way similar to Case (1). The details are left to the reader as an exercise.

0

IX-3. Orbital stability In this section, we introduce a concept of stability (respectively asymptotic stability) in a sense different from the stability (respectively the asymptotic stability) of Chapter VIII (cf. Definitions VIII-1-1 and VIII-1-3). We denote again by C(rf the orbit passing through fl. Also, assume that every solution p(t, if) is defined for

t>0. Definition IX-3-1. An orbit C(sjo) is said to be orbitally stable as t -+ +00 if, for any given positive number e, there exists another positive number b(e) such that distanee(p"(t, rl"), C(rlo)) < c for t > 0 whenever 14 - rlo1 < b(e).

This definition is independent of the choice of the point ijo on the orbit C(ilo). If C(i-lo) = C(#,) and if the condition of this definition is satisfied in terms of ilo, then the same condition is satisfied also in terms of , with a different choice of b(e). The following example illustrates these new concepts.

Example IX-3-2. The orbit of the system d.

[y2] =

passing through the point ij = I

[c?].

[ - 2]

] is given by

yj = r(f[)cos{r(i))t + 9(t)}, where it =

yl +y2

y2

= r(ij)sin{r(if)t + 9(f))),

It is easy to show that if r(n) - r(q) = b > 0 and

9(ili) = 9(ij2), then distance(p(t, f2), C(#,)) = b. Hence, the orbit C(ij) is orbitally stable as t -+ +oo for every i ) . However, i h ) - p (b ff) I = r(il') + r(')2 ) since 1r(ffi)t+9(ili)1-1r(il2)t+9(il2)j = 6t (cf. Figure 1). Therefore, every nontrivial solution p"(t, i)) is not stable as t +oc in the sense of Definition VIII-1-1.

Definition IX-3-3. An orbit C(fo) is said to be orbitally asymptotically stable

as t - +00 if (i) C(ilo) is orbitally stable as t +oo, (ii) there exists a positive number 6o such that lim

C(i )) = 0

whenever {il - iloj < 60.

This definition is independent of the choice of the point 4 on the orbit. However, the choice of 6o depends on rjo.

Consider a system of differential equations (IX.3.1)

= f (y,

IX. AUTONOMOUS SYSTEMS

284

where the entries of the R"-valued function f is continuously differentiable on the entire y-space W1. Assume also that the solution At, rjo) is periodic in t of period 1 (i.e., #(t + 1, ito) = p1t, i"Jo) for -oo < t < +oo) and that f'(p1t, ijo)) 54 0. The system of linear differential equations di7

(IX.3.2)

=

Of

is called the first variation of system (IX.3.1) with respect to the solution pit, rlo). The coefficients matrix of (IX.3.2) is periodic in t of period 1.

Let pi, p2, ... , p be the multipliers of (IX.3.2) (cf. Definition IV-4-5). Since 2 p-(t,

(pit, o)) d p1t, ijo), linear system (IX.3.2) has a nontrivial periodic

solution fjt, im). This implies that one of the multipliers must be 1. Set pl = 1. The following theorem gives a basic sufficient condition for orbitally asymptotic stability.

Theorem IX-3-4. If n - 1 multipliers p2,... , p, satisfy the condition ipiI < 1 (j = 2,... , n), then the periodic orbit C(rjo) is orbitally asymptotically stable as

t-4+oo. Prof. We prove this theorem in five steps. It suffices to find an (n - 1)-dimensional manifold M in a neighborhood of the point so that (1) M is transversal to the orbit C(i) at ilo; i.e., the tangent of C(yo) is not in the tangent space of M at ilo, (2) there exist two positive numbers K and r such that

Iu(t,n - p1t,i )I <- KIn -

iole-°:

for

i EM and t>0

(cf. Figure 2).

FIGURE 1.

FIGURE 2.

Step 1. There exists an invertible n x n matrix P(t) whose entries are real-valued, continuously differentiable, and periodic in t of period 1 (or 2) such that the transformation (1X.3.3)

6 = P(t)v"

3. ORBITAL STABILITY

285

changes (IX.3.2) to (IX.3.4)

dt =

A = [ 0 B,

AV,

where B is a constant (n - 1) x (n - 1) matrix (cf. Theorems IV-4-1 and IV-4-3). Since the absolute values of n -1 multipliers are less than 1, there exist two positive numbers Co and a such that for t > 0. Iexp[tB] I < Coe-tot (IX.3.5) A fundamental matrix solution of (IX.3.2) is given by

exp[tA] = [1

1/(t) = P(t) exp[tA],

o

0 1 exp[tBj J

This implies that the first column vector of P(t) is a periodic solution of (IX.3.2) and all the other columns of W(t) are not periodic. Hence, assume that

P(t) _ [d AtA),

(IX.3.6)

Q(t),

where Q(t) is an n x (ii - 1) matrix. Note that det[P(t)] 56 0.

Step 2. Change (IX.3.1) by (IX.3.7)

y = z' + At, rlo)

to Tt

(IX.3.8)

8

ay(p1t,

dt =

no))z + h(t, z

where

h(t,z-) = f(z + pit,t)o)) - I(plt,no)) -

y(pit,qo))z

It is easy to show that h(t, 6) = 0 and az (t, 6) = 0.

Step 3. Change (IX.3.8) by the transformation i = P(t)w". P(t)-1 18 (p-(t, jo))P(t) - d (,t)

,

Since A =

it follows that

L

(IX.3.9)

dt = Aw + #(t, tip),

where g(t, w) = P(t)-'h(t, P(t)w). Note that g(t, 6) = 0 and 8w (t, 0') = 0. This implies that for any given positive number r, there exists another positive number

K(r) such that (IX.3.10)

19(t, w1) - g"(t, w2)I < K(r)[w1 - t62I

whenever kw"II < r and I w2I < r and that (IX.3.11)

lim K(r) = 0.

r-.0

= ti and w2 = 0. Then, (IX.3.10) becomes (IX.3.12) jg"(t,w)I < K(r)Iti'j for t > 0. Set ti

for

t>0

IX. AUTONOMOUS SYSTEMS

286

Step 4. Let us write (1X.3.9) in the form (IX.3.13)

_

dv

= 9i(t,19),

dt

dt

By + g2(t, w),

where v and g2 are (n - 1)-dimensional vectors w" _ V and g = I _ J . System J

111

(IX.3.13) can be changed to the system of integral equations t

u(t,

C))ds,

+ J00

(IX.3.14)

v(t, ) =

p ( tB1

+ f exp [(t - s)BI g2(s, w(s, ))ds, 0

where 1; is an (n -1)-dimensional arbitrary constant vector. In this step, a solution w(t, t) of (IX.3.14) will be constructed in such a way that ie-ot

ko 1

(IX.3.15) for

t>0

(IX.3.16)

and

1t1 < 60, where ko and bo are suitable positive numbers. First fix three positive numbers ko, ro, and 6o so that Ico > 2, kobo < ro, and K (r) < 2 for 0 < r < ro. Then, from (IX.3.5), (1X.3.10), (IX.3.11), (IX.3.14), and a (IX.3.15), it follows that + t

(I)

< K(ko[

t9i (s,+u(s,&ds

JL0

K(a

itt

e-O°ds

If1)

and

jexE(t -

exp (tBJ ( +

e-2otlrj + S

t f eo"ds 0

Se

of

I1+

Next, set lIJt ({) = sup

K(

(eot 1

(1 +

J+9i(s,t,ii(s,())ds

< K(ko

< ko1 1e-OL

(t, 4)1 : t > 0) if the entries of an R"-valued func-

tion t(i(t, £) are continuous for (IX.3.16) and Then,

I

)

-

l}} < kolle-ot for (1X.3.16).

ft 9i(s,tG2(s,&ds l +oo

at

< 2IItLI -

1211(&-ot

287

3. ORBITAL STABILITY and

f exp[(t-s)B]92(s,))ds t

f+ exp((t-s)B](s,(IV)

- 2[[i1 -

<

<

Q

&IOe-oc

for (IX.3.16) if the entries of R'-valued functions 1(t, )) and (t, t; are continuous (j = 1,2) for (IX.3.16). for (1X.3.16) and that J ,(t,£)I < Let us define successive approximations as follows: ko1&-OC

15m(t, = L

gm(t,

where %ao(t, = 0 and f=

ao

g1(s,(s,))ds,

exp[tB) +

J0t

exp[(t-s)B]92(s, '

Then, it can be shown without any difficulties that

m

exists

lim >im(t,

m --+oc

uniformly for (1X.3.16) and the limit %i(t, { is a solution of integral equations (1X.3.14) satisfying condition (IX.3.15) for (IX.3.16). Note that i(O,t;) 0

where a(t) = f gl (s, z/i(s, })ds. This implies that 00

Step 5. Set

00,6 = Then, (IX.3.18)

Plt,vio)

is a solution of system (IX.3.1) such that

At, f) - pjt, i o)[ 5 Ko{f [e-°`

for

(IX.3.16),

where Ko is a positive constant. Define an (n - 1)-dimensional manifold M by

M = (g=4(0'6: 01<<601Using (IX.3.6), (IX.3.17), (IX.3.18), and (1X.3.11), we obtain

(0, f) = P(0)W (0, a + i?o =

dP (0,

det dp(0, 'b), Q(0)]

-76

40) + Q(O) 0.

+ rlo,

IX. AUTONOMOUS SYSTEMS

288

Thus, the manifold M is transversal to the orbit C(fjo) at rp. Note that (t, t At, ¢(0, 6). Moreover, by continuity of the solutions of (IX.3.1) with respect to the initial conditions, there exist two positive numbers 61 and ro such that for i satisfying the condition 11 -401

there is a real number r(q such that 1r(q-)l < ro and p(r(g71,'7) E M. Therefore, (IX.3.18) implies that

1pit+r(17),q - P-(t,il0){
for

t > 0.

Thus, C(ilo) is orbitally asymptotically stable as t --+ +oc.

Remark IX-3-5. If the entries of an R'-valued function f (t,1/) is continuously differentiable on the entire f.-space lRn and system (IX.3.1) has a periodic solution p ( t , i l o ) of period 1 such that f (p(t, i )) j4 0, there exist n - 1 vectors 4j(t) C

Rn (j = 2, ... , n) such that (i) the entries of vectors 9'J (t) (j = 2,... , n) are continuously differentiable and periodic in t of period 1,

(ii) n vectors

dpig0),

q"2(t), ...,

form an orthogonal system.

Proof.

If n = 2, it is easy to find 42(t). For n > 3, there is the recipe. f(At, 40)) dpdtilo} Note first that f(p1t,ilo))00. Set g1(t)= f (pit, no)) f (pit, ilo)) where u" b denotes the usual dot product. Since fl(t) is smooth, there exists a constant vector i71 such that nl ill = 1 and that ili + qi (t) A 0 for 0 < t < 1. Choosing an orthonormal set {ill.... , in I, where n, are constant vectors, define

-

Q'n(t) = T)n -

+ai(t) (rJi + ji(t)) on(t)

(1 = 2, ... ,

n),

where ah(t) _'h - ji(t). 0 For this construction, see, for example, (U).

Example IX-3-6. The orthogonal system of Remark IX-3-5 is useful to study solutions of (IX.3.1) in a neighborhood of a periodic solution. To illustrate this application, let us look at a system dy dt

=f

where f is an 1R3-valued function whose entries are continuously differentiable on the entire. y-space R3. Assume that (SI) has a periodic solution p(t, irjo) of period

I such that f(plt,r'b)) 54 0. In this case, there exist two vectors j2(t) E IR3 and 4fi(t) E L3 such that

289

3. ORBITAL STABILITY

(i) the entries of vectors 4"4(t) (j = 2,3) are continuously differentiable and periodic in t of period 1,

(ii) three vectors d

Note that

dp(ttrlo)

Q , 4'2(t), and 43(t) form an orthogonal system.

= f (plt, no)).

For any fixed real non-negative number r, the set

P(T) = {l7-,m) + u142(r) + u2Q3(T) (ul,u2) E R2} at piT,, o). Letting t, u1, and

is the plane which is perpendicular to the orbit u2 be three functions of r to be determined, set (IX.3.19)

f(t) = p( r, no) + u1Q2(r) +

u243(T)

From (IX.3.19) and the given system (S1), we derive dt

du1

{/,-/ f(f(t))dT = f(P(r>io)) +

This yields (IX.3.20) df11

Wdue

and

dT

42(r) +

y' dt

2(T) f(y(t))dr dt

-

(

2(t)'

((t)

due _ dT 93(r)

+ u1

42(T) 1 dr l u1 dQ2(T)\

dr

dge(r)

+ (f(p1r,TIo))

d9d(T)) uz]

dT

((t)

d93(r)1

u1 - 93(t)

dg3(r))

= [f(PIT'770)) f(PIT,, )) + (fcvlr,no)) (IX.3.21)

+ u2 dQ3(r)

d-,

dT J u2, dr

u1

x [f(Ar, *lo}).N(t))} -', f(

where a" b denotes the usual dot product and we assumed that

q"1(r} = 1 (j =

2, 3).

Note that

AM)) = RAT, 7lo))+ul

89

(PIT,r, ))42(r)+

u2

f(YlT,r/o))93(T)+O(ItII +Iu2I) 09

Hence, from(IX.3.20) and (IX.3.21), we derive dt 77.

= 1 + 00U11 + Iu21)

IX. AUTONOMOUS SYSTEMS

290

and (IX.3.22) dul

dr

42(r) '

- (o) d7-

93(T) '

ul + [i(T).

l

.

_d_r)

W

Jut -

(0)

(FFIr, 1I0))42(r) 1

d9drr)) -

ul +

(r))1 u2 + O(lul l + 1U21),

[i(r).

(i(t). ddrr)) ul - (t(t). d (r)

U2

r, 90))93(r) 11 U2

J U2 + O(lulI + 1U21)

Let Q(t) be the 3 x 3 matrix whose column vectors are (at, qo)), fi(t), and ,fi(t), i.e., Q(t) = [ f (p1t, rlo)) 6(t) q"3(t)! . The transformation w = Q(t)v changes the linear system dtr, (S2)

_

L(PIt,rl *6

to

dv'

=

0

/31(t)

32(t)

0

atl(t) a12(t)

0

a21(t)

16,

a22(t)

Using these notations, write (IX.3.22) in the form dul d7-

(IX.3.23) 1

= all(r)ul + a12(r)u2 + 91(T,ul,u2),

dug

dr

= a21(r)u1 + a22(r)u2 + 92(T,u1,u2),

where

K(r)(lul-V11+1u2-V21)

191(r,u1,u2) -

(i = 1,2)

with K(r) > 0 and limK(r) = 0 for lull + lull < r and Ivll + 1v21 < r. Observe r-O that the two multipliers of the linear system

j

du'

= [all(t) a12(t) a21(t)

a22(t)

U

are also multipliers of system (S2). Therefore, using system (IX.3.23), Theorem IX-3-4 can be proven. In general, we obtain more precise information concerning the behavior of solutions in a neighborhood of a periodic solution in this way. The materials of this section are also found in 1CL, Chapter 13, §2, pp. 321-3271.

4. THE POINCARE-BENDIXSON THEOREM

291

IX-4. The Poincare-Bendixson theorem In this section, we explain the structure of C+(i-) on the plane. Consider an R2-valued function f (y) of y E R2 such that the entries of f (y7) are continuously differentiable on the entire g-plane R2. Denote again by pit, ill the unique solution of the initial-value problem dg = f (y-), y(0) = i. The main result of this section is the following theorem due to H. Poincare [Poll] and 1. Bendixson [Ben2].

Theorem IX-4-1. Suppose that the solution pit, qo) is bounded for t > 0 and that C+(ilo) contains only a finite number of stationary points. Then, there are the following three possibilities: (i) C+(i-Io) is a periodic orbit,

(ii) C+(i o) consists of a stationary point, (iii) C+(i)) consists of a finite number of stationary points and a set of orbits each of which tends to one of these stationary points as I ti tends to +oo. To prove this theorem, we need some preparation.

Definition IX-4-2. A finite closed segment t of a straight line in R2 is called a transversal with respect to f if f (y1 0 0 at every point on t and if the vector f (y) is not parallel to t at every point on t (cf. Figure 3). Observation IX-4-3. For every transversal t and every point i , the set tnC+( contains at most one point (cf. Figures 4-1 and 4-2). I

PU,

FIGURE 3.

FIGURE 4-1.

FIGURE 4-2.

The following lemma is the main part of the proof of Theorem IX-4-1.

Lemma IX-4-4. If pit, qo) is bounded fort > 0 and if there exists a point i7 E C+(7) such that C+ (11) contains a nonstationary point, then C+(i ) is a periodic orbit.

Proof.

We prove this lemma in three steps. Since C+(qo) is invariant and rj1 E C+(7 ), it follows that p1t,i7l) E C+(jo) for all t. Furthermore, C+(ijt) C C+(io) since C+(ilo) is closed.

Step 1. Let rj be a nonstationary point on C+(t7 ). Also, let t be a transversal with respect to f that passes through I. Then, t n C+(10) = {77} since rj E G+(ijl) C C+(no).

292

IX. AUTONOMOUS SYSTEMS

Step 2. Since tj E C+(r)l ), there exists a sequence {tk : k = 1,2.... } of real lira tk = +oo and p1tk, ill) E t (k = 1, 2, ... ). Note that numbers such that m+oo p(tk, ill) E C+(ilo). Therefore, p(tk, m) = #(k = 1,2.... ). This implies that there exist two distinct real numbers rl and T2 such that it = p(rl, ill) = (r2, ill) and hence p(t, rl = p(t, p(ri , ili )) = p1t, p(r2i ill )). This, in turn, implies that p(t + T1, ill) = p(t + r2i ill) for t > 0. Therefore, the orbit C(ill) is periodic in t of period Irl - r21. Furthermore, C(ill) C C+(rjo). Note that there is no stationary point on C(qj ).

Step 3. Since C+(ilo) is connected, it follows that distance(C(ijl), C+(ilo)-C(ni)) = 0. Hence, if C(ijl) # L+ (10), there exists a sequence { k E C+(i o) : k = 1, 2.... } such that tk C(rj,) (k = 1, 2, ...) and lim £k = { E C(ill ). Assume that k-.+oo

there exists a transversal t such that E e and lk E t (k = 1, 2.... ). Note that l; E C(ill) C C+(ilo). Then, k = t c C(ill) (k = 1, 2, ... ). This is a contradiction. Thus, it is concluded that C+(rlo) = C(iji). Now, we complete the proof of Theorem IX-4-1 as follows. Proof of Theorem IX-4-1If C+ (, ) does not contain any stationary points, then (i) follows (cf. Lemma IX-4-4). If C+(no) consists of stationary points only, we obtain (ii), since C+(ilo) is C+(7-) for connected. If f-+ (Q contains stationary and nonstationary points, then any point it E C+(ilo) does not contain nonstationary points (cf. Lemma IX-4-4). This is true also for t < 0. Hence, (iii) follows.

Observation IX-4-5. In cases (i) and (iii), the set 1R2 - C+(ilo) is not connected. Furthermore, if an orbit C(i)) is contained in C+(i o), two sides of the curve p1t, tl") belong to two different connected components of R2 - C+(t ). In fact, if we consider a simple Jordan curve C which intersects with the orbit C(tl at n transversally, then the curve p(t,,o) intersects with C in a neighborhood of two distinct points on C infinitely many times (cf. Figure 5).

Theorem IX-4-6. If C(ijo) n C+(ilo) 0 0, then C+(ijo) = C(rjo) and either no is a stationary point or the orbit C(ilo) is periodic. Proof In this case, C(ilo) C C+(ilo). If C+ (8o) contains nonstationary points, the orbit C(i o) consists of nonstationary points. Choose a transversal a at ilo. Then, it can be shown that C(ilo) is periodic in a way similar to the proof of Lemma IX-4-4, since r1o E C+(tlo).

Observation IX-4-7. If C(ilo) n L+(no) = 0, then lim distance(plt,ilo) and t-+oo

,C+(Q) = 0. It follows that if C+(ilo) consists of a stationary point then lim p(t, irjo) If C+(ijo) is a periodic orbit, then C+(rjo) is called a limit t+oo cycle (cf. Figures 6-1 and 6-2).

293

5. INDICES OF JORDAN CURVES

FIGURE 6-1.

FIGURE 5.

FIGURE 6-2.

The materials of this section are also found in [CL, Chapter 16, §§1 and 2, pp. 391-3981 and (Har2, Chapter VII, §§4 and 5, pp. 151-1581. In [Hart[, the PoincareBendixson Theorem was proved without the uniqueness of solutions of initial-value problems.

IX-5. Indices of Jordan curves In this section, we explain applications of index of a Jordan curve in the plane to the study of solutions of an autonomous system in 1R2. Consider again an R'-valued function f (y-) of y E 7,k2 whose entries are continuously differentiable on the entire y-plane R. Denote also by pit, J) the unique solution of the initial-value problem

dg dt

= f (y1, yi(0)

To begin with, let us introduce the concept of indices of Jordan curves. Let C be a Jordan curve y = il(s) (0 < s < 1) with the counterclockwise orientation (cf.

Figure 7). Assume that &(s)) # 0 for 0 < s < 1. Set u(s) = f ('l(s))

If(s))1

(0 <

s < 1). Then, il(s) is the unit vector in the direction of f'(il(s)). There exists a real-valued continuous function 0(s) defined on the interval 0 < s < I such that a (s) (s)

_ Isin0(s)I'

Definition IX-5-1. The index of the Jordan curve C with respect to the vector 0(1) - 8(0) .

field f (y) is given by I f{C) =

2r,

This definition is independent of the choice of a parameterization rj(s) of C and the function 0(s). Let us denote by D the domain bounded by C (cf. Figure 7).

Observation IX-5-2. If the domain D is divided into two domains DI and D2 by a simple curve £, then 1 f{C) = l f{",) + If-(8D2) if f (rl) 0 on £ U C, where 8D? (j = 1, 2) denote the boundaries of domains DI and D2, respectively, as the portions of I f{8Di) and I j{d Dt) along £ canceled each other (cf. Figure 8).

Observation IX-5-3. If All) A 0 on C U D, then I f{C) = 0. To show this, divide V into sufficiently small subdomains, use the fact that the vector field 19(s) has no change on a small subdomain, and apply Observation IX-5-2.

IX. AUTONOMOUS SYSTEMS

294

Observation IX-5-4. Assume that a point d E V is a stationary point, i.e., f(d) = 0. Assume also that f(,) 96 0 on C U V except at a. Then, If(C) depends only on aa. Therefore, we define the index of an isolated stationary point d with respect to f by Ilea) = I f(OV), where V is a neighborhood of as such that there are no stationary points in V other than a.

Observation IX-5-5. If D contains only a finite number of stationary points N

dl, d2, ... , dN, then I j
Observation IX-5-3.

Observation IX-5-6. If f (y) # 0 on C and if f(,7(s)) is tangent to C at every point i(s) (0 < s < 1) of C, then I1{C) = 1 (cf. (CL, Chapter 16, §4, Theorem 4.31).

Proof.

It is evident that the index of C with respect to f and the index of C with respect

to the tangent vector

to C are the same, i.e.,

If4C) = hn)(C)Assume without any loss of generalities that # ( 0 )=# ( I ) ) = 0

and

i

(s) > 0

for

0 < s < 1,

where rig (s) (j = 1, 2) are the entries of the vector r"(s) (cf. Figure 9).

-

__11

(0.0)

FIGURE 7.

FIGURE 8.

n2=0

FIGURE 9.

For 0
V(T,s) =

if

T = s,

if

T < s (mod 1),

1

(T, s) _ (0, 1).

Then, v(T, s) is continuous in (T, s) for 0 <_ r < s < 1. Hence, studying how 16(0, s)

changes from s = 0 to s = 1 and how i6(T,1) changes from T = 0 to T = 1, we obtain Ij.(.)(C) = 1. 13

5. INDICES OF JORDAN CURVES

295

Observation IX-5-7. If C is a periodic orbit of the autonomous system di dt

(IX.5.1)

f(+1+

then 1 f-(C) = 1.

Observation IX-5-8. If C is a periodic orbit of autonomous system (IX.5.1), then the domain D contains at least a stationary point (cf. Observations IX-5-3 and IX-5-7).

Observation IX-5-9. Suppose that f(it(s)) is tangent to C only at a finite number of points ij(rl ), #(7-2),. .. ,1t(TN) on C. Assume also that at each point it(rk), either (1) p(t,it(rk)) V V for ItI < 5k, or

(2) p(t, it(rk)) E D for 0 < ItI < bk,

... , SN. Let us call rj(rk) an exterior (respectively interior) contact point in case (1) (respectively (2)) (cf. Figure 10). Let us denote by E (respectively H) the total number of interior (respectively exterior) contact points among N points it(-r1), ij(r2), ... , #(T-N). Then E+H = N and for some sufficiently small positive numbers 51,

(IX.5.2)

1 f-(C) =

E-H+2 2

Sketch of proof.

Let rj(r,) and fj(rk) be two consecutive contact points such that T. < rk. Then, (i) if both of these two points are exterior contact points, then the tangent to C

changes 7r in angle more than the vector field f does from the point #(T.) to the point fj(rk) (cf. Figure 11-1), (ii) if both of these two points are interior contact points, then the vector field f changes it in angle more than the tangent to C does from the point #(T,) to the point it(rk) (cf. Figure 11-2), (iii) if these two points are an exterior contact point and an interior contact point, then the tangent to C and the vector field f change in the same amount in angle (cf. Figures 12-1 and 12-2). Since the total amount of change of the tangent to C in angle is 27r (cf. Observation IX-5-6), we arrive at formula (IX.5.2). 0

q(

FIGURE 10.

FIGURE 11-1.

FIGURE 11-2.

IX. AUTONOMOUS SYSTEMS

296

Observation IX-5-10. Let d be an isolated stationary point. Assume that a neighborhood V of d is divided into a finite number of sectorial regions S1, S2,... , SN

by a finite number of orbits C(rji ), C(il3),... , C(iv) in such a way that (a) rli, Tt2,

-

, nN E 8V,

(Q) either p1r,rjk) E V fort > 0 and tends to 99 as t -+ +oo, or p5r,ilk) E V for t < 0 and tends to d as t - -cc (cf. Figures 13-1 and 13-2).

FIGURE 12-1.

FIGURE 12-2.

FIGURE 13-1.

FIGURE 13-2.

Let us assume also that each sectorial region Sk satisfies one of the following four conditions: (I) if it E Sk and {il - d] is sufficiently small, then C(r)) C Sk and pit, q-) tends to +oo (cf. Figure 14), d as Itl (II) if i) E Sk and Jil - dl is sufficiently small, then pit, i)) E Sk only on a finite

t-interval ak < t < Yk (cf. Figure 15),

FIGURE 15.

FIGURE 14.

(111) if it E Sk and ail - dl is sufficiently small, then p1t,i) tends to dd in Sk as t +oc, but plt, q") does not tend to dd in Sk as t -+ -oo (cf. Figure 16-1), (IV) if ij E Sk and Jrj - al is sufficiently small, then p1t,71) tends to ad in Sk as t -. -oc, but plt, q+) does not tend to dd in Sk as t - +oo (cf. Figure 16-2). The sectorial region Sk is said to be elliptic (respectively hyperbolic) in case (I) (respectively (II)). In cases (III) and (IV), the sectorial region Sk is said to be parabolic. Let us denote by E (respectively H) the total number of elliptic sectorial regions (respectively hyperbolic sectorial regions) among St, ... , SN. Then, it is known that (IX.5.3)

E--H+2 2

This result is similar to formula (1X.5.2) of Observation IX-5-9. For a proof in detail, see (Har2, Chapter VII, §9, pp. 166-172).

297

5. INDICES OF JORDAN CURVES

Observation IX-5-11. Suppose that the vector field f (y e) depends on a parameter e E 0 continuously, where 0 is a connected set. In this case, if f (if(s), e) # 0 at every point if(s) on the Jordan curve C for all e E A, then the index I f(.,()(C) is a constant independent of e. In fact, in this case, the integer I f( E)(C) depends on e continuously, and hence it is a constant. Example IX-5-12. If an isolated stationary point d is a node, a spiral point, or a center, then E = H = 0. Hence It{a") = I.

Example IX-5-13. If an isolated stationary point d is a saddle point, then E = 0 -1 (cf. Figure 17). and H = 4. Hence

FIGURE 16-1.

FIGURE 16-2.

FIGURE 17.

Example IX-5-14. More general cases of isolated stationary points a are shown by Figures 18-1, 18-2, and 18-3. In fact,

1E = 0, H = 2 and hence II-(a") = 0 E = 2, H = 0 and hence It-(d) = 2 E = 1, H = 1 and hence l1-(d) = 1

in the case of Figure 18-1, in the case of Figure 18-2, in the case of Figure 18-3.

Example IX-5-15. Let p(z) be a polynomial in a complex variable z with complex coefficients.

Set y' = t(zj and y2 = !3(zJ (i.e., z = yl + iy2) and regard the

differential equation (IX.5.4)

ddt'

dz

= p(z) as a system of two differential equations

_ 9(p(yl + iy2)],

dtY2 = (p(yl + iy2)]

on the y"-plane. It is easy to see that, if zo = 171 +ir12 is a zero of p(z) of multiplicity 111 m, then if#?) = m, where AM _ R(p(yl + iy2)] and i) = [112, . For example, %P(YI + iy2)] if p(z) = iz2, system (IX.5.4) becomes

dtl

-2y1Y2,

ddt2

= y1 -

y2

The point 0 is an isolated stationary point and II
e > 0 (cf. Observation IX-5-11).

[!1], then, 1-

(0) = 2 for

IX. AUTONOMOUS SYSTEMS

298

Example IX-5-16. Assume that f ( = fl (pl'Y2) satisfies the condition f (arty) f2(y142)

= aP f (rte, where A is a real variable and p is an integer. Set y = r

os0 I. Then,

= f(it) can be written in the form

the autonomous system

dr

r dO

dt = rPFI (0),

dt

rPF2(0),

where

F1(0) = f 1(cos 8, sin o) cos 0 + f2(cos 0, sin 0) sin 0, Sl F2(0)

= -f1(cos0,sin6)sin0 + f2(cos0,sin0)cos0

(cf. Exercise VIII-3). If a ray to is defined by F2(0) = 0, then to is an orbit of the system dff = f (r))). Thus, the entire y"-plane can be divided into sectorial regions by those orbits QB determined by equation F2(0) = 0. For example, in the case of system (IX.5.5), we obtain Fl (0) = - sin 0 and F2(0) = cos 0. Observe that

dr dt

>0

for

-r<0<0,

=0

for

0=-r and 0,

<0

and

>o

for

=0

for

<0

for

0 = - and 2

dt

for 0<0
-2 <0<2, 2<0<

.

2'

3 1r

Hence, E = 2 and H = 0 (cf. Figure 19). Therefore, I f-{0) = 2 (cf. Example IX-5-15).

FIGURE 18-1.

FIGURE 18-2.

FIGURE 18-3.

FIGURE 19.

The materials of this section are also found in [CL, Chapter 16, §§4 and 5, pp. 398-402] and [Har2, Chapter VII, §§2 and 3, pp. 146-151, and §§8 and 9, pp. 161-1741. For a geometric and topological treatment of indices, see (Mil.

EXERCISES IX

IX-1. For each of the following three systems, using the given function V (x, y), show that all orbits are bounded as t -. +oo .

299

EXERCISES IX

_

(1)

Y2;

V(x,y) = 3x2 +4

_ -yx2 + 3x,

-xy2 - 4y,

Y2;

= -x3 - y,

= y,

(2)

= y,

V(x,y) =

dy

V(x,y) = x4 +2

= - (xs - 3x4 + 2x3 + 120x2 - 23x + 5) - (1 + x2)y, + J (ss - 3x4 + 2x3 + 120x2 - 23s + 5)ds.

2

0

IX-2. Show that every solution y(t) and its derivative

(t) of the differential L(t)

equation j2 + dt + y3 =0 tend to 0 as t -i +oo. IX-3. Consider an autonomous system yj

(S1)

__

Wt [y2]

fj(11j,112)

[f2(yi,y2)

'

where f j and f2 are continuously differentiable on the entire (yj, y2)-plane. Assume

that (1)

fl(yl,y2)>0 fi(yi,y2) < 0

for y2>0 and -oo
(2) fj(yi,1) > 1 and fi(yi, -1) < -1 for -oo < yl < +00, (3) f2(yi,1) = 0 and f2(yi, -1) = 0 for -oo < y1 < +oo, (4)

1

f2(11j,y2) <_ YI(112 2 - 1)

f2(Y1,Y2) ? Y1 (Y22 - 1)

for

jy2j < 1

and

for

jy21 < 1

and

0 < yi < +oo, - oo < yi < 0,

Find G+((n1,n2)) for (771,772) such that In2t < 1.

Hint. There are two possibilities: Case 1. The solution (yi(t),y2(t)) of (Si) that satisfies the initial condition (C)

yi(0) = 771

and

112(0) = 172

is bounded as t -+ +oo.

Case 2. The solution (y1(t),y2(t)) of (SI) that satisfies the initial condition (C) is unbounded as t +oo. In Case 1, G+((nj, n2)) is either {(0, 0)} or a periodic orbit. In Case 2, G+((nj, 92))

is{(x,±1):-oo
IX. AUTONOMOUS SYSTEMS

300

Examples. (a) Every orbit in Iy21 < 1 of the system with fl(yl,y2) = y2 and f2(yi,y2) _ yl(y2 - 1) is periodic.

(b) The stationary point (0,0) is a stable spiral point if fl(yl,y2) = y2(2 sin(yly2)) and f2(yj,y2) = 2yi(y2 - 1). (c) The stationary point (0, 0) is an unstable spiral point if ff(yi, y2) = y2(2 +

-

sin(yiy2)) and f2(yi,y2) = 2y, (y22 1). Verify these statements by using the function V(yl, y2) = yi - ln(1 - y2).

IX-4. Let us consider a system (52)

dt

-

W= [y2 ]

,

f(U _ [f2(Y-)

where f, (y-) and f2(y-) are continuously differentiable with respect toy in a domain Do C R2. Assume that (i) system (S2) has a periodic orbit p-(t, ik) of period 1 which is contained in the domain Do,

(ii) RA t' v)

6,

(1 (

(5t, o))) dt is negative. 0 Show that the periodic orbit p(t, jo) is orbitally asymptotically stable as t -' +00. (iii) the integral

i (pjt, qo)) +

,9y

Hint. This is called Poincnre's criterion. Look at 49f

dzV

dt

=

Then, I det Y(1)I < 1 for the fundamental matrix solution Y(t) of this system such that Y(0) = 12 (cf. (4) of Remark IV-2-7). Hence, an eigenvalue p of Y(1) must satisfy the condition dpi < 1. (The other eigenvalue of Y(1) is 1.) Now, use Theorem IX-3-4.

IX-5. Assume that two functions f (x) and g(x) are continuously differentiable in

x E R. Assume also that the differential equation dt2 + f (x)

drt

+ g(x) = 0 has

a nontrivial periodic solution x(t) of period I such that f f (x(t))dt > 0. Show 0

that (x(t), x'(t)) is an orbitally asymptotically stable orbit as t phase plane.

+00 in the (x, x')

Hint. Use Exercise IX-4-

IX-6. Show that there exists a nontrivial periodic orbit of the system dI7

=f (y),

y=

Ly2J,

W) Lf2h (Y-)

where (1) the entries of the 1R2-valued function f (y-) is continuously differentiable on the entire y-plane,

301

EXERCISES IX

and f(y)

(2) f(6) (3)

ifil96

of

(O) = i, eft (0) = 8,

ay1

OW

liM

,

2Z-2 (0)

+oo(ft(Y1,Y2) + yr) and

(4) Ivll+l

(p)

= -2 and

ay1

= 1, +oo(f2(yt,N2) +

1Y11+liM

yz) exist.

IX-7. Given that Y1 - y2

y2

find the total number E of elliptic sectorial regions, the total number H of hyperbolic sectorial regions, and the total number of parabolic sectorial regions in the = f (y, E) in the neighborhood of the isolated stationary point 0 of the system following two cases: (i) E = 2 and (ii) E _

dt

2. Also, find I y(0) for e 96 0.

IX-8. Let us consider a system

dt = f (y),

(S3)

where the entries of the l 3-valued functionf is continuously differentiable on the entire y-space R3. Assume that (S3) has a periodic orbit pit,ip) of period 1 such that f (r'(t, o)) 0 0. Assume also that the first variation of system (S3) with respect to the solution p(t,' o), i.e., d9

Of (r(t,io))u,

dt

has three multipliers p1 = 1, p2i and p3 satisfying the condition: 1P21 < 1 and jp3j > 1, respectively. Construct the general orbits pit, r) of (53) such that distance(At,17),

C(s'7o)) tends to0ast-+too. IX-9. Show that the differential equation d-t2 + (x + 3)(x + 2) does not have nontrivial periodic solutions.

drt

+ x(x + 1) = 0

Hint. Two stationary points are a node (0,0) and a saddle (-1,0). Furthermore, x2

setting f(x1,x2) =

, we obtain divf(x1,x2) _

-(xt + 3)(x1 + 2)x2 - xt(xt + 1) J -(x1 + 3)(x1 + 2) < 0 if x1 > -1. Also, use index in §IX-5. IX-10. For the system I

= f(x,y) = x(1 - x2 - y2) - 3y,

= g(x,y) = y(1 - x2 - y2) + 3x,

dt

(1) find and classify all critical points, (2) find dt

= f (x, y)

+ 9(x, y)

.-

5

IX. AUTONOMOUS SYSTEMS

302

for the function V (x, y) = x2 + y2, dV = 01, (3) find the set S = dt (4) examine if S is an invariant set, (5) find the phase-portrait of orbits.

{(x):

IX-11. For the system dx

dt = dt = dz

dt =

f(x,y,x) = - x(1 - x2 _

y2)2

+ 3xz + y,

9(x,y,z) = - y(1 - x2 - y2)2 + 3yz - x,

h(x,y,z) = - z -

3(x2 + y2),

(1) find all critical points and determine if they are asymptotically stable, (2) find 8V + h(x, y, x) 8V dV = f (x, y, x) 8x 8V + g(x, y, z) 8y 8z dt for the function V(x, y, z) = x2 + y21+ z2, dV (3)

find thesetS=((x,y,z):

=0},

find the maximal invariant set M in S, find the phase portrait of orbits. IX-12. Find the a phase portrait of orbits of the system (4) (5)

dt = y,

dt = y + (1 - x2)x2(4 - x2).

IX-13. Let f (x, y) and g(x, y) be real-valued, continuous, and continuously differentiable functions of two real variables (x, y) in an open, connected, and sim-

ply connected set D in the (x, y)-plane such that 8f (x, y) + Lf (x, y) j4 0 for all dxt

(x, y) E D. Show that the system = f (x, y), d = g(x, y) does not have any nontrivial periodic orbit that is contained entirely in D. IX-14. Let p(z) be a polynomial in a complex variable z and deg p(z) > 0. Set x = 3t(z] and y = Q [z]. Verify the following statements. (i) In the neighborhood of each of stationary points of system (A)

d = R[p(z)],

dt =

`3`[6 (x)],

there are no hyperbolic sectors. Also, system (A) does not have any isolated nontrivial periodic orbit. (ii) In the neighborhood of each of stationary points of system (B)

= 3R[p(z)], L = -`3'[p(x)],

dt there are no elliptic sectors. Also, system (B) does not have any nontrivial periodic orbits.

303

EXERCISES IX

IX-15. Find the phase portrait of orbits of the system

d = x2 - y2 -3x+2,

d _ -2xy + 3y.

Hint. See Exercise IX-14 with p(z) = (z - 1)(z - 2). dx

dy

IX-16. Find explicitly a two-dimensional system dt = f (x, y), d t= 9(x, y) so that it has exactly five stationary points and all of them are centers. IX-17. Consider a system dy dt = f (y1,

(S4)

where the entries of the R'-valued function f are analytic with respect toy in a domain Do C R'. Assume that (i) system (S2) has a periodic orbit pit, ijo) of period 1 which is contained in the domain Do, (ii) AP-Tt,no)) 0 (iii) for any open subset V of Do which contains the periodic orbits p(t, i"p), there

exists an open subset U of V which also contains p'(t, o) such that for any point i in U, the orbits p'(t, i of (S4) is contained in V and periodic in t. Show that if U is sufficiently small, for any point i in U, we can fix a positive period is bounded and analytic with respect to i in any simply of p1t, y) so that connected bounded open subset of U.

Hint. Apply the following observation. Observation. Let Do be a connected, simply connected, open, and bounded set in 1Rk and let T2 (j = 1, 2, ...) be analytic mappings of Do to l . Suppose that, for any pointy E Do, there exists a j such that T3 [yl = y, where j may depends on Y. Then, there exists a jo such that Tjo [yj = y for all y E Do. Proof.

Set E j _ {y E D o : T j [ y 1 = Y - ) .

j = 1, 2, ... .

Then, (1) E. is closed in Do, +00

(2) Do = U E,, 3=1

(3) Do is of the second category in the sense of Baire.

Hence, for some jo, the set Ego contains a nonempty open subset (cf. Baire's Theorem). Since Tea is analytic, we obtain Tao [ 7 = y" for all y" E Do. O For Baire's Theorem, see, for example, [Bar, pp. 91-921.

CHAPTER X

THE SECOND-ORDER DIFFERENTIAL EQUATION dt2 + h(x)

+ g(x) = 0

In this chapter, we explain the basic results concerning the behavior of solutions of a system 112

[-h(yi)112 - 9(111)

as t -- +oo. In §X-2, using results given in §IX-2, we show the boundedness of solutions and apply these results to the van der Pol equation (E)

x + E(x2 - 1)

d +x=0

(cf. Example X-2-5). The boundedness of solutions and the instability of the unique

stationary point imply that the van der Pol equation has a nontrivial periodic solution. This is a consequence of the Poincar&$endixson Theorem (cf. Theorem IX-4-1). In §X-3, we prove the uniqueness of periodic orbits in such a way that it can be applied to equation (E). In §X-4, we show that the absolute value of one of the two multipliers of the unique periodic solution of (E) is less than 1. The argument in §X-4 gives another proof of the uniqueness of periodic orbit of (E). In §X-5, we explain how to approximate the unique periodic solution of (E) in the case when a is positive and small. This is a typical problem of regular perturbations. In §X-6, we explain how to locate the unique periodic solution of (E) geometrically as e - +oo. In §X-8, we explain how to find an approximation of the periodic solution of (E) analytically as a +oc. This is a typical problem of singular perturbations. Concerning singular perturbations, we also explain a basic result due to M. Nagumo [Na6] in §X-7. In §X-1, we look at a boundary-value problem

y = F (t, y, d i )

y(a) = o,

,

y(b) = 8.

Using the Kneser Theorems (cf. Theorems 1II-2-4 and III-2-5), we show the existence of solutions for this problem in the case when F(t, y, u) is bounded on the entire (y, u)-space. Also, we explain a basic theorem due to M. Nagumo [Na4] (cf. Theorem X-1-3) which we can use in more general situations including singular perturbation problems (cf. [How]). For more singular perturbation problems, see, for example, [Levi2], [LeL], (FL], [HabL], [Si5], [How], [Wasl], and [O'M].

304

1. TWO-POINT BOUNDARY-VALUE PROBLEMS

305

X-1. Two-point boundary-value problems In this section, first as an application of Theorems 111-2-4 and 111-2-5 (cf. [Kn]), we prove the following theorem concerning a boundary-value problem (X.1.1)

d2

= F t, y, dt)

y(a) = a,

,

y(b) = Q

Theorem X-1-1. If the function F(t, yt, y2) is continuous and bounded on a region 11 = {(t, Y1, y2) : a < t < b, lyt I < +oo, Iy2I < +oo}, then problem (X.1.1) has a solution (or solutions). Proof.

For any positive number K, the set Aa = {(a, or, y2) : Iy21 < K} is a compact and connected subset of ft We shall show that A0 satisfies Assumptions 1 and 2 of §111-2 for every positive number K. In fact, writing the second-order equation (X.1.1) as a system (X.1.2)

dyt

dt

_

- Y2,

dye

dt

= F(t, yt, y2),

we derive y1(t) = y1(a) + J y2(s)ds, a

y2(t) = y2(a) + JF(s,yi(s),y2(s))ds. ` Hence, if (a, y1(a), Y2 (a)) E A0, we obtain

Jy2(t)I < K + M(b - a), ly1(t)I < lal + [K + M(b - a)](b - a), where I F(t, yi, Y2)1 < Al on Q. Therefore, Ao satisfies Assumptions 1 and 2 of §111-2. Thus, Theorem 111-2-5 implies that SS is also compact and connected for every c on the interval Te = {t : a < t < b}. We shall prove that if K > 0 is sufficiently large, the set Sb contains two points (771,772) and (t;t, (2) such that (X.1.3)

n, < 0 < (1.

In fact, by using the Taylor series at t = a, write yt (b) in the form

yt(b) = a + y2(a)(b - a) +

2 dt

where c is a certain point in the interval 10. Since

(c) (b - a)2, I ddt2

(c)

M, the quantity

yI(b)I can be made as large as we wish by choosing Iy2(a)I sufficiently large. Thus, there are two points (nt, n2) and ((1, (2) in Sb such that (X.1.3) is satisfied. Since the set Sb is compact and connected, there must be a point (Q, () in the set Sb. This implies the existence of a solution of problem (X.1.1). 0

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

306

Example X-1-2. Theorem X-1-1 applies to the following two problems: (X.1.4)

y(a) = a,

dt2 + sin y = 0,

y(b)

and

y(b) = Q

0, y(a) = a, + p-+-, = However, Theorem X-1-1 does not apply to

(X.1.5)

dt2

(X.1.6)

y(b) = Q.

y(a) = a,

d 22 + y = 0,

For more general cases, the following theorem due to M. Nagumo (Na4] is useful.

Theorem X-1-3. Assume that (i) a real-valued function f (t, x, y) and its derivatives az and

09 f

are continuous

in a region V = {(t, x, y) : (t, x) E A, -oo < y < +oo}, where 0 is a bounded and closed set in the (t, x) -space; (ii) in the region D, the function f satisfies the condition

1f(t,x,y)1 : 0(IyI),

(I)

where 0(u) is a positive-valued function on the interval 0 < u < +oo such that

+°° udu

(II)

+00;

(iii) two real-valued functions wi (t) and w2(t) are twice continuously differentiable

on an interval a < t < b and satisfy the conditions for wi(t) < w2(t) a < t < b, Ao

= {(t x)-a
_(t) <

x
and

d2wi (t) 2

(IV)

d2dt

(t)

2

> f 1 t, wi (t), d


dt t) )

d t, W2 (t),

dt t))

,

for a < t
(iv) two real numbers A and B satisfy the condition (V) wi(a) < A < w2(a), and wi(b) < B < w2(b). Then, the boundary-value problem

d 22 = f I t, x,

')

,

x(a) = A, x(b) = B,

has a solution x(t) such that (t, x(t)) E Ao for a < t < b, i.e., wi(t) < x(t) < w2(t), for a
The main tools are the following two lemmas.

b.

307

1. TWO-POINT BOUNDARY-VALUE PROBLEMS

Lemma X-1-4. Let x(t, to,rl) be the solution of the initial-value problem d2x

= f (t, x,

d .) ,

x'(to) = n,

x(to)

where a < to < b, (to, l;) E Ao. Then, for any given positive number M, there exists a positive number a(M) such that l x'(t, to, l;, il) j < a(M) if (X.1.7)

jr71 < M and (T, x(r, to, e, rl)) E Do for to < r < t or t:5 r < to.

Proof.

Letting L be a positive number such that (X.1.8)

w2(t) - w1(t) < L

a < t < b,

for

choose a(M) > 0 for any given positive number M in such a way that a(M) > 111 and a(M) udu

(X.1.9)

q5( u)

+!

> L.

Suppose that there exist ri and r2 such that to < Ti < r2 < t and that

x'(r1) = M < x'(r) < x'(r2) = a(M)

for

r1 < r <

T2,

where x(T) = x(r, to, t ,17). Then, since x'(r) > 0 for ri < r < r2, it follows that xO(X'x (T

Hence,

f

))) <

x/ (7-)

for

T1 < -r < -r2.

a(M) u/du

0u) -

This contradicts the choice of L by (X.1.9). Therefore, Lenuna X-1-4 is true for to < r < t. We can treat the case t < r < to similarly, since if we change t by -t, the differential equation x" = f (t, x, x') becomes x" = f (-t, x, -x'). 0

Lemma X-1-5. Set

7 = rim {a(ba)I wi(a)+e, -w2' (a)-eI, where e is an arbitrarily fixed positive number. Let also x(t, c) be the solution to the initial-value problem d2x

= f (t, x,

)

,

x(a) = A, i (a) =

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

308

Then,

(1) two curves x = x(t, c) and x = w2(t) meet for some t on the interval a < t < b

if c>'Y; (2) two curves x = x(t, c) and x = wl (t) meet for some ton the interval a < t < b if c < -ry. Proof.

For part (1), by virtue of Lemma X-1-4, x'(r,c) > a < r < t. The proof of part (2) is similar.

La

if (r,x(t,c)) E Ao for

Proof of Theorem X-1-3.

Now, let us complete the proof of Theorem X-1-3. The main point is that when two curves x = x(t,c) and x = w2(t) or two curves x = x(t,c) and x = w, (t) meet, they cut through each other. So look at Figure 1.

(b, B) (a. A)

t=a

r=b FIGURE 1.

Example X-1-6. Theorem X-1-3 applies to the boundary-value problem (X.1.10)

d2x dt2

_

x(0) = A,

x(1) = B

Ax'

if A is a positive number. In fact, assume that 0(u) is a suitable positive constant. If

wi(t) = sinh(ft) -a and w2(t) = sinh(ft)+$ with two positive numbers a and ,3 such that -a < A < j3 and sinh(/) - a < B < sinh(VrA_) +,Q, all requirements of Theorem X-1-3 are satisfied.

If A is negative, Theorem X-1-3 does not apply to problem (X.1.10). Details are left to the reader as an exercise.

2. APPLICATIONS OF THE LIAPOUNOFF FUNCTIONS

309

X-2. Applications of the Liapounoff functions In this section, using the results of §IX-2, we explain the behavior of orbits of a system

as t

[yj

d

(X.2.1)

Y2

__

y[-h(yi)y2

dt

- 9(yi)

l)y2- 9(y1) and f (yl = [_h(yi

+oo. Set y = I yl

JLet us assume that

J

h(x), g(x), and dd(x) are continuous with respect to x on the entire real line R. Also, we denote by p(t,r) the solution of (X.2.1) satisfying the initial condition y(0) = n Fi rst set V (y) = + G(yl ), where G(x) = fox g(s)ds. Then, 2Y2

09

= [9(yi ), Y21,

Set also, S = { g : l

8y'

-

f (y-)

= -h(yi)

y22.

R Y) = 0 y. Then, U E S if and only if either h(yl) = 0 or JJJ

Y2 = 0.

Observation X-2-1. Denote by M the set of all stationary points of system (X.2.1), i.e., M = {9: g(yl) = 0, y2 = 01. Then, M is the largest invariant set in S if the following three conditions are satisfied:

(1) h(x) > 0 for -oo < x < +oc, (2) h(x) has only isolated zeros on the entire real line IR, (3) g(x) has only isolated zeros on the entire real line R.

The proof of this result is left to the reader as an exercise (cf. Figure 2, where 0 and 0). By using Theorem IX-2-2, we conclude that lim p(t, y) = 17 E M if conditions t-+00 (1), (2), and (3) are satisfied and if the solution p(t,g) is bounded for t _> 0. Note that G+(rt) is a connected subset of M. In Observation X-2-1, the boundedness of the solution p(t, i) for t _> 0 was assumed. In the following three observations, we explore the boundedness of all solutions of (X.2.1). Set 0,

G(x) =

o

g(s)ds

and

H(x) =

Jo

f

X

h(s)ds.

o

Observation X-2-2. Every solution p(t, q) of (X.2.1) is bounded for t > 0 if (i) h(x) > 0 for -oc < x < +oo and (ii) lim G(x) = +oo. 1x1

+00

This is a simple consequence of Theorem IX-2-3. In fact,

lim V (Y-) = +oo. 191-.+00

310

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

Observation X-2-3. Every solution pit, of (X.2.1) is bounded for t > 0, if (i) h(x) > 0 for -oo < x < +oc, (ii) urn IH(x)l = +oo, and (iii) xg(x) > 0 I=t-+o0

for -oc < x < +oo. Proof Change system (X.2.1) to (X.2.2)

d

[zl]

Z2 - H(z1)

dt

22

-g(z1)

by the transformation

Y2 = z2 - H(zi).

Y1 = Z1,

Denote by qqt,() the solution of (X.2.2) such that z(0) _ . Set V, (z-) = zz2 + G(z1). Then, L9 V,

ay -

[g(zl), z21,

8z"

,P = -g(z1)H(zl)

Note that g(x)H(x) > 0 for -oo < x < +oo and that dt V1(ggt,C))

Hence, setting q"(t, S) =

- z(g-(t,C)) F(glt,S)) S 0 z1(t,

for

t > 0.

= c > 0, we obtain

and V,

2[z2{t,S)]2 <

c

for

t>0,

since G(x) > 0 for -oo < x < +oo. Therefore, Figure 3 clearly shows that q1 t, C) is bounded for t > 0. This implies that all solutions of (X.2.2) are also bounded for

t>0.

{s2, 0)

z2=0

Y2=0 (43,0)

(41.0)

di-1

FtcuRE 2.

z1=0

F1cuRE 3.

2. APPLICATIONS OF THE LIAPOUNOFF FUNCTIONS

311

Observation X-2-4. Every solution p(t, >)1 of (X.2.2) is bounded for t > 0 if (i)

=

lim H(x) = +oo, (ii) = --Cc lim H(x) = -oo, (iii) g(x) > a for x > as > 0, +00

and (iv) g(x) < -a for x < -ao < 0, where a and ao are some positive numbers. Proof.

In Observation X-2-3, the Liapounof function

G(zi) = 2[y2 + H(yl)]2 + G(yi)

Vi(z) =

was used. Now, let us modify Vl to a form

V2(yl = 2[y2 + H(yi) - k(yi)J2 + G(yl) Then, OV2

09

and

_

[[y2

- dyyt )1 + 9(yi ),

+ H(yi) - k(yi )J {h(yi)

(Y ')

az

dy,

1

Y2 + H(1h) - k(yi) I

J {y22

J

+ [H(yi) - k(yi)J y2}

- 9(yi) [H(yi) - k(yl)J Using (i) and (ii), three positive numbers M, a, and c can be chosen so that

fa [H(x) - cJ > M

x > a > ao,

for

-a [H(x) + c] > M

for

x < -a < -ao.

Also choose a function k(x) so that for x > 2a, for x!5 -2a,

{c (1)

(II}

k(x) =

Jk(x)J < c

-C dk(x)

and

>0

dx

for

- on < x - +oo

and dk(x)

dx

>

m > 0

for

JxJ < a

for some positive number m (cf. Figure 4). k

k=c

x

k=-c x=-2a

x=-a

x=O

x=a

FIGURE 4.

x=2a

,

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

312

If a positive number b is chosen sufficiently large,

OV2 .f

0

for

or 1y21 > b.

lyil ? 2a,

In fact,

-g(yl) [H(bi) - k(yl))

(A)

- a IH(yl) - cI < -M < 0

for yl > a,

a IH(y1) + cj < -M < 0

for yl < -a

and

Y2 + IH(yj) - k(yi )11/2 ? 1

(B)

for

1yi 15 2a, 1y21 > b

if b > 0 is sufficiently large. Therefore, (i)

av2

' f = -g(yi) IH(yi) - k(yi)1 < 0

(u)

f<

09

--

(y1

<0

for

11/21 < +00,

Iyi I >- 2a,

for a < 1yi I G 2a,

1y21 ? b,

and OV2

(iii)

f < - m {y22 + IH(yi) - k(yi)1 y2} g(yt)IH(yi) - k(yi)I < 0

forlyil < a,

1y21 ? b

if b > 0 is sufficiently large. Since lim G(x) = +oo, Theorem IX-2-3 implies that every solution of (X.2.2) 1XI

+00

is bounded.

Example X-2-5. For the van der Pol equation 2 + e(x2 - 1) dt + x = 0,

where a is a positive number, h(x) = e(x2 - 1) and g(r) = x. Hence,

G(x) = 2x2

and

H(x) = e (3x3 - x)

Therefore, conditions (i), (ii), (iii), and (iv) of Observation X-2-4 are satisfied. This implies that every solution of the van der Pol equation (

X.2.3 )

d

yl

dt

y2

y2

-E(Y - 1)y2 - yi,

is bounded for t > 0. System (X.2.3) has only one stationary point 6. It is easy to see that 0 is an unstable stationary point as t -+ +oo. Therefore, using the Poincaare-Bendixson Theorem (cf. Theorem IX-4-1), we conclude that there exists at least one limit cycle. In §X-3, it will be shown that system (X.2.3) has exactly one periodic solution.

3. EXISTENCE AND UNIQUENESS OF PERIODIC ORBITS

313

Example X-2-6. For a given positive number a, the system [yi

(X.2.4)

d

_

y2

[

Y2 J

-aye - sin (yl )

satisfies three conditions (1), (2), and (3) of Observation X-2-1. But, (X.2.4) does not satisfy conditions of Observations X-2-2, X-2-3, and X-2-4. Therefore, in order to prove the boundedness of solutions of (X.2.4), we must use some other methods.

[ij.

In fact, using the Liapounoff function V (y) = -yZ - cos (yl ), we obtain

ev. f =

WY=

2Since V (y-)

-aye < 0, where f (y =

t

a

lim

+oo

exists for every solution p-(t, ii) of (X.2.4). Now, observe that (1) We must have a < 1. Otherwise we would have y2(t,,)2 > 2(a+cos(yi(t,rte))

for t > 0. This implies that dt V (pl t, r7-')) < -2a(o - 1) < 0. This contradicts (X.2.5).

(2) If -1 < a < 1, the solution p(t, 7) must stay in one of connected components of the set {y" : V(y) < a + e < 1} for large positive t. Those connected components are bounded sets. (3) In case a = 1, we can show the boundedness of p(t, t) by investigating the behavior of solutions of (X.2.4) on the boundary of the set {g: V(y-) < 1).

X-3. Existence and uniqueness of periodic orbits In this section, we prove the following theorem (cf. (CL, p. 402, Problem 51).

Theorem X-3-1. Assume that (i) two real-valued functions h(x) and g(x), and X < +00,

dg

(x) are continuous for -oo <

(ii) g(-x) = -g(x) and h(-x) = h(x) for -oo < x < +oo, (iii) g(x) > 0 for x > 0, (iv) h(0) < 0,

(v) H(x) =

f h(s)ds has only one positive zero at x = a, 0

(vi) h(x) > 0 for x > a,

(vii) H(x) tends to +oo as x -- +oo. Then, the system (X.3.1)

d [Yyj dt

[-h(yi)y22- 9(yi)

has exactly one nontrivial periodic orbit and all the other orbits (except for the stationary point 0) tend asymptotically to this periodic orbit as t +oo. Proof.

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

314

Change system (X.3.1) to

zi)

d

(X.3.2)

dt Lz2J

Lz2-

by the transformation Y2J

Lz2

-zH(zi),

Setting

V(z) =

+ G(z1),

2

where

G(x) =

J0

z g(s)ds

and

z=

[z2] look at the way in which the function V(z) changes along an orbit of (X.3.2). For example, dt V

(zI = 9(zl)[z2 - H(zi)] - z29(z1) = -g(z1)H(z1)

along an orbit of (X.3.2). Hence, dz2

g(zl)

z2 - H(zl)'

dz2

dV

9(z1)H(z1)

dV

dz1

z2 - H(z1)'

dz2

dzi

z2 - H(zi)

dzi

9(z1)

_

H(zi)

along an orbit of (X.3.2).

z1(t' a)be the orbit of (X.3.2) such that z(0, a) Observation 1. Let z(t, a) = IZ24,a)]

[0].

Then, V (i(0, a)) = 2 a2. There exists exactly one positive number ao such h

that [

z1(ao, ao) =

0]

and

z2 (t, ao) >0 for 0:5t < oo

for some positive number co, where a is the unique positive zero of H(x) given in condition (v) (cf. Figure 5).

Observation 2. Since

0 when z2 = H(z1), there exist two positive numbers

r(a) and Q(a)such that

z(r(a), a)

0

-Q(a)

and

0 < zi (t, a) :5a for 0< t < r(a)

if 0 < a < ao. Also, since H(x) < 0 for 0 < x < a, we obtain dtV(z'(t,a)) = -g(zi(t,a))H(zi(t,a)) > 0

for 0 < t < r(a)

3. EXISTENCE AND UNIQUENESS OF PERIODIC ORBITS

315

except for a = ao and t = oo. Therefore,

V(z"(r(a),a)) - V(z(0,a)) > 0

(A)

for

0 < a < a0 (cf. Figure 6).

(0.ao)

zI = 0

FIGURE 6.

FIGURE 5.

Observation 3. If ao < a, there exists a positive number r0(a) such that J z1(ro(a), a) = a,

0 < z1(t, a) < a for 0 < t < ro(a), for 0 < t < ro(a)

10 < z2(t,a) - H(zl(t,a))

(cf. Figure 7). In particular ro(ao) = Co (cf. Observation 1). If the variable t is restricted to the interval 0 < t < 7-0 (a), the quantity z2(t, a) can be regarded as a function of z1(t,a), i.e.,

z2(t,a) = Z(zl(t,a),a), where Z(x, a) is a continuous function of (x, a) for 0 < x < a and a > a0, and continuously differentiable for 0 < x < a and a > ao except for x = a and a = a0. Furthermore, Z(x, al) < Z(x, a2) for 0:5 x:5 a if a0 < a1 < 02 (cf. Figure 7).

Set 2(x, a) = I Z(z, a) J . Then, d ,

dx

V(Z(x,a))

-

g(x)H(x)

Z(x,a) - H(x) >

for

0

0
and, hence,

3jV(Z(x,a1)) > ±V(i(x,a2))

for

0
if 00 < 01 < 02. Thus, we obtain (I)

V(zro(al),al)) - V(z(O,al)) > V(E(ro(a2),a2)) - V(E(O,a2)) > 0

for go
X. THE SECOND-ORDER DIFFERENTIAL EQUATION

316

Observation 4. If a0 < a, there exists a positive number ri(a) such that ri(a) > ro(a), zl (r1(a), a) = a, and zi(t, a) >a for ro(a)
(0. a)

I

(O.a0)

(a. z2(r0(a), a))

O,a ( )

z2=H(zt)

z2=H(z0

(O.ao)

z2=0

zi=a

zi =0

=0

Z, =0

FIGURE 8.

FIGURE 7.

Note that z2(ri(a),a) < 0 and that H(zi(t,a)) > 0 for ro(a) < t < 7-1(a) since zi (t, a) > a on this t-interval. Regarding zi (t, a) as a function of z2(t, a) for z2(r1(a), a) < z2 < z2(ro(a), a), we obtain

(II) 0 > V(zi(ri(ai),ai))-V(zlro(ai),ai)) > V(z(ri(a2),a2))-V(z(ro(a2),a2)) for ao < al < a2 in a way similar to Observation 3 (cf. Figure 8).

Observation 5. If ao < a, there exists a positive number r(a) such that r(a) > 71(a), z1(r(a), a) = 0, and 0 < zi (t, a) < a for r1(a) < t < r(a) (cf. Figure 9). Note that z2(t, a) < H(zi (t, a)) < 0 for r2 (a) < t < r(a). Again, regarding Z2 (t, a) as a function of zi (t, a) in the same way as in Observation 3, we can derive

(III) V(z(r(ai),al))-V(z-'(ri(ai),ai)) > V(i(r(a2),a2))-V4flri(a2),a2)) > 0 if a0 < aI < a2 (cf. Figure 9).

Observation 6. Thus, by adding (I), (II), and (III), we obtain (B)

V(z(r(aI),at)) - V(z(0,a1)) > V(zr(a2),a2)) - V(z0,a2)) > 0

if a0 < al < a2. This implies that the function G(a) defined by

g(a) = V((r(a),a)) - V(z(0,0 = Zz2(r(a),a)2 - 2a2 is strictly decreasing for a > ao as a -+ +oo. Also, 9(a) > 0 for 0 < a 5 a0 (cf. (A)).

Observation 7. Since dV

lim jz21-+oo dzi z1lim00

a

=0

uniformly

= +oo uniformly

for 0 < z1 < a, for

- oo < z2 < +oo,

3. EXISTENCE AND UNIQUENESS OF PERIODIC ORBITS

317

it follows that lim (V(i(ro(a),a)) - V(z(0,0J = 0, a+oo

lim

(V(z(rl(a),a)) - V(z(ro(a),a))J = -oo,

lim

JV(z(r(a),a)) - V(i(ri(a),a))J = 0.

a+oo a-.+00

Therefore, lim Cg(a) = -oo.

(C)

a-'+00

Thus, we conclude that g has exactly one positive zero a+, i.e.,

(X.3.3)

9(a)

Zz2(r(a), a)2

-

2a2

> 0,

0
= 0,

a = a+,

< 0,

a>a+

FIGURE 10.

FIGURE 9.

From (X.3.3) and symmetric properties (ii) of the functions h(x) and g(x), we conclude that a(t, a+) is the only periodic orbit, and all the other orbits tends to z"(t, a+) asymptotically since

Iz2(r(a), a)I > a

if 0<0
Iz2(r(a),a)I < Of

if a> a+.

Thus, we complete the proof of Theorem X-3-1.

0

Remark X-3-2. Condition (iv) of Theorem X-3-1 can be replaced by the following condition: (iv') there exists a positive number 6 such that H(x) < 0 for 0 < x < S.

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

318

X-4. Multipliers of the periodic orbit of the van der Pol equation In Example X-2-5, we looked at the van der Pol equation

LX + e(x2 - 1) d + x = 0,

(X.4.1)

where a is a positive number. Using Observation X-2-4, it was shown that every orbit of the system d dt

(X.4.2)

/

i

[y112

-

112

] [ -E(y1 1)112 - 111 is bounded for t > 0. It was also remarked that 0 is the only stationary point of system (X.4.2) and that the stationary point 0 is not stable as t - +oo. In fact, the linear part of the right-hand side of (X.4.2) at y = 0 is Ay, where A = [ 01

[i]. Y21-

Since trace[A]=f and det (A) = 1, the stationary point

and

is an unstable

node for e > 2, while 0 is an unstable spiral point for 0 < e < 2. Therefore, using the Poincare-Bendixson Theorem (cf. Theorem IX-4-1), we conclude that there exists at least one limit cycle. Now, Theorem X-3-1 implies that system (X.4.2) has exactly one limit cycle and all the other orbits except for the stationary point 6 approach this limit cycle asymptotically as t - +oo. In fact, since h(x) = e(x2 -1), 3

H(x) = e I

3

- x]

,

and g(x) = x, the seven conditions (i) - (vii) of Theorem X-3-1

are satisfied. In particular, the positive zero of H(x) is a = J > 1. In this section, we prove the following theorem concerning the multipliers of the unique periodic solution x = x(t, e) of the van der Pol equation (X.4.1). Theorem X-4-1. The multipliers of the periodic solution x(t, e) of (X.4.1) at 1 and p such that )pI < 1. Proof.

If we! set v =

z

xz

11

2

, wwhere 11 =

that eJ (x(t)2 - 1)dt = - /

d , then dt = -e(x2 - 1)y2. This implies

. Now Now look at Figure 11. Y

FIGURE 11.

5. THE VAN DER POL EQUATION FOR A SMALL PARAMETER

319

On each of two curves Cl and C2, let us denote y as a function of v by yl (v) and y2(v)

respectively. Note that x2 > 1 on C1, but x2 < 1 on C2. Hence, yl(v)2 < y2(v)2. This implies that fT (x(t)2

e

- 1)dt > 0,

where T is a period. Therefore, we can complete the proof by using the following lemma.

Lemma X-4-2. Assume that two functions f (x) and g(x) are continuously differentiable in R. Assume also that the differential equation d2

2 + f(x) dt + g(x) = 0

has a nontrivial periodic solution x(t) of period 1 such that

r1

J0

f (x(t))dt > 0. Then,

the multipliers of the periodic solution x(t) are 1 and p such that jpI < 1.

Remark X-4-3. If system (X.4.2) has more than one periodic orbit, then at least one of them must be orbitally unstable. Therefore, the proof of Theorem X-4-1 is another proof of the uniqueness of periodic orbit of (X.4.2).

X-5. The van der Pol equation for a small e > 0 In this section, we explain a method to locate the unique periodic orbit of differential equation (X.4.1) (or system (X.4.2)) for a small e > 0. Set e = 0. Then, system (X.4.2) becomes (X.5.1)

dt[y2J

LyYi

Every orbit of (X.5.1) is a circle and of period 2n in t. We expect that the periodic orbit of (X.4.2) must be approximated by one of those circles as e -+ 0. The main problem is to find the radius of the circle which approximates the periodic orbit of (X.4.2) for a sufficiently small e > 0.

Since the periodic orbit and its period are functions of e, we normalize the independent variable t by the change of independent variable

t = (I +eW)T

(X.5.2)

so that the period of the periodic orbit becomes 27r for every e > 0. Transformation (X.5.2) changes differential equation (X.4.1) to x722

+ e(1+eW)(x2-1)d + (1+&,d)2x = 0

that can be written in the form (X.5.3)

d 2X 22

+ x = e(1 + ew)(1 - x2)

- e(2W + ew2)x.

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

320

Observation X-5-1. In the case when a real-valued function f (r) is continuous and periodic of period 27r in r, the general solution of the linear nonhomogeneous differential equation d2x -T' + x = f(r) d

(X.5.4)

is given by (X.5.5)

/'r+2n x(r) = K cos(r + m) + a- J

sf(s)sin(r - s)ds

r

as it is shown with a straight forward calculation, where K and 0 are arbitrary constants. This solution is periodic in r of period 2ir if and only if 2w

2w

f (s) sin(s)ds = 0

(X.5.6)

and

J0

10

f (s) cos(s)ds = 0.

Observation X-5-2. In the case when condition (X.5.6) is not satisfied, set 2, f (S)

27r

(X.5.7)

f(s) sin(s)ds and C[J] = 1

S[ f ] = 1 fo i

7t

cos(s)ds.

J

o

Then, v(T) = Kcos(r + 0) jr+21r

(X.5.8)

1

s f (s) - S[ f ] sin(s) - C[ f ] cos(s)Jsin(r - s)ds

+ 2a

is the general solution of the differential equation dr2 + x = f(r) - S[f]sin(r) -

Furthermore, v(r) is periodic of period 27r.

Observation X-5-3. Set f ( X,

,W, E)

_ (I + (w)(I - x2)

- (2W + EW2)x.

Letting K be a parameter and a function v(r, K, w, f) be periodic in r of period 27r, set

S(K,w,) _

I

2w

1

j

C(K,, E) = 7r

2w

f f

(v(sKw,_(s,K,WE)WJf

(1T

cos(s)ds.

5. THE VAN DER POL EQUATION FOR A SMALL PARAMETER

321

Now, let us consider an integral equation +2 a

v(r, K, w, E) = K cos(t) + (X.5.9)

2n

s I f lV,

fr

dr, w, EL

- S(K, w, c) sin(s) - C(K, w, f) tos(s)] sin(T - s)ds. Solutions of (X.5.9) satisfy the differential equation d 2V

(X.5.10)

d-,r2

+ V = E [f( V,

Va

, w, E I - S(K, w, c) sin(r) - C(K, w, E) cos(r)

as long as v is periodic in r of period 2ir. This integral equation can be solved by using successive approximations in such a way that the solution v(r, K, W, E) is a convergent power series in c: (X.5.11)

EmVm(T,fi,w) = Rcos(T)+E vl(r,K,w)+

v(r,K,w,E) _

,

m=0

where vm(r, K, w) are polynomials in (K, w) with coefficients periodic in r of period 2n. For any given positive numbers Ko and wo, there exists another positive number Eo(Ko,wo) such that series (X.5.11) is uniformly convergent for

I K1 < K0,

IwI

<- wo,

IEI

<- Eo(Ko,wo),

-00
This implies that S(K, w, c) and C(K, w, E) are also convergent power series in c: 00

0C

(X.5.12) S(K,w,E) = E EmS,,,(K,w) and C(K,w,E) _ F, E"'Cm(K,w), m=0

m=0

where Sm(K,w) and C,,,(K,w) are polynomials in (K,w) with constant coefficients. These two power series also converge uniformly for I K1 <- Ko,

IwI < wo,

IEI

Eo(Ko,wo)

Observation X-5-4. Inserting series (X.5.11) and (X.5.12) into system (X.5.10), we obtain d2vi

dr2

+ vi = (1 - K2cos2(r))(-Ksin(r)) - 2wKcos(r) - So(K,w) sin(r) - Co(K,w) cos(r) = K3 cos2(r) sin(r) - K sin(r) - 2wK cos(r) - So(K,w) sin(1T) - Co(K,w) cos(r)

= 4 K sin(3r) +

1

- K - So(K, w)J sin(r)

4 - [2wK + Co(K, w)1 cos(r).

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

322

Since v1 is periodic in r of period 2,r, we must have

So(K,w) = I K3 - K

Co(K,w) _ -2wK.

and

This implies that S(2, 0, 0) = 0 , C(2, 0, 0) = 0, and

8K

x(2,0,0)

(2,0,0)

8C200

200

_

2

-10

0 41

--8.

Therefore, the system of equations S(K, w, e) = 0, C(K, w, e) = 0 has a solution K(e) = 2 + 0(e), w(e) = 0(e). The functions K(E) and w(e) are power series in e which converge if Iej is sufficiently small.

Observation X-5-5. Set x(t, e) = v (T+ f1w(E) I K(E), w(E), e/f Then, x(t, e) is a periodic solution of (X.5.3) and

x(t, e) = K(e) cos

+ 0(e)

t

(1+Ew(e)

as

a --+ 0.

From the fact that y2(t, E) _

(t, f) =

-

1

t + 0(e) sin C 1 + ew(e))

K(f)

as

e -+ 0,

we obtain the following conclusion.

Conclusion X-5-6. The unique periodic orbit of (X.4.2) tends to the circle yl +

Y22 = 4 ase-.0+.

X-6. The van der Pol equation for a large parameter In this section, we consider the van der Pol equation (X.4.1) for a large e. Let us write (X.4.1) in the form

(`1

_

(X.6.1)

dt [z21

-z1 3

=z2-ef 3 - zl).

by setting x=z1 and

- I Eu'

21 Set {z2]

and t = er. Then, system (X.6.1) becomes I

d [$2

(X.6.2)

dr 1

where a = E

Il

6. THE VAN DER POL EQUATION FOR A LARGE PARAMETER

323

Observation X-6-1. Note that, along an orbit of (X.6.2), (X.6.3)

dud dwl

Q2w1

01 3

- wtl

Therefore, if 3 > 0 is sufficiently small, the slope dw, of any orbit is small at a point (wl, w2) far away from the curve C : w2 = 3 - w!. This implies that every orbit moves toward the curve C almost horizontally (cf. Figure 12). Observation X-6-2. From Observation X-6-1 a rough picture of orbits of (X.6.2) is obtained (cf. Figure 13).

FIGURE 12.

FIGURE 13.

Actually, defining the curve Co by Figure 14, we prove the following theorem.

Theorem X-6-3. For a sufficiently small positive number J3, the unique periodic orbit of (X.6.2) is located in an open set V(/3) such that the closure of V(f3) contains the curve Co and shrinks to CD as 0 -+ 0+. Proof of Theorem X-6-8. In eight steps, we construct an open set V(13) so that (1) the closure of V(O) contains the curve Co, (2) the closure of V(J3) shrinks to the curve Co as O 0+,

(3) if an orbit of (X.6.2) enters in the open set V(!3) at r = ro, then the orbit stays in V(/3) for r > ro. Step 1. Fixing a number a(3) > 2, we use the line segment

C1(a) = j wi, a(3)3 - a((3) I

:

0 < wi <_ a(/3)}

as a part of the boundary 8V(i3) of V(/3) (cf. Figure 15).

FIGURE 14.

FIGURE 15.

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

324

Note that the slope of an orbit given by (X.6.3) is negative on C1 (O) and decreasing

to -oo as wt increases to a(f3). Also,

dw2

is 0 on C1(f3).

Step 2. Let us consider a curve

CO) = j (wi, ws) : w2 = 3' - wl - Op(wi, 0),

1 < wl < a(fl)}

,

where

(i) µ(w1,3) is continuous for 1 < wl < a(0) and continuously differentiable for 1 < wl < a($), (ii) µ(w1, 0) > 0 for 1 < w1 < a(0), and (iii) µ(a(,3), 0) = 0. On the curve C2(f3), 3

\

31

= -0µ(w1,13)

- wl

and, hence,

_

Owl

/32w1

l

W2 _ f wl _ wl /f

3

µ(w1, 0)

On the other hand, the slope of the curve C2($) is

_ w1 2 dEc(w1,f) 1 ,8 dwl dw2

dwl

This implies that if µ is fixed by the initial-value problem (X.6.4)

1,

dwl

µ(a(0),,8) = 0,

we obtain dw2

/32w1

W2 -

C 3

-

1

dwl

for

1 < wl < a($),

w'/

where dw2 denotes the slope of C2(0). The unique solution to problem (X.6.4) is i given by

a(/3)2 - wl. Thus, we choose the curve C2(O) _ {(wt, wz) : u,2 =

3

- wl - O a(0)2 - wi, 1 < w1 < a(0) }

as a part of the boundary OV(#) of V(f3) (cf. Figure 16). Note that on the curve C2(,8),

w2 =

-3 - Q

a(f3)2 - 1

at w1 = 1.

6. THE VAN DER POL EQUATION FOR A LARGE PARAMETER

325

Step 3. We choose the curve C3(/3) =

{(wIw2): W2 = -3 - Q a(N)2 - w1, 0 < wl < 11

as a part of the boundary OV(/3) of V(/3) (cf. Figure 17). C

C1(p)

FIGURE 17.

FIGURE 16.

On the curve C3(/3),

3 - 31 - wl l - /3 a(/3)2 - wi 2

1w3 W2

wl)

31

and, hence,

0'w1

/32w1 (,w3

wl/

31

/

a(Q)2

- w1 + 3

On the other hand, the slope of the curve C3(/3) is

+

(31 - wl

dw2

dw,

1

implies that

02W,

w2-

(u, -wI ) 3


a(Q)2 - wl

This

for 0
dw,

denotes the slope of C3(0) (cf. Figure 17). Note that on the curve awl C3(3), W2 = - 3 - Qa(Q) at wI = 0. where

Step 4. Now, let us fix the positive number a(/3) > 2 by the equation (X.6.5)

2

3

+ 3a(3) = a(3) - a(13).

Denote by C+(/3) the curve consisting of Cl(/3), C2(13), and C3()3), i.e., C+(Q) _ C1(Q) UC2(/3) UC3($). Let C_(0) be the symmetric image of C+(Q) with respect to the point (0, 0). Then, C+(3) U C_ (/3) is a closed curve if a(,Q) satisfies condition (X.6.5). We use the closed curve C+(/3) UC_(/3) as a part of the boundary 8V(/3) of V(/3) (cf. Figure 18).

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

326

b(3) 3

Step 5. Fixing a positive number b($) < 2 so that 3 =

- b(j3) +,A(0), we

choose the curve

r!(,3) = {twiw2) : w2 =

b(3)3-

b(j3) +,6b($)2 - w;, 0 < w, < b(13)}

as a part of the boundary 8V(j3) of V($) (cf. Figure 19). Note that, on r,(j3), 2 w2=3 at w, = 0.

FIGURE 18.

FIGURE 19.

On the curve r, (,B), 02w1

/32w1

Q b(B)2 - wl +

w2

>-

b(3) 3

- b(8) -

(3'

w,)

Q wl

b(3) - w,

On the other hand, the slope of the curve 171(3) is

dullw2

ap)2'-

wi

Step 6. We use the two curves

r2(0) = {(wiw2):

3

w2

r3($) = {(wltw2): U2

3

-wi, 1
0 < wl < -2, 3

ll

JJ

as parts of the boundary 8V(j3) of V(8) (cf. Figure 20). Note that on r3(Q), the slope of an orbit given by (X.6.3) is positive and dd is negative.

Step 7. Denote by r+(j3) the curve consisting of r,(8), r2(8), and r3(8), i.e.,

r+w) = r1(Q) u r2(a) u r3(8). Let r_ (j3) be the symmetric image of r+(j3) with respect to the point (0, 0). Then, r+(p) U r_(8) is a closed curve. We use the closed curve r+(j3) U r_ (f3) as a part of the boundary OV(0) of V(8) (cf. Figure 21).

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

328

(iii) the function Y(x) is a solution of the differential equation

F (x, Y, and

d'Yx(x)

I = 0,

M for a positive constant M on the interval 0 < x < t.

Let y(x) be any solution of the differential equation

(A>0)

AL2 + f (x,y,LY) = 0

satisfying the conditions y(O) = Y(0) and ly'(0) - Y'(0)l l< p. Then,

ly(x) - Y(x)I < {h, + on the interval 0 < x <

a(L

+

k)}exp[ Lx]

if c and A are sufficiently small.

Proof.

We prove this theorem in four steps.

Step 1. Setting

y=u+Y(x)

-__v+dY),

and

change the equation

ad.z + f (x, y, d-T ) = 0

(A > 0)

to the systern

du

(X.7.1)

dx

dv

= v,

F(X, u, v, a),

where

F (x, Y(x), v + d

)

I + F(x, u, v,.)l < (e + AM) + Kf ul,

as long as (x, u, v) is in the regi/on

l

Do = {(x, u, v) : 0 < x < t, Jul <- a(x), lv) < pe-`= + b(x)}. Also,

F (riY(x)v + d ()) F x, Y(x). v + d

> Lv

())

< Lv

for

v > 0,

for

v < 0,

in Do. Hence, in Do, (X.7.2))

F(x, u, v, A) < -Lv + Klul + (e + AM) F(x, u, v, A) > -Lv - Klul - (E + AM)

for

v > 0,

for

v

0.

329

8. A SINGULAR PERTURBATION PROBLEM

Step 2. Suppose that two functions wl (x, .A) and w2(x, A) satisfy the following conditions: (X.7.3)

0 < w2(x,A) < pe-' + b(x), 10 < wl(x,.A) < a(x), w2(0,.1) > p, wl(0,A) > 0, wi (x, A) > W&, A),

Aw2(x, A) > -Lw2(x, A) + Kwl (x, A) + (E + AM)

on the interval 0 < x < e. Then, as long as (x, u, v) is in the region

Dl = {(x,u,v): 0 <_ x < e, IuI < wi(x,A), IvI < w2(x,A)}, it holds that IvI < wi (x, A), .F(x, u, w2(x, A), A) < Aw2(x, )1),

F(x. u, -w2(x, A), .A) > -aw2(x, I\).

Look at the right-hand side of (X.7.1) on the boundary of V. Then, it can be easily seen that if a solution of (X.7.1) starts from Dl, it will stay in Dl on the interval 0 < x < e. Step 3. Show that two functions

Jwi(x,A) =

J0

w2(x A) =

+ 62(1+1),

A)l

x

pe-Lx/A + Alex:/L

ApA e + AM + K62(e + 1) where bl = L2 + , satisfy the requirements (X.7.3) if f, A, L and a positive constant 62 are sufficiently small. Observe that two roots of AX2 +

LX -K=0 are -L -(o and (o = L

+O(A).

Step 4. Note that

wl (x, A) = (L (1 - e-Lx/A) +

bK I (eKx/L - 1) + 82(x + 1).

/

To complete the proof, look at Jul < wl (x, A) as 62 -4 0.

The inequality 2( a, )as 62 I VI v< wr dy dx

Note that

0, yields the following estimate of

dY(x) < pe-Lx/A +

fE

dx

+

L

lim a-Lx/A

0

=0

L

if

(h + M1 ] ex'/L. L

x > 0.

J

d

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

330

X-8. A singular perturbation problem In this section, we look at behavior of solutions of the van der Pol equation (X.4.1) as a --+ +oo more closely. Setting t = Er and A = c2, let us change (X.4.1) to d2x 2 J1dr2 + (x -

1)dx

dr

+x=

0.

Set A = 0. Then, (X.8.1) becomes

(x2 - 1) d + x = 0.

(X.8.2)

Solving (X.8.2) with an initial value x(O) = xo < -1, we obtain x = 0(r), where 2 )2 - In I0(r) I = -r + 2 - In(-xo).

Observe that d0(r)

0(r) >0 0(7)2-1

dr

if

d(r) < -1.

2

Note also that setting ro = 2 - In(-xo) -- 2 > 0, we obtain ,(ro) = -1 and 45(7-)2

- 1 > 0 for 0:5 r < ro. The graph of 0(r) is given in Figure 22.

FIGURE 22.

(i) Behavior for 0 < r < 7-o -bo, where do > 0: Let us denote by x(r, A) the solution to the initial-value problem (X.8.3)

d2z

(x - 1) dx x = 0, dr- + 2

a dr2 +

x(O, A) = xo,

40,A)

= 7-7,

where the prime is d7- and q is a fixed constant. Using Theorem X-7-1 (Na6J, we derive

I

Ix(r, A) - O(T)I 5 AK, Ix'(r, A) - 4'(r)l 5 I7-7 -

0'(0)le-P'1x

+ AK,

331

8. A SINGULAR PERTURBATION PROBLEM

for 0 < T < To - 60, where K and a are suitable positive constants.

(11) Behavior for lx+1l< o: First note that lim p0'(T) _ +oo. Set ¢'(ro-2bo) _

1Pi > 0. Then, there exists T1(A) for sufficiently small A > 0 such that { O
llm T1(A) _ 7.0 - 280, X

x(T1(A),A) _ O(Tp - 26p),

A) >

x'(T,

for

r1(A) < T < TO - 60.

2P1

Set p =

1A)

Then,

(X.8.4)

ALP

= p2(x2 - 1) + p3x,

regarding p = p(x, A) as a function of x and A. Note that

0 < p(x, A) < 2p'

for

¢(ro -26o) < x < x(ro-b0, A) < -1,

0 < A < AO,

where A0 is a sufficiently small positive number. It is important to notice that if we make 60 small and if we make A0 also small accordingly, we can make p1 small. Set (X.8.5)

o = -1 - ¢(T0 - 26o)

M0 = max Ix2 - 11.

and

I1+xI
Then, (X.8.6)

o > 0,

lien o = 0,

Furthermore, 0 < p1 < Mo since P1

=

and

1 - ((TO - 25p)2

lim Mp = 0.

0-+0+

Therefore,

0(TO - 260)

0 < p(z, A) < 2Mo

for

¢(r0-25o) < x < x(ro - 50, A), 0 < A < A0.

Now, we shall show that (X.8.7)

0 < p(x, A) < 2Mo

for

If (X.8.7) is not true, there must exist a p({, A) = 2M0,

Ii + xt < o,

0 < A < A0.

such that

0 < p(x, A) < 2Mo for -1-a:5 x<

Then, this implies that

5

A)2(Mo +

A)2(1 + g) < 0.

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

332

This is impossible.

(iii) Behavior for -1 + or < x < 2 - al: To begin with, it should be remarked that

J(2_1)d

[..-_]2 i

(X.8.8) 1)d. =

=

8

-2- ( -3+1J

-x)-3<0, for

C3

= 0,

-1
Fix a positive number al so that

j(2 - 1)4 < - y

(X.8.9)

-1+a< x < 2-al,

for

where y is a sufficiently small positive constant. Note that ry --+ 0 as do

0+ and

al -0+-

x,,\-

Integrating (X.8.4), it follows that

p(z )

= P( I, A) - J

(C2 - 1)

(I)P(

-

f

agd

1

x<0. Hence, if we choose \0>0

so that

A

y

7

0 < A < A0,

for

< 4K

where

K=

f

2

0

w e obtain 0 < p(x, A) < 9K for -1+a
(II) p(x A) > y

-K max

0< Then, 0 < p(x, A) <

A) for 0 < x < 2 - al. Suppose that

2K 2

0<

for

< x, 0 < A < A0.

< L. Thus, we proved that 2K

(X.8.10) 0 < p(x,A) < 2K

for

-1+a < x < 2 - al,

0 < A < A0.

(iv) Behavior for 2 - al < x < 2 + a2: First look at

p(2 - at, A)

2&0), A) -

1) +

p(, )ld

333

8. A SINGULAR PERTURBATION PROBLEM

Notice that p(2 - al, A) is independent of 5o. Then, it can be shown that A

0+

as

and A

a, -+ 0+

0+.

p(2 - a,, A)

For a fixed positive number a2, assume that

0 < p(x, A) < 1

for

2 - al < x < 2 + a2i 0 < A < Ao.

Then, 2+02

P(2 + 2, A)

12-,,

p(2 -- o , A)

g2 - 1) + P( A)fl

> Q.

Hence, 2+0a

J -o,

g2 - 1) +

< p(2 - Aal, A)

for 2 - al
This is a contradiction if al and A are small. Therefore, if Ao > 0 is sufficiently small, there exists an x(A) such that p(x(A),A) = 1

and

2 - al < x(A) < 2 + a2

for

0 < A < Ao.

It can be shown that ao* Urn x(A) = 2. Setting r(x(A)) = r(A), it can be shown that liM +r(A) = ro. Now again, apply Theorem X-7-1 (Na6) to the initial-value problem

AT2 + (x2 - 1)d + x = 0,

x(r(A)) = x(A),

x'(r(A)) = 1.

Figure 23 shows the general behavior of x(r, A) for small positive A.

TO

FIGURE 23.

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

334

EXERCISES X

X-1. (1) Show that if F(t,yl,y2) is continuous and bounded on a < t < b, [yl[ < +oo, 1y2[ < +oo},

Il = {(t, Y1, Y2)

the boundary-value problem

F (t, y, &)

dt2 =

!LY (a)

()

= a,

y(b)

has a solution. (2) Does the boundary-value problem dtY dt2

= F t, y, dY dt

dy(a) dt = a'

dY (b)

dt

_3

have a solution? (3) Show that the boundary-value problem d2x dt2

x(-2) = A, x(3) = B

= tx + x3,

has a solution for any real numbers A and B.

Hint. A counterexample for (2) is

d2

= 0,

(n) = 0,

(b) = 1.

where -a For (3), apply Theorem X-1-3 [Na4j with w,(t) = a and w2(t) and 3 are sufficiently large positive constants. X-2. Find the global phase portrait of each of the following two differential equations: d2x

(i)

+ x2(1 - x)2 dt +

(x - 1)2(x + 1)x = 0;

2 d2x dx2

+

dx dt

-

1

l + x2

= 0.

X-3. Suppose that (i) f (x, y, t) is continuously differentiable everywhere in the (x, y, t)-space (i.e., in p3), (ii) g(x) is continuously differentiable in -oo < x < +oo, (iii) e(t) is continuous on 0 < t < +oo, (iv) f (x, y, t) 0 for x2 + y2 > r2 and t > 0, where r is a positive number r, r>

(v) G(x) = J g(t)dC --' +oo as jx[

+oo,

0

(vi) 1 je(r)Idr is bounded for 0 < t < +oo. 0`

Show that every solution (x(t), y(t)) of the system dy = dx _ -f(x,y,t)Y dt -

Y

'

df

is well defined and bounded for 0 < t < +oo.

- g(x) + e(t)

335

EXERCISES X

Hint. Set V (x, y) = 2 + G(x). Then, d = - f (x, y, t)yz + e(t)y. There exists a positive number ro > r such that V(x, y) > 4 for x2 + y2 > ro. Hence, if an orbit (x(t), y(t)) satisfies conditions that x(t)2 + y(t)2 > ro for to < t < ti, we obtain

V(x(t),y(t))

for

to
V(x(to),y(to)) + ft., (e(r)!dr

for

to
d V(x(t),y(t)) < Ie(t)Iiy(t)I <2(e(t)j This yields

V(x(t),y(t)) <

X-4. Show that the differential equation d yi _ ill dt [y2 ] [ -E(yi - 1)y2 - i/i - Eyi has exactly one periodic orbit, where a is a positive parameterX-5. Find an approximation for the unique periodic orbit of (E.1) for small e > 0 and for large e. X-6. Considering the system of two differential equations d yi y2 (E.2) dt [ y2] [ -e(yi - 1)y2 + yi - byl where a and 6 are positive numbers, (a) show that every orbit is bounded for t > 0, (b) determine whether each stationary point is stable or unstable, examining every possibility, (c) using the function ( E.1 )

-

! y,

-y2 + E(-yi + 6YNY2 + J

(-s + bs3J[1 + E2(si)]ds

0

and Theorem IX-2-2, show that if 0 < 6< 3, every orbit of system (E.2) tends to one of the stationary points as t -' +oo, (d) discuss the uniqueness of periodic orbits. X-7. Show that the differential equation d2x

+ f (x)

d+ 9(x) = 0

dz has at most one nontrivial periodic solution if fix) and g(x) are continuously differentiable in !R and satisfy the following conditions

<0

(i) (ii)

(iii)

f (x) { > 0 9(x)

<0

for

> 0

for

J

for

1x3 < 1,

for

IzI > 1,

-2 0,

g(x)dx = 0.

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

336

Hint. See [Sat]. The main ideas are as follows: (1) If there are more than one nontrivial periodic orbits, then at least one of them should not be orbitally asymptotically stable.

(2) For a nontrivial periodic orbit (x(t), x'(t)) of period T > 0, the inequality

fT

f (x(t))dt > 0 implies that this orbit is orbitally asymptotically stable o

(cf. Exercise IX-5). (3) Set

a(t) = V(x(t), x'(t)) = Then,

da(t) dt

(

x'(t))2 2

f

+

_ -f(x(t))(x'(t))2

x(t)

.

Therefore, we obtain

dA J f(x( t) )dt = - Jr (x '

(I)

)

2

Now, investigate the quantity (x'(t))2 as a function of A along the periodic orbit (x(t), x'(t)). For a fixed value of A, compare (x'(t))2 for different values of x(t), using the assumptions on f and g. Step 1. First the following remarks are very important. (a) If we set y = x', the given differential equation is reduced to the system

dt = -f (x)y -

dt = y,

(S)

A careful observation shows that the index of the critical point (0, 0) is 1, while the index of the critical point (-2,0) is -1. There are only two critical points of (S). (c) The periodic orbit and any line {x = a constant) intersect each other at most twice.

(d) The critical point (-2,0) should not be contained in the domain bounded by the periodic orbit. To show this, use the fact that the index of any periodic orbit is 1. These facts imply that the periodic orbit should be confined in the half-plane x > -2. 2

Step 2. A level curve of V (x, y) = 2 +

r:

J0

is a closed curve if the curve is

totally confined in the strip jxj < 1. In particular, (1, 0) and (-1, 0) are on the same

level curve V (x, y) = / 0

1

Io

_ 1 g(C)d{. Since A(t) is increasing as long as

jx(t)I < 1, the periodic orbit cannot intersect the level curve V(x,y) = / 0

,

Therefore, the periodic orbit must intersect the lines x = 1 (as well as the line x = -1) twice. Denote these four points by (1,s1(A)), (1,-rt(B)), (-1,-q(C)), and (-1, 77(D)), where r)(A), n(B), q(C), and q(D) are positive numbers.

337

EXERCISES X Step 3. Set

= V(1, rl(A)), AB = V (1,

AA

-n(B)),

Ac = V(-1, -n(C)), AD = V (-1, rl(D))

Then, Set

AA > AD-

AC > AD,

AC > AB,

AA > AB,

Z(A) = { A : AA > A > max(AB, AD) }, Z(B) = { A : AB < A < min(AA, )'C) }, Z(C) = { A : Ac > A > max(AB, AD) }, Z(D) = { A: AD < A < min(AA, Ac) }.

Note that the interval lo = JA: min(AA, Ac) > A > max(AB, AD)} is contained in Z(A) n Z(B) n Z(C) n I(D). Now,

(a) on the arc between (1, rl(A)) and (1, -rl(B)) of the periodic orbit, regard y as a function of A and denote it by yi (A), where AB < A 5 AA, (Q) on the arc between (1, -rl(B)) and (-1, -r1(C)) of the periodic orbit, regard y as a function of A and denote it by y2 (A), where AB < A < AC, (ry) on the are between (-1, -rl(C)) and (-1, rl(D)) of the periodic orbit, regard y as a function of A and denote it by yi (A), where AD < A < AC, (6) on the arc between (-1, rl(D)) and (1, rl(A)) of the periodic orbit, regard y as a function of A and denote it by yz (A), where AD :5 A < AA. Then, it can be shown that yi (A)2 < y2 (A)2 yi (A)2 < y2 (A)2

on on

I(A),

yi (A)2 < y2 (A)2

on

I(C),

yi (A)2 < y2 (A)2

on

Z(D).

Z(B),

Step 4. Now, fixing a AO E Z(A) n Z(B) n 1(C) n T(D), evaluate the integral (I) as follows: rT

J

{JA8 f(s(t))dt = -

rc

dA

w yi (A)2 +

J

dA

y2 (A)2

+'Ac

where dA

J\A rac

dA

_

ra°

AC

dA

Joe y2 (A)2 = Jaa y2 (A)2 I"

dA

ra°

+

(A)2,

dA y2(A)2,

Jib rAc

dA

dA

yi (A)2 = - JaD yi (A)2 - ao yi ao ys (A)2

aD y2 (A) 2

Jao

IA a

dA

yi

lao

as yi (A)2

AA

yl (A)2 +

r"

dA

yi (A)2

dA

rAD

y2

(A)2.

(A)2,

dA

1

y2 (A)2 T ,

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

338

Thus, we conclude that

T f (x(t))dt > 0.

J

This implies that every periodic orbit is orbitally asymptotically stable. Hence, there exists at most one periodic orbit.

X-8. For a nonzero real number c and a real-valued, continuous, and periodic

function f (t) of period T which is defined on -oo < t < +00, find a unique solution 0(t, E) of the differential equation

E

dt = -y + f (t)

which is periodic of period T. Also, find the uniform limit of 0(t, c) on the interval

-oo
X-9. Assuming that X(t) satisfies the condition

(X(t)2 -

X(t) = 0

and

IX(t)I > 1

on an interval 0 < t < t, where a is a positive constant, and that x(t, A) is the unique solution to the initial-value problem

22 + (x2 -1) dt + x = 0,

find lim d2x(2' a-.o+

x(O, A) = X (O),

x'(O, A) = X'(O),

as a function of X (t) for 0 < t < Q.

dt

X-10. Assuming that y(t, E) is the solution to the initial-value problem d2

Y t2 + 2ydy = 0, dt2 dt

y(0) = a, y'(0) =

Q

where e is a positive parameter, or is a real number, and ,8 is a positive number, find lim y(t, e) for any fixed t > 0. f 0+

Hint. Look at E dt = c - y2. Then, c = Q + a2 > a2. Hence, y2 increases to c very quickly. Then, use Theorem X-7-1 [Na6], or find the solution explicitly (cf. Exercise I-1).

X-11. Find, if any, solution(s) y(t, c) of the boundary-value problem

4

dt2 + 2ydty

= 0,

y(0) = A, y(1) = B,

in the following six cases: (1) 0 < B < A, (2) B = A, (3) B > IAA, (4) B = -A > 0, (5) CBI < -A, and (6) B < 0 < A, assuming that e is a positive parameter. Also, find lim y(t, e) for 0 < t < 1 in each of the six cases. e

0+

EXERCISES X

339

Hint. Use explicit solutions together with the Nagumo Theorems (Theorems X1-3 and X-7-1) on boundary-value and singular perturbation problems. See also Exercises X-10 and X-12, and [How].

X-12. Let f (x, y, t, e) be a real-valued funcion of four real variables (x, y, t, e). Assume that (i) 0(t) is a real-valued, continuous, twice-continuously differentiable function

on the interval Zo = {t

:

0 < t < 1) and satisfies the conditions 0 =

f (¢(t), 4'(t), t, 0) and 4(1) = B on Zo, where B is a given real number, (ii) the function f (x, y, t, e) and its partial derivatives with respect to (x, y) are continuous in (x, y, t, e) on a region R = {(x, y, t, e) : Ix - ¢(t)I < rl, IyI < +00, t E Zo, 0 < E < r2}, where r1 and r2 are positive nmbers, (iii) If (4(t), 4'(t), t, e) I < Ke for t E Zo, where K is a positive number,

(iv) there exists a positive number µ such that Of (x, y, t, e) < -p on R, (v) there is a positive-valued and continuous function '(s) defined on the interval J+00

+oo and that If (x, y, t, e)I < +'(IyI) on

0 < s < +oo such that R,

(vi) A is a given real number. Then, there exists a positive number co such that for each positive f not greater than co, there exists a solution x(t, E) of the boundary-value problem d2x

f d#2 = f (X,

d , t, E) ,

x(0, e) = A, x(1, c) = B

such that Ix(t,E) - di(t)I < IA - h(0)led=te-ft/` +C2e on Zo, x'(t, e) - 4'(t)

C3e e

+ co for 0 < c5e <_ t < 1,

where c1 i c2, c3, c4, and cs are positive numbers.

Hint. Cf. Theorem X-1-3. See also [How].

X-13. Let a(t), b(t), and f (t) be real-valued, continuous, and continuously differentiable functions of t which are defined on the interval 0 < t < 1. Also, let 4o(t) be a real-valued solution of the differential equation a(t) dt + b(t)z = f (t). For a positive number e, denote by y(t, e) the unique solution of the initial-value problem d3y

+ d-t2 + a(t)

dy

+ b(t)y = f (t),

3

y(0) ='7o(E), y'(0) = 171 W, y"(0) = 12(E)-

Show that ly(t, e) -00(t)I+ Iy'(t, e) -40 (t)I +Iy"(t, e) -40 (t) I tends to zero uniformly

on the interval 0:5 t < 1 if

e-0+.

4o(0)I + Iti1(E) - 40(0)I + Irn(E) - 0011(0)1 --+ 0 as

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

340

X-14. Let Y E lR", y E Rm, t E R, and e E R. Also, let ft f, y, t, e) and g(x", y, t, e) be respectively R'-valued and 1R'"-valued functions of (x, y, t, e). Assume that (i) an R"-valued function fi(t) and an R'-valued function t%i(t) are continuous and continuously differentiable on an interval Zo = {t : a < t < b} and satisfy the system of equations dx"

dt

= f (Y, i, t, 0),

0 = g(x, y, t, 0)

on Zo,

(ii) f (Y, y, t, e), g(x', y, t, e), and their partial derivatives with respect to (x, y-) are continuous on a region R = {(:e, y, t, e) : I x - m(t) I < rl, Iy - j(t) I < r2, t E lo, 0 < e < r3 }, where rl, r2, and r3 are positive numbers,

(1(t), ti(t), t, 0) are less than a negative num-

(iii) all eigenvalues of the matrix ber M on .70.

Show that the initial-value problem d:5

T = f (:

, y, t, E),

a

Y

dt =

§(Y, 9, t, f),

x(a) = ((e),

y(a) = #(f)

has a unique solution (2(t, e), y(t, f)) for t E Zo and 0 < e < r4 if e + IC(e) (a)I + Ir(e) - t'(a)I is sufficiently small on 0 < e < r4i where r4 is a positive number. Also, show that Ii(t, e) - ¢(t)I + I y(t, e) - tj,(t)I - 0 uniformly on Zo as

e + IC(E) - (a)I + I#(E) - (a)I - 0. Hint. See [LeL].

X-15. Let x E lR", y E R"', t E R, and e E R. Also, let t, e) and g(i, y, t, e) be respectively R"-valued and R'-valued functions of (i, y", t, e) which are periodic in t of period T. Assume that (i) an R"-valued function fi(t) and an R"'-valued function t%'(t) are continuous, continuously differentiable, and periodic of period T on the interval Zo = it :

-00 < t < +oo} and satisfy the system of equations

d = f (:F, y", t, 0), 0 =

g"(x, y, t, 0) on Za,

(ii) f(i, y", t, e), g(i, y", t, e), and their partial derivatives with respect to (x, y') are continuous on a region R = {(i, y, t, e) : Ix - m(t)I <- ri, I y - 7G(t)I < r2, t E Zo, IEI < r3}, where rl, r2, and r3 are positive numbers,

(iii) there exists a real m x m matrix P(t) such that (iiia) the entries of P(t) and P(t)-1 are real-valued, continuous, continuously differentiable, and periodic of period T[Bat on Zo, (iiib)

P(t)8b(m(t),1(t),t,0)P(t) =

B2(t)], where Bi(t) is a real

m1 x ml matrix with eigenvaules havingnegative real parts on 10i while B2(t)

is a real (m - ml) x (m - ml) matrix with eigenvalues having positive real parts on Zo,

(iv) the system

EXERCISES X

d,F= .iF

341

8iq(t), (t),t,0) - Of (fi(t), G(t), t, 0) [

w(t), +G(t), t,

0)J

ON(* it), t, 0) z

does not have nontrivial periodic solutions of period T. Show that the system

di dt - f(s,y,t,f),

dy"

adt

=9'(2,y,t,e)

has a unique periodic solution (a(t, e), y(t, e)) of period T on To if 0 < Iej is sufficiently small. Also, show that jT(t, e) - (t)I + lc(t, e) - iJJ(t)j 0 uniformly on To

Hint. See [FL]

CHAPTER XI

ASYMPTOTIC EXPANSIONS

In §§V-1 and V-2, we defined formal solutions of a system of analytic differential equations. Formal solutions are not necessarily convergent. For example, as we mentioned it in Remark V-1-4, the divergent formal power series 00

f=

(-1)m (m! )x'"+1 is a formal solution of x2 m=0

dy

+ y - x = 0. This equation

r_

has an actual solution f (x) = el/=J t-1e-hIldt for x > 0. Integrating by parts, we N

obtain f =

0

tNe-l'tdt. Since 0 <

(-1)m(m!)xm+1 + (- 1)N+1((N + 1)!)e1/x fox

M=0

el/=10,7 tNe-l/tdt = xN+2 - (N + 2)el/= r stN+1e-l/tdt < xN+2, we conclude that f (X)

-

0

N

(_l)m(m!)xm+l < ((N + 1)!)xN+2 for x > 0. This is an example of M=0

an asymptotic representation of an actual solution by means of a formal solution. In this chapter, we explain the asymptotic expansions of functions in the sense of Poincare and in the sense of the Gevrey asymptotics. In the Poincare asymptotics, flat functions are characterized by the condition lim E f) = 0 for all positive intexm gers m, whereas in the Gevrey asymptotic, flat functions are characterized by the condition If (x) I exp(cIxI-k) < M as x -i 0, where c, k, and M are some positive numbers. Generally speaking, the Poincare asymptotics is too general for the study of ordinary differential equations. A motivation of the Gevrey asymptotics is also given by the Maillet Theorem (cf. Theorem V-1-5). In §XI-1, we summarize the basic properties of asymptotic expansions of functions in the sense of Poincare. The Gevrey asymptotics is explained in §§XI-2-XI-5. For more information concerning the Poincare asymptotics, see, for example, [Wasl]. The Gevrey asymptotics was originally introduced in [Wat] and further developed in [Nell. To understand the materials concerning the Gevrey asymptotics of this chapter, [Ram 1], (Ram 21, [Ram3], [Si17, Appendices], [Si18], and (Si19] are helpful.

XI-1. Asymptotic expansions in the sense of Poincare In this section, we explain the asymptotic expansions in the sense of Poincare. Let x = a be a point on the extended complex x-plane. Consider a formal power series 00

(XI.1.1)

P(x) = E c,(x - a)m. M=0 342

1. ASYMPTOTIC EXPANSIONS IN THE SENSE OF POINCAR$

343

Let D be a sector in the x-plane with vertex at x = a and Do be a neighborhood of x = a in D. Assume that f (x) is defined and continuous in Do. Definition XI-1-1. The formal series (XI.1.1) is said to bean asymptotic (series) expansion of f (x) as x -k a in V if for every non-negative integer N, there exists a constant KN such that N

(XI.1.2)

f(x)->2cm(x-a)m

< KN Lx - aIN+r,

N = 0, 1, 2,- ..

M-0

for all x in Do. Such an asymptotic relation is denoted by f(x)

(XI.1.3)

p(x)

as x

a in D.

This definition of an asymptotic expansion of a function was originally given by H. Poincare [Poi2J. Before we explain some basic properties of asymptotic expansions, it is worthwhile to make the following remarks.

Remark XI-1-2. The vertex x = a can be x = oo. In that case, the asymptotic on

C nx-"'.

series is in the form m=0

Remark XI-1-3. Assume that f (x, t) is a function defined and continuous in (x, t) for x in D and t a parameter in a domain H in the t-plane. A formal power series N

F c,,,(t)(x-a)'", where c,(t) is a function oft, is said to bean asymptotic (series) M=0

expansion of f (x, t) as x 0 in V if for every non-negative integer N, there exists a function KN(t), independent of x, such that N

f(x, t) -

Cm(t)(x - a)m
N=0,1,2,...

M=0

for all x in Do. If, moreover, KN(t) are independent of t, then the asymptotic expansion is said to be uniform with respect to t. Remark XI-1-4. If f (x) is holomorphic (i.e., analytic and single-valued) in a neighborhood of x = a, by virtue of the Taylor's Remainder Theorem, f (x) admits its Taylor's series expansion as its asymptotic expansion in any sector with vertex

at x = a. Theorem XI-1-5. For a continuous function f (x), there is at most one asymptotic expansion as x -+ a in a sector with vertex at x = a.

Proof Assume that there are two asymptotic expansions of f (x) at x = a 00

(XI.1.4)

c,,(x - a)14

f (x) ^_M=0

as x -a in D

XI. ASYMPTOTIC EXPANSIONS

344 and

7m(x-a)'

f(x)^,

as x -i a in D.

M=0

Then, for every non-negative integer N, there exist two constants KN and LN such

that N

N+i , f (x) - > cm(x - a)m < KN Ix - al

N = 0, 1, 2, .. .

m=0

N

N=0,1,2,...

,1.(x - a)m
.f (x) m=0

for all x in a neighborhood Do of x = a in D. For N = 0, we have Ieo - 7ol < (Ko + Lo)lx - al for x in Do. Let x

a in D; then, we obtain co = 70

Now, assume that Ck = 7k for k = 0, 1,... , N - 1. Then, from (XI.1.5) and (XI.1.6), it follows that ICN -7NIIx-aIN < (KN+LN)Is-alN+l for x in Do. Let x -+ a in D. Then, cN = 7N. Thus, cm = 7m is true for every non-negative integer

m. O

Theorem XI-1-6. Assume that f (x) and g(x) are two continuous functions such that (XI.1.7) 00

f (x) > cm(x - a)'

and g(x) ^-' E 7m(x - a)"'

as x --+ a in D.

m=0

M=0

f (x) ± g(x)

00

(cm ± 7.)(x - a)m

as x -+ a in D

M=0

f(x)g(x)

1

Ch7k) (x - a)m

as x -+a in D.

m=0 h+k=m Proof. N

Fix a non-negative integer N and put f (x) _ > cm(x - a)'+Ei(x)(x-a)N+i M=0

N

and g(x) _ > 'ym(x - a)m + E2(x)(x - a)N+1 Then, there exists two constants m=0

KN and LN such that (XI.1.10)

IE1(x)I <_ KN,

IE2(x)I <_ LN

1. ASYMPTOTIC EXPANSIONS IN THE SENSE OF POINCAR$

345

for x in Do. Therefore,

N

If (x) ± 9(x)] - E I(Cm ± 1'.](x - a)m < (KN + LN)I x - aIN+1

(XI.1.11)

M=0

for x in Do. Thus, (XI.1.8) holds. Also, it is easy to see that there exists a positive

constant k such that f(x)9(x) -

(

Ch7k) (x - a)m

< KIx - aIN+1

m=0 h+k=m

for x in Do. Thus, (XI.1.9) holds.

Theorem XI-1-7. Suppose that f (x) is a function holomorphic in 0 < Ix - al < 8 and admits an asymptotic expansion (XL1.4) with V = (0 < Ix - at < b}. Then, the asymptotic series actually converges to f (x) in 0 < Ix - at < S. Proof.

ra

Since lim f (x) = co, x = a is a removable singularity. Thus, f (x) is holomorphic in Ix - al < b. Therefore, the asymptotic series agrees with the Taylor's series (cf. Remark XI-1-4 and Theorem XI-1-5). Thus, the asymptotic series converges in Ix - aI < 8.

For a vector z E cm with entries (z1,z2,... zm) and p = (p1,p2,... Pm) with non-negative integers p, (j = 1, 2,... , m), we define I6oI = P1 + p2 + + p,,, z' = zr z2 ... zm , and Iz7 = max{Iz1I, IZ21, ... , Ixml}. Theorem XI-1-8. Suppose that F(x, z) is a function with power series expansion 00

F(x, ) =

F.(x)zY, which converges uniformly forx E D, I zl < bo, where Fp(x) Ip1=o

00

is continuous in D and admits an asymptotic expansion Fp(x) c 1: Fpk(x - a)k k=0

as x -, a E D. Define the formal power series in (x, 00

4i(x, 2) = E 4ip(x)iV,

(XI.1.12)

lpI=0

where

0 4ip(x) _ E Fpk(x - a)k.

(XI.1.13)

k=0

If f (x) is a continuous Cm-valued function with entrywise asymptotic expansions 00

f (x) ^-- > f (x - a)' __ p"(x) as x -' a in D, 1=0

XI. ASYMPTOTIC EXPANSIONS

346

then 45(x, pjx) - fo) defines a formal power series of (x - a) and (XI.1.14)

as x -+ a in D.

F(x, f (x) - fo) = 4'(x, p(x) - fo)

Proof.

Choose p so small that I f (x) - fol < 6o for Ix - al < p. Fix a positive integer N and put N

00

F(x,f(x) - fo) = E F,(x)(f(x) - fo)p +

> Fp(x)(f(x)

- fo)p.

Ipl=N+1

Ip1=0

Then, by virtue of Theorem X1-1-6, we obtain N

N

/.-y

F,,(x)(f(x) - f0)p

Ip1=0

o)'.

1p1=0

00

E Fp(x)(f(x) - fo)p = O(Ix - alN+1). Ipi=N+1 Set

N(x) =,D(x,Plx) - fo) = >2 Hm(x - a). M=0

Then, N

N

(XI.1.17)

Hm(x - a)m +O((x - a)N+1)

op(x)Q3(x) - f0)p = 1p1=0

m=0

N

N

and, hence, (XI.1.18)

E Fp(x)(f(x) - fo)p = E Hm(x - a)m +O(Ix - aIN+1). m=o

Ip1=0

Since this is true for all positive integers N, the theorem follows immediately. 0 Theorem XI-1-9. Suppose that f (x) is a continuous function with asymptotic ex-

Let p(x) denote the formal power series (X7.1.1). Then, Pox) defines a formal power series in (x - a) and f(x) ox) as x -+a in D. pansion (XI.1.4) and co

34

0.

Proof.

Consider F(z) =

I

co+z

.

Then, the theorem follows from Theorem X1-1-8.

0

1. ASYMPTOTIC EXPANSIONS IN THE SENSE OF POINCAR$

347

Theorem XI-1-10. Suppose that f (x) is a continuous function with asymptotic expansion (XL 1.4) as x -' a in D. Let p(x) denote the formal power series (XI. 1.1).

Ten, J

1

f(t)dt

as x --* a in D,

m+1cm(x-a)m+1

where the path of integration is the line segment joining a and x, i.e.,

as x -a in D.

f a=P(t)dt

f aaZ f(t)dt

Proof N

Set pN(x) _ E c, (x - a)'. From (XI.1.4) and (XI.1.5), it follows that m-0

If (t) -PN(t)Idtl 5

J 0x_al If(C)

PN(t)l ds

Js-al

KN

J0

sN+1

for every non-negative integer N, where s = It - al.

= KN Ix - al N+2 N+2 O

Theorem XI-1-11. Suppose that f (x) is a holomorphic function with asymptotic expansion (X1.1.4). Let p(x) denote the formal power series (XI.1.1). Let D be a

proper subsector of D with vertex x = a. Then, I dp

= E mr,,,(x 00

a)m-1

ax as x -- a in D, where

.

m=1

Proof.

For a non-negative integer N, put f (x) = pN(x) + EN(x). Then, there exists a positive constant KN such that

EN(x)I 5 KNIx - alN+'

(XI.1.19)

for x in a neighborhood of x = a in D (cf. Definition XI-1-1). Let the radial boundaries of D be arg(x - a) = 6t and those of D be arg(x - a) = 0± (9_ < 0_ < 0+ < 9+). Let 0 = min{6+ - 0+, 0_ - 9_ }. Let x be a point of D. Consider a circle I' _ {t1 It - xI = 2Ix - al sin 9 }. Then, r and its interior are contained in D. Using Cauchy's integral formula, we obtain df (x) dx

=

1

r

f

2rri Jr (t - x)2

__

1 r PN (t) 2rri Jr (t x)2

-

I + 2rri

Ir (t -

EN (C)

x)2

dt.

XI. ASYMPTOTIC EXPANSIONS

348 ei4'

2Ix-al sin 0.Then,

(

r_(rll>I= e'4'2Ix-aIsin0,0<0<27r l

and (XI.1.19) implies 1

1 EN ()

tai

r (x)2

11

I fEN()drl 172

_

1

2x

1

EN(e)e"'dcb = 0 (Ix - aIN)

7

.

Thus, the theorem is proved.

Theorem XI-1-12. Suppose that {fn(x)In = 1,2,3,...) is a sequence of continuous functions such that they admit unifonn asymptotic expansions 00

a)'

fn(x) ^-- >

(XI.1.20)

(n = 1, 2, 3, ...) as x --+ a in D.

M=O

Assume that {f(x)In = 1,2,3.... } converges uniformly to a function f (x) in a subsector Do of D with vertex x = a. Then, lira Cnm = Cm

00

f (x) ^- E cm(x - a)'

as x -+ a in Do.

M=0

Proof The assumption implies that for each non-negative integer N, there exists a positive constant KN, independent of n, such that N

(XI.1.23)

a)'1

I

(n = 1,2,3,... )

:5

M=0

for x in a neighborhood of a in D. Furthermore, for each pair of positive integers (j, k), there exists a positive constant bjk such that (XI.1.24)

If,(x) - fk(x)I S bjk

for x in Do, where (XI.1.25)

bjk-+0

as j, k -+ oo.

Put N

(XI.1.26)

pjN(x) =E cjm(x - a)n' M=0

(j = 1, 2, 3, ... ).

1. ASYMPTOTIC EXPANSIONS IN THE SENSE OF POINCARE

349

Then, (XI.1.27)

(j = 1,2,3,...).

Ifj(x) -P)N(x)I 5 KNIx-alN+1

Thus, (XI.1.28)

IP) N(x)-PkN(x)I 5 5)k+2KNIx-alN+l

(j,k=1,2,3,...).

In particular, (XI.1.29)

Ic)o - ckol < b)k+2Kolx-al

(j,k= 1,2,3,...)

for N = 0 and x in Do. Therefore, (XI.1.25) and (XI.1.29) imply that lim c)o = CO exists.

Assume that lim c),,, = c,,, exist for m < N. Then, from (XI.1.28), it follows 1

that

00

N-1

E (C),,, - Ckm)(x - a)m + (CjN - CkN)(x - a)N M=0 < S)k + 2KNIx - alN+1

(j,k = 1,2,3,...)

for x in Do. Thus, IC)N - CkNI Ix - aIN < e)k + 61k + 2KNIx - alN+1 where e)k

k=1 ,2 , 3 . .

. .

0 as j, k -. oo. Hence, Ic)N -CkNl :5-

aIN +2KNIx-al

11

Iz

(j,k= 1,2,3,...).

Therefore, lim c)N = cN. Consequently, lim c),, = c,,, for all m. ) --00 l-00 Furthermore, since (XI.1.27) holds independently of j, we obtain N

f(x) - > cm(x - a)m < KNlx - alN+1 M=0

for x in Do. Thus, (XI.1.22) holds.

The following basic theorem is due to E. Borel and J. F. Ritt.

Theorem XI-1-13 ((Bor( and [Ril). For a given formal power series in (x - a) (XI.1.30)

P(x) = > c,,,(x - a)m m=0

and a sector with vertex x = a

(XI.1.31)

D={xIO
XI. ASYMPTOTIC EXPANSIONS

350

them exists a function f (x) which is continuous in D and holomorphic in the interior of D, and 00

as x - a in D.

f (x) ^_- E c,,,(x - a)'

(XI.1.32)

M=0

Proof.

Without loss of generality, assume that the sector D contains the ray arg(x -

a) = 0. Construct functions a,,, (x) for m = 1, 2.... such that f (x) = co + 00

E ca,,,(x)(x - a)m is continuous in D, holomorphic in the interior of D, and m=1

satisfies (XI.1.32). Let bm and 0 be positive numbers, where bm are to be specified later, whereas 0 is chosen to satisfy 0 < 0 < 1 and

for x E D.

01 arg(x - a)I < 2

(XI.1.33)

Consider r

1

am(x) = 1 - exp l -

(XI.1.34)

Then, am(x) are holomorphic in a the fact that

2m1,m(x - a)0 J ,

`sector

m = 1, 2, ... .

containing D. Also, from (XI.1.33) and

I1-esl=l jetdt
for

z<0,

it follows that bm

(XI . 1 . 35)

lam(x)I < - 2mymlx - aIe'

m= 1 , 2 ,...,

in D. Put bm =

(XI.1.36)

I Icml-'

if c,,, 0 0, if Cm=0.

0

Then, 00

00

Icmam(x) (x - a)mI < E m=1

M=1

Ix - alm-e 2m.ym

for x E D.

Hence, Oo

f (x) = co +

E c,nam(x) (x - a)m

m=1

converges uniformly in D. Consequently, f (x) is continuous in D and holomorphic in the interior of D.

1. ASYMPTOTIC EXPANSIONS IN THE SENSE OF POINCARE

351

To show that f (x) satisfies (XI.1.32), let N be a positive integer and observe that N+1

N+1

AX) -0D - E cm(x - a)m

=EC

m=1

m=1

exp

2--y-(Z -

ax - a)m 7)01(

00

+ (x - a) N+1

i cmam(x)(x -a)

m-N-1]

.

Lm=N+2

Since (XI.1.33) implies that r = 0. 1, 2,... , N,

2'n7m(x - a)e ] Ix - aj-"I,

iexp [

are bounded for x E D, there exists a positive constant H1 such that N

,,=1

r c-exp _ I

m-1

bm

2mrym(x


- a)e] (x - a)m

Also, (XI.I.35) implies that there exists a positive constant H2 such that 00

E cmam(x) (x -

1

a)m-N-1

-aim-N-1

Ix

[00

(27)N+1Ix - aj0 m=N+2

m=N+2

- i (2y)N+2 +2 I

{x

- aI - H2

(2'r)m-N-1

for X E D.

2y

Thus, there exists a positive constant KN such that N

f(x)-

c,n(x-a)m
for xED.

Therefore, (XI.1.32) is satisfied. 0 Similarly to Theorem XI-1-13, we can prove the following theorem.

Theorem XI-1-14. For a given formal series p(x, t) _

00

cm(t)(x - a)m in pour

m

ers of (x - a and a sector D given by (XI.1.31), where q. (t) (m = 0, 1, ...) are holomorphic and are bounded in a domain S2, there exists a function f (z, t) which is holomorphic with respect to (x, t) in D x S2 and satisfies the conditions (XI.1.37)

c,,,(t)(x - a)m

A x , t) M=0

as x -+ a in D

XI. ASYMPTOTIC EXPANSIONS

352 and

00

( X,

(XI.1.38)

dt

as x -. a in V

dt

m=o

uniformly fort in Il. Proof. I dcd't(t)

As

6m =

{

are bounded in Sl (m = 0,1, ... ), choose

I ax 1cm(t)I]

0

+I

I d d (t) J }-1

t r:

0,

ifC,,,(t)-0.

`

L

if c -(t)

Set

00

Cm(t)am(x)(x - a)m,

f (x, t) = co(t) + m=1

where a,,, (x) are given in (XI.1.34). Then, it can be shown that f (x, t) is holomorphic with respect to (x, t) in D x 1 and satisfies (XI.1.37) and (XI.1.38) in a manner similar to the proof of Theorem XI-1-13. O Examples XI-1-15. The followings are two examples of functions admitting asymptotic expansions. 1. Let r(z) denote the Gamma function and Log z denote the principal branch of the complex natural logarithm of z. If 0 = arg z satisfies IOI < 7r, a real number w satisfies Iw I < 2 , and 10 + w l < 2 , then Log[r(z)]

rN

_ (z -

Logz - x + M=1

+ eN(z) where IeN(z) <

(

m-1

2m(2m

-(2N+1)

BN+1

2(N+1)(2N+1)z

(

cos 8 + w ))2N+1I sin(2w (see, for example, [01, p. 294]).

)I

and the B,,, are the Bernoulli numbers

2. The exponential integral function Ei(z) = N

J

ettdt has the following form: zo0 Z

Ei(z) = er J(k - 1)!z-k + N!100 ett-N-ldt. k=1

From this observation, it follows that N

e-ZEi(x) - E (k -

1)!z-kI

< {N! + (N + 1)!(7r + (N + 1)-1))

IzI-N-1

k=1

for I arg(-z)I < 7r. The asymptotic expansion is valid for I arg(-z)I <

example, (Wasl, p. 31[).

32

(see, for

2. GEVREY ASYMPTOTICS

353

XI-2. Gevrey asymptotics The Gevrey asymptotics is based on the following two definitions.

Definition XI-2-1. Let s be a non-negative number. A formal power series p = +oo

E anx' E C[[x]) is said to be of Gevrey order s if there exist two non-negative m=0

numbers C and A such that (XI.2.1)

[am[ < C(m!)'Am

for m = 0,1,2,... .

We denote by C([x]], the set of all power series of Gevrey order s.

Definition XI-2-2. A function 0 of x is said to admit an asymptotic expansion of Gevrey orders (s > 0) as x -+ 0 on a sector D(r, a, b) = {x : 0 < fix[ < r, a < arg(x) < b},

where a and b are two real numbers such that a < b and r is a positive number if (i) ¢ is holomorphic on V (r, a, b), +00

(ii) there exists a formal power series p = > a,,,x"' E G[[x]], such that an inm equality

N-1

(XI.2.2)

0(x) -

E amxm < Kv,a,A(N!)'(Bv,a,3)8[x[N M=0

holds on D(p, a, 0) for every positive integer N and every (p, a, 03) satisfying

the inequalities 0 < p < r and a < a <,3 < b, when KP,Q,o and Bp,,,,o are non-negative numbers determined by (p, a, /3).

We denote by A, (r, a, b) the set of all functions admitting asymptotic expansions

of Gevrey order s as x -' 0 on the sector D(r, a, b). We also set J(4) = p for 0 E A. (r, a, b). In §XI-3, we shall explain the basic properties of ar[[x]] A. (r, a, b), and the map J : A. (r, a, b) -+ G[[x]],. In the example given at the beginning of this chapter, the formal solution p = 00

E (-1)m(m!)x"+1 of X2!Ly +y-x = 0 is of Gevrey order 1 and the solution f (x) = M=0

eL/z Jxt-1e-lltdt admits an asymptotic expansion of Gevrey order 1. Furthermore, JJa

J(f) = p. Also, the Maillet Theorem (cf. Theorem V-1-5) states that any formal solutions of an algebraic differential equation belong to C[[x]], for some s, where s depends on each solution.

It is well known that if a complex-valued function 0(x) is holomorphic and bounded on a domain 0 < [xj < r, where r is a positive number, then 0 is represented by a convergent power series in x. The Gevrey asymptotic expansions arise

XI. ASYMPTOTICS EXPANSIONS

354

in a similar but more general situation. To explain such a situation, let us consider N sectors

St={x: 0
(1=1,2,...,N)

N

which satisfy the condition USt = (x: 0 < IxI < r}. The set {S11S2,... SN} is t=1

called a covering at x = 0. Also, a covering {S1, S2, ... , SN } at x = 0 is said to be good if (i) at < at+l (t = 1, 2, ... , N), where aN+l = al + 27r, (

(ii) bt - at < x (1 = 1, 2,... , N), 0 (1 = 1, 2, ... , N) and Stn Sk = 0 otherwise if 1,,E k, where ) S t n St+1 SN+1 = S1. The following theorem is the most basic fact in the Gevrey asymptotics.

Theorem XI-2-3. Assume that a covering {Si, S2, ... , S1 %r) at x = 0 is good and that N functions 01(x), ¢2 (x), ... -ON (x) satisfy the conditions (1) 4t(x) is holomorphic on St, (2) dt(z) is bounded on St, (3) it holds that (XI.2.3)

on St n St+1,

I¢e(x) - 01+1(x)I < 'Yexp [-

where -y > 0, A > 0, and k > 0 are suitable numbers independent of 1. Set

(X1.2.4) +00

Then, there ex,sts a formal power series p = E a ..xm E C[[x)]. such that ¢t E M=0

A. (r, at, bt) and J(41) = p for each 1. There are various situations in which Gevrey asymptotic expansions arise. To illustrate such a situation, let ¢(x) be a convergent power series in x with coefficients

in C. For two positive numbers r and k, set k

x

rr

J0

0(t)e-('1x)}tk- 1 dt.

This integral is called an incomplete Leroy transform ofd of order k. +oo

Theorem XI-2-4. For every ¢ =

cx"j E C{x}, it holds that m=0 R

if

4,k(0) E Al/k P,-2k'2k

355

2. GEVREY ASYMPTOTICS and

+00

r 1 1 + k Cm2'",

J(tr,k(4)) _ M=0

where k and p are any positive numbers and r is any positive number smaller than the radius of convergence of 0. Proof.

The following proof is suggested by B. L. J. Braaksma. For an arbitrary small positive number e, let {Sl (e), S2(e), ... , SN(c)} be a good covering at x = 0, where

St(E) _ {x : I argx - dtI < 2k - e, 0 < jxj < r} ir, and set

with real numbers dt such that -ir < d 1 < . - - < dN

J OtW _

(t = 1, 2, ... , N).

I. 4t(x) = tr,k(0t)(xe-d`) In particular, choose dt = 0 for some t. Then, {Sj(e),S2(e),... ,SN(e)) and {01, 02, . . , ON} satisfy conditions (1), (2), and (3) of Theorem XI-2-3. In fact, (1) and (2) are evident. To prove (3), note that -

k Qe(x) =

zk

/ re'd'

J

r

0(a)e-(o/s) o

'da,

0

where the path of integration is the line segment connecting 0 to re{d'. Therefore, of (x) - bt+1(x) =

k X

j

0(a)e-(ol=)'ak-1d,

t

where the path ryt of integration is the circular arc connecting re'd'+1 to re'd'. Statement (3) follows, since

e-(°/=)`

/

exp - t fx

cos[k(arg a - arg x)J

and kj arga-argxj < 2 -ke for a E ?Y and x E St(e)f1St+1(E) Since a is arbitrary, it follows that

61 E Allk I P, de - r,-, dt +

2k

(t = 1, 2, ... , N).

Furthermore, J(01) can be computed easily (cf. Exercise XI-13). +00

Observe that a power series p = 1: a,nxm E C[[xjj belongs to C[[xJ), if and only m=0

+ao

if d(x) =

I'(l

sm)xm corollary of Theorem XI-2-4.

+

belongs to C{x}. Therefore, we obtain an important

XI. ASYMPTOTICS EXPANSIONS

356

Corollary XI-2-5. For any p E G[[x]], and any real number d, there exists a function ¢(x) E A. d - 2 , d + ) such that J(O) = p. (r,

2

This corollary corresponds to the Borel-Ritt Theorem (cf. Theorem XI-1-13) of

the Poincar6 asymptotics. Also, this corollary implies that the map J , d + 2) -+ c[[x]], is onto. A8 (r, d - s7r 2 Theorem XI-2-3 is a corollary of the following lemma.

Lemma XI-2-6. Assume that a covering {St : e = 1, 2,... , N} at x = 0 is good and that N functions 61(x), 62(x), ... , 6v (x) satisfy the following conditions: (i) 6t is holomorphic on St n St+1, 7exp[-Alxl-k] on St n St+1, where -y > 0, A > 0, and k > 0 are (ii) I6t(x)I suitable numbers independent of e. Define s by (XI. 2-4). Then, there exist N functions ), (x), t.b2(x), ... ,'IPN(x) and a +oo

formal power series p = >

E C[[x]], such that

m=o

(a) 4,t E A. (r, at, bt) and J(4't) = p, where St = {x : 0 < I x[ < r, at < arg(x) < bt} (e = 1, 2, ... , N), (b) 6t(x) _ t(x) - i/'e+l(x) onSt n St+1. Let us prove Theorem XI-2-3 by using Lemma XI-2-6. Proof. Set

5t(x) = 4t(x) - 41+1(x)

(e = 1, 2, ... , N).

Then, there exist N functions 4111(x), 02(z), ... , ON (x) satisfying conditions (a) and (b) of Lemma XI-2-6. In particular, (b) implies that

Oe(x) - 4,t+1(x) = t t(x) - 4Gt+1(x)

(e = 1, 2,... , N)

on St n St+1. This, in turn, implies that

01(x) - 4't(x) = 4,1+1(x) - 4Gt+1(x)

(e = 1, 2,... , N)

on St n St+1. Define a function 0 by

4,(x) = 4t(x) - 4Vt(x)

on St

(e = 1, 2, ... , N).

Then, 0 is holomorphic and bounded for 0 < IxI < r. Therefore, 0 is represented by a convergent power series. Since .01 = -01 + 0, Theorem XI-2-3 follows immedi-

ately. 0 We shall prove Lemma XI-2-6 in §XI-5. Because the Gevrey asymptotics of functions containing parameters will be used later, we state the following two definitions.

3. FLAT FUNCTIONS IN THE GEVREY ASYMPTOTICS

357 00

Definition XI-2-7. Lets be a positive number. A formal power series

am(u-')em

m=0

is said to be of Gevrey order s uniformly on a domain V in the u-space if there exist two non-negative numbers Co and C, such that (XI.2.5)

Iam(i )I < Co(m!)sCr

for u E D and rn = 0, 1, 2, .... Set V = D(60, a0, Qo) = {e : 0 < Ie] < 5o, ao < arg e < (30} and W = D(b, a, Q).

Definition XI-2-8. Let s be a positive number. A function f (u e) is said to admit 00

an asymptotic expansion

on D if

am(u")em of Gevrey order s as c

0 in V uniformly

m=0

(i) > am(i )em is of Gevrey order s uniformly on D, m=0

(ii) for each W such that a0 < a < 0 < X30 and 0 < b < 60, there exist two non-negative numbers KW and Lµ such that N

(X1.2.6)

f (u, e) - > am('tL)cm <

1)!]8L

M=0

fori ED,eEIV andN=1, 2, .... Theorem XI-2-3, Theorem XI-2-4, Corollary XI-2-5, and Lemma XI-2-6 can be extended in a natural way so that we can use them for functions containing parameters. We leave such details to the reader as an exercise. The materials of this section are also found in [Rain 11, [Ram 21, [Si17, Appendices], and [Si18).

XI-3. Flat functions in the Gevrey asymptotics In the next section, we shall show that C[[x)), and A. (r, a, b) are differential

algebras over C and the map J : A, (r, a, b) C[[x)), is a homomorphism of differential algebras over C. In §XI-2, it was shown that the map J is onto if b - a < sir (cf. Corollary XI-2-5). In this section, we explain the basic results concerning the nullspace of J. To begin with, we introduce the following definition.

Definition XI-3-1. A function f (ii, e) is said to be flat of Gevrey order s as e - 0 in a sector

V = D(r,a,b)={e: 0
p = E am (u")em of Gevrey orders as f

0 in V uniformly on D and the expansion

m=0

off is 0, i.e., all the coefficients a, (ii) of p are equal to zero. The following theorem characterizes flat functions of Gevrey order s.

XI. ASYMPTOTICS EXPANSIONS

358

Theorem XI-3-2. A function f (u", c) is flat of Gevrey order s as e -- 0 in a sector D(r, a, b) uniformly on a domain D in the ii-space if and only if for each W = D(p, a, Q) such that 0 < p < r and a < c< < ft < b, there exist non-negative numbers KW and Aw such that

If(u,e)I
(XI.3.1) where

I.

(XI.3.2)

k = s

Proof.

Suppose that for a fixed W, there exist two non-negative numbers C and A such

that I f ( u " , e)j < C(m!)"Am jelm

(11,f) E D x W and m=0,1,2,....

for

Then, by virtue of Stirling's formula

r`k)

2m

(XI-3.3)

m! = 2nm mme'm [i l

If(u,e)I < Com"'Ag'jem

+0(01,

(u, e) E D x W and m = 0, 1, 2, .. .

for

For a given e E W, choose a non-negative integer m so that k

me <

(;i)

< (m+1)e.

Then, 1

I f(u'

e) 5

Comm/k (me)

Hence, I f (u", e) I

and A =

e

= Coe-"'/k = Co exp I ke {e - (m + 1)e}]

< K exp[-A ej-k] for every (6,e) ED x W, where K = Coexp[s]

. The converse is evident. 0 0

3. FLAT FUNCTIONS IN THE GEVREY ASYMPTOTICS

359

Remark XI-3-3. It is known that a function f (11, e) is identically zero on D x V if

(i) f is holomorphic in a on V for each fixed ii in D, (ii) f is flat of Gevrey order s as e --+ 0 in V uniformly on D,

This implies that the homomorphism J :A, (r, a, b) - C[[xBJ, is one-to-one if b - a > sa. Remark XI-3-3 is a consequence of the following lemma.

Lemma XI-3-4 ([Wat]).

Let V = D A - T ,

), when p is a positive number.

2k 2k

If f (x) is holomorphic in V and If (x)1 < Cexp[-Blxl-kJ for x E V for some positive numbers k, B, and C, then f = 0 identically in V. Proof

Fora>k,set F(x,a) = f(x)

B

P

cos(k )

x_k,2a

Then, for each fixed value of a, F is holomorphic in V and

IF(x,a)l < Cexp - B C1 -

coss((k9)

(k7r)) IxE-k I

for

x E V,

where 0 = argx. In particular,

IF(x,a)l < C

for

argx = ±7r,

IF(x,a)1 < Cexp IB

( cos-E () - 1 r-k

x E V, for

Ixj = r
1

Therefore, by virtue of the Phragmen-Lindelof Theorem (cf. Lemma XI-3-5), we conclude that

IF(x,a)I < Cexp [B

(COS I

cos(U')

for

I arg x1 <

,

x E V.

In particular,

If (x)I < C exp [BI

i)p-k - COS

) s k]

for

argx = 0, x E V.

Letting a -+ k, we derive f (x) = 0 if argx = 0 and x < p. This completes the proof of Lemma 111-3-5.

Since use was made of the Phragmdn-Lindelof Theorem, we shall state and prove the theorem precisely (cf. [Ne2, pp. 43-44J).

XI. ASYMPTOTICS EXPANSIONS

360

Lemma XI-3-5 (Phragmen-Lindelof).

Let W be a closed sector in C given by

W = {x : a < arg x < ,Q, 0 < IxI < p(arg x) }, where p(w) is a positive-valued and continuous function on the interval a < w < Q. Assume that (1) Q - a < A , where k is a positive number, (2) f is continuous on W, 0

0

f is holomorphic in W, where W is the interior of W, I f(x)I < Kexp[AIxI-k] for x E W, where A and K are positive numbers, If (x) I < M for x on the boundary of W except for x = 0 , where M is a

(3) (4) (5)

positive number. Then,

If (x)I < M

(XI.3.4)

for

x E W.

Proof.

Choose I > k so that Q - a <

0= a

Q

a P

< 7r. Set g(x) = f (x) exp[-e(e-'sx)-r], where .

1

and e> 0. Then,

2 I9(x)I = If(x)Iexp[-ecos(t(argx - 0))Ixl-`] < Kexp[-IxI-r(ecos(f(argx - 0)) - AIxIt'k)]

for x E W. Note that eI arg x - 01 < f (3- a)J < 2 for x E W. Therefore, 2 lim g(x) = 0 and I9(x)I 5 If(x)I for x E W. Since g is holomorphic in W, it

Izl-u follows that Ig(x)I < M for x E W. Therefore,

If (x)I < 11f lexp(e(e-sex)-t)I

for

xEW.

Letting e - 0, we derive (XI.3.4). We can also prove the following theorem without any complication. Theorem XI-3-6. If f is holomorphic in e on D(r, a, b) for each fixed u" in V and f is flat of Gevrey orders as e - 0 in D(r, a, b) uniformly on a domain D in the u"-space, then a, b)

j

f (u, a)do and of (a, e) are also flat of Cevney orders as e -' 0 in Of

uniform'

ly on D.

The proof of this theorem is left to the reader as an exercise. For more information see [R.aml], [Ram2] and [Si17, Appendices].

XI-4. Basic properties of Gevrey asymptotic expansions In this section, we summarize the basic properties of the Gevrey asymptotic expansions. First we prove the following theorem.

4. BASIC PROPERTIES OF GEVREY ASYMPTOTIC EXPANSIONS

361

Theorem XI-4-1. Assume that (i) a C-valued function F(ul,... , un, e) is holomorphic in (ul,... , un, e) in a do-

main D(rl,... ,rn) x D(r,a,b), where D(rl,... ,rn) = {(u1,... ,un) : Iu3f < r,, j = 1, ... n}, (ii) flu l, ... , un, e) admits an asymptotic expansion p(ul, ... , tin, E) _ +oo

Ean,(u1 i ... , un)E' of Gevrey order s as a -' 0 in D(r, a, b) uniformly mc0

on D(rl,... , r.), where the coefficients a,n(u1 i ... , un) are holomorphic in

D(ri, . . . , rn), (iii) n C-valued functions 61 (v", e), .... On(v, e) are holomorphic in a domain D x D(r, a, b), where D is a domain in the 6-space, (iv) for each j, the function 03 (v, e) admits an asymptotic expansion p, (v, e) _ +oo

E b3,,n(v')e'n of Gevrey order si as a -+ 0 in D(r, a, b) uniformly on V, where m=1

the coefficients bj,,n(v) are holomorphic in D. Regarding the coefficients a,n(u1,... , un) as power series in ( u1 , formal power series in a by

, u.), define a

Pi-(Of"

P(16, c) = p(p1(i', e).... M=0

where the coefficients P,n(v") are holomorphic in D. Then, for any (a, 0) satisfying

the condition a < a < Q < b, there exists a positive number p(a, 3) such that 0 < p(a, (3) < r and that the function F(Oi (6, e), ... , thin (6,t), c) is holomorphic in D x D(p(a, 0), a, Q) and admits the formal power series P(U, e) as its asymptotic expansion of Gevrey order max(s, si , ... , sn) as e 0 in V(p(a, 03), a, 03) uniformly on the domain D.

Remark XI-4-2. As a consequence of this theorem. we conclude that under the same assumptions as in Theorem XI-4-1, the formal power series P(v, e) in a is of Gevrey order max(s, s1, ... , sn) uniformly on D. Proof of Theorem XI-4-1.

Fixing a pair (a, (3) satisfying the condition a < a < 0 < b, choose a suitable good covering {S1, ... , SN } ate = 0 and N(n + 1) functions Ft(u1, ... , un, e), ¢1,e(vU, e), ... , QSn,t(U, e) (e = 1,... , N) such that

(1) for each e, the function Ft(ui,... ,un,e) is holomorphic in the domain D(r1 i ... , r,,) x St and admits the formal power series p(u1,... , Unit) as its asymptotic expansion of Gevrey order s as e -' 0 in St uniformly on D(r1,... , rn), (2) for each (j, e), the function 6,,e(17, e) is holomorphic in the domain V x St and admits the formal power series pj (u", e) as its asymptotic expansion of Gevrey order sJ as e -+ 0 in St uniformly on V. (3) there exist two positive numbers K and A such that

,un,E)

- Ft+1(ul,... Un,()! < Kexp

a - Iefk

XI. ASYMPTOTICS EXPANSIONS

362

on D(rl,... , rn) x (St n St+l ), where k = 1, (4) for each j = 1, ... , n, it holds that 101,1 (V,e) -o,,e+1(V,e)I

Kexp -

k

on

D x (St n St+1),

JEI

where k, =

1

,

Si

n) for some fo. (5) $t = D(p, a, E3), Ft. = F, and O,,& We can accomplish this by virtue of Corollary XI-2-5 and Theorem XI-3-2.

Observe that Fe+1(&,e+1(v,e),... Fl(41,t(V, E), ... , ¢n,[(ve E), E) (v, E), ... , 41n,,+1(v, e), e) I ,1+1(V,e), e)1 + JFt(¢1,1+1(Lt,E), +dn,[+1(v,e),E) - F1+1(dl.[+1(6,e), tl

n

< LKexp [-I -1

E Ik

+ Kexp [-( E

IC`J'

for (6, e) E V x (St n S[+1), where L, K, and A are suitable positive numbers. Therefore, using Theorem XI-2-3, we complete the proof of Theorem XI-4-1. 0 As a corollary of Theorem X1-4-1, the following result is obtained without any complication.

Theorem XI-4-3. C([x]], and A. (r, a, b) are commutative differential algebra over C, t.e.,

(i) f + g E C((x]], (respectively A. (r, a, b)) if f and g E C((x]], (respectively A, (r, a, b)),

(ii) f9 E C([xJ], (respectively A. (r, a, b)) if f and g E C((xi], (respectively A. (r, a, b)), (iii) cf E C[(x]], (respectively A. (r, a, b)) if c E C and f E C((x]J, (respectively A. (r, a, b)),

(iv) 1 E C((xJ], (respectively A, (r, a. b)) if f E C((x)J, (respectively A. (r, a, b)), (v)

/ Z f dx E

C((x]],

(respectively A, (r, a, b)) if f E C((xf, (respectively

A. (r, a, b)) Furthermore, the map J : A. (r, a, b) --t C((xi], is a homomorphism of differential algebra over C.

Remark XI-4-4. Using Theorem XI-3-6, we can prove (iv) and (v) of Theorem XI-4-3 in a way similar to the proof of Theorem X1-4-1. Also, it can be shown

that if f (x) E A, (r, a, b) with J(f) = >+"Qa,,,x' and ao 54 0, we obtain m=o

1

f

E

5. PROOF OF LEMMA XI-2-6

A. (p(a, 8), a, 0) and J

\I/

363

= J{f) , where a < a < /3 < b and p(a, /3) is a

suitable positive number depending on (a,fl). The materials in this section are also found in [Ram1], [Ram2], and [Si17, Appendices].

XI-5. Proof of Lemma XI-2-6 In order to prove Lemma XI-2-6, choose N line segments C1, C2, ... , CN so that Ct E St n St+1 (t = 1, 2,... , N), i.e., Ct : z = teie1 (0 < t < r), where for each t, Bt is a fixed number satisfying the condition at+1 < 8t < bt. These N line segments divide the open punctured disk V = {x : 0 < IzI < r} into N open sectors S1, S2, ... , SN, where

St = {x : 9t_ 1 < arg(x) < 01, 0 < txI < r}. Set

N lr Zt(x) =

1

21rt

h=1

for x E St, t = 1, 2,... , N. The functions +Gt can be continued analytically onto St by deforming Ct_1 and Ct without moving any of their endpoints. In doing this, we do not change other line segments Ch (h i4 t - 1, t). Thus, the function tyt is holomorphic on St, t = 1, 2,... , N, respectively. Now, assuming that x E St n St+1, compute t/t(x) - tPt+1(x). To do this, write 'Pt and tLt+1 in the following forms respectively:

tt't(x) _ Ot+1 {x) =

{2 =)

27ri

/ ft

e", t - x

(2 =)

Ln2 E C" h#lJ

+ 2Ri

-x

c,

where the paths Ct and Ct+1 of integration are obtained by deforming C, without moving either of its endpoints so that (1) Ct C St n St+l and Ct+l c St n St+1, (2) -Ct + Ct+1 is a simple closed curve whose interior contains x. Thus, (b) follows, i.e.,

0&) - 01+1(x) = tai

fe, + e«,

at(x)

on

St nSt,.1.

Let us derive asymptotic properties of ib1. To do this, fix an t and a closed sector W contained in St. Let Ct (respectively Ct_1) be a path obtained by deforming Ct (respectively CL_1) without moving the endpoints so that W is contained in the

XI. ASYMPTOTIC EXPANSIONS

364

interior of the simple closed curve Ct_ 1 + 7t - C1, where 1't is a circular arc joining two points reist-' and rest. For x E W, it holds that

{-1)

ot (x) = - 27ri

f

(-1)

sew x

+ 2?7

f

bt-1(1)

x

f

(-1)

d{

+ 27ri

x

It may be assumed that the path Ct is given as a union of a line segment L1 and a curve rt defined by

Lt : x = te'"

(0 < t < r1 < r), (0
rt : x = Ftt(T)

where we is a constant such that 01 < we < bt, and

r1 < litt(r)I < r (0 < 7- < 1).

pt(i) = re'B',

pt(0) = r1e'",

It can be also assumed that there exists a positive number a < 1 such that JxI < ar1 8t(-) dC is represented by a convergent power series in x for x E W. Then,

1 fr, c - x 1J 21ri

whose radius of convergence is not less than r1.

Let us estimate the integral

bt()

Since

L, c - x

27ri

mM+1

1

x 1

1

we obtain

1

Q)m

x

M

6t{f)

1

[Ymx"'

L-x

27ri

x,

+

xM+lEM+1(x),

m=0

where

am

__

1

r 5()

(m > 0) and EAi+I(x) =

-

21ri JL, Cm+1

1f

27ri

61W <. £M+1({ - x)

Now, estimate the a,,, as follows: lQm

=

I fr'

1

ebt(Teu.")eudJ

27r

J

7 27r

I

f

-I-

/r'

I

+oo

r_m-1e_ar-

\A-1lk\m

(

dr =

\_7r

7 2k-,r

f

Tm+1 dT

l

l A-Ilk! m r(\ k 1

27)1-k (v') -m/k (m!)l",

where the Stirling's formula (XI.3.3) was used as m -' +co.

365

EXERCISES XI

To estimate EN+i(x), note that there exists a positive number 0 depending on W such that for t E L1, x E W. it - xJ > Jtj sin(0) Hence, 1

JEN+i(x)J

r

^ye-ar

.ya-(N+1)/k

' rN+2sin(0)dr

2a Jo

<

2kirsin(0)

r(

N+ 1 k

7/N+llity+l)/k

< k sin(O) ` )eke ) We can estimate

-11

and

2ri c,_, f - x

-1

2iri Jc - x

(h94 Q,t-1)

in a similar manner. Thus, the proof of Lemma XI-2-6 is completed. The material of this section is also found in [Si18].

EXERCISES XI

XI-1. Let S be an open sector D(ro, a, b), i.e.,

S = {x : 0 < JxI < ro, a < argx < b}, where a and b are real numbers and ro is a positive number. Denote by (a) A(S) the set of all functions which are holomorphic in S and admit asymptotic expansions in powers of x as x -. 0 in every open subsector

W = {x : 0 < JxJ < r < ro, a < a < argx < Q < b} of S,

(b) J[f] the asymptotic expansion for f E A(S), (c) Ao(S) the set of all those functions f (x) in A(S) such that f (x) ^-- 0 as x 0 in every W (i.e. AO(S) = If E A(S) : J[f] = 0}). Show that C[Jxll. A(S), and Ao(S) are differential algebras over the field C and that the map J : A(S) C[[x] is a homomorphism of differential algebras over the field C.

XI-2. Using the same notations as in Exercise XI-1 and considering a linear difN

ferential operator P = E ak(x)Dk with coefficients ak(x) which are holomorphic k-o

in a disk JxJ < r containing S, where D =

d , show that

XI. ASYMPTOTIC EXPANSIONS

366

(i) P defines three homomorphisms

P1 = P : Ao(S) - Ao(S),

P2 = P : A(S) -- A(S), P3 = P : C[[x]]

C[[x]]

of vector spaces over the field C, (ii) two vector spaces A(S)/Ao(S) and C[[x]] are isomorphic,

(iii) dimensions of three vector spaces Nullspace ofP1i Nullspace ofP2, and Nullspace ofP3 over C are not greater than the order of P, (iv) two vector spaces {Nullspace of P2}/{Nullspace of P1} and J(Nullspace of P2] are isomorphic, (v) there exists a homomorphism T : Nullspace of P3 .A0(S)/P1 [A0(S)] of vec-

tor spaces over C such that Nullspace of T = J[Nullspace of P2] and T[Nullspace of P3] = {P2[A(S)] nAo(S)}/P1[Ao(S)]

Hint. (ii) J is onto (cf. Theorem X1-1-13) and .Ao(S) is Nullspace of J. (iii) Use Theorem IV-2-1 for Nullspace of P1 and Nullspace of P2; use an idea similar to the proof of Theorem V-I-3 for Nullspace of P3. (iv) Nullspace of the homomorphism J : Nullspace of P2 -+ J[Nulspace of P2] is Nullspace of P1. Also, this homomorphism is onto. (v) Assume that P3[¢] = 0 in C{[xJI. There exists a b E A(S) such that J[4] = 4 (cf. Theorem XI-1-13). Such b is not unique. Actually, ¢ + also satisfies the condition J[4, + '] = 4, if and only if iP E A0(S). So define T by T[4,] = P2[4J (mod P1[Ao(S)]). Note that J[P2[4,]] = P3[4'] = 0. This implies that P2[O] E Ao(S).

X1-3. For each of the following three power series, find s such that f E C[[x]],. cc

(a) f(x)

_

(mx)m,

,n=o 00

00 r(1+

(b) f (x) =m_or E 1+

2)

3

m

2m

(c) f (x) _ Exm 11 (5+ V5-h). m=o

h=m

XI-4. For two real numbers s > 0 and A > 0 and for a formal power series +00

f = 1: a,nxm E C[[x]1, set

+00

I I f 11 &,A = E ( M=0

Denote by E(s, A) the set of

all f E C[[x]] satisfying the condition IIf II..A < +oc. Show that 7=0 (i) E(s, A) is a Banach space with the norm IIf II.,A, (ii) IIf9II.,A <_ IIf II..A I19II..A if f E E(s, A) and g E E(s, A), (iii) if B > A, then E(s, A) C E(s, B), and II f II,,a < IIf II,,A for f E E(s, A),

(iv) if B > A, then Il f II.,e - If (0)I < [IIf II.,A - If (0)I] for f E .6(s, A), where f (0) = a0 if f = E.m>o a. xm, B (v) for f E E(s, A) and an integer q such that sq > 1, it holds that

df 11

eA

5 -LIIfII.,A

EXERCISES XI

367

Comment. The formal power series f E C([x]] is of Gevrey order s if and only if f E E(s, A) for some positive number A. XI-5. Show that f (e) = 0 identically in a sector S if f is holomorphic and flat of Gevrey order 0 as e --+ 0 in S.

Hint. I f (e)I < KA' IeIr' fore E Sand N = 0,1,2,..., where K and A are positive numbers independent of N.

XI-6. Show that C[[x]]o = C{x}.

Hint. f = 00E

E C[[x]]o if and only if Ic,.. I < KAt If I' (m = 0, 1, 2.... ),

=G

where K andmA are positive numbers independent of m. X1-7. Consider a system of differential equations

(S)

xy+t dy

= f0 + bg + Ip!>_o

where q is a positive integer, fo and fp E C[(x]]n, 4> is an n x n matrix whose entries are formal power series in x, p = (pl, ... , p,>) with n non-negative integers pt, ... , Pn, IpI = Pt + - - - + p,,, and yam' = yP' - - - VP-. Assume that sq > 1 and that

(i) fo and f. E E(s, A)n, and the entries of 0 belongs to E(s, A) for some A > 0, (ii) the power series F, IIfpIIs,A9P is a convergent power series in y", where Ipl>2

IIY1Ia,A = max{IIy,II5,A :1 < j < n},

(iii) fo(0) = 0, (iv) (3(0) is invertible. Show that there exists a unique power series E C[[x]]n satisfying system (S) and the condition ¢(0) = 0. Furthermore, E (C[[x]],)n. Hint. Write system (S) in a form y" = go + x9+1 1P dx Banach fixed-point theorem in terms of the norm II IIs.A-

+ E y'gp and use the ipl>2

XI-8. Let D(r) = {z : IzI < r}, S(r, p) = {z : 0 < IzI < r, I arg zI < p}, and a power +w

series f (z, e) = E fn(e)zn is convergent uniformly in a domain D(ro) x S(r, p), n=o

where ro, r, and p are positive numbers. Assume that the coefficients fn(e) of the series f are holomorphic in S(r, p) and admit asymptotic expansions in powers of e as e -' 0 in S(r, p). Suppose also that limo fo(e) = 0 and limo f, (c) 0 0. Show C1-

that there exist a positive number rt and a function O(e) such that (1) O(e) is holomorphic in S(rt, p) and admits an asymptotic expansion in powers of a as e -+ 0 in S(rt, p), (2) l ¢(e) = 0, and (3) f (c(e), e) = 0 identically in S(rt, p).

XI. ASYMPTOTIC EXPANSIONS

368

X1-9. Consider a formal power series

#,

_

F(x, y,

(P)

y

fa,,n2,

ip,l+IpI>o yi

where

, fn:,as E

%' _ yn

(p21,

z1

C[[x]]

"

, pi = (pll,... ,pln), and Pz =

xm

... , p2",) with non-negative integers P1k and p2k. Denote by Fg(x, y z) the

Jacobian matrix of ! with respect to W. Assume that power series (P) satisfies the conditions (1) f;(0,6,6) = 0, (2) Fy(0, 0, 0') is invertible, (3) fn, a, E E(s, A)" for some s > 0 and A > 0, and (4) the power series i[ fn, a, Ij, Aye' is a ia,I+?P2I?0

convergent power series in y and F. Show that there exists a unique power series 92 in (x, z) such that (i) drr,, E C[[x]]", (ii) b(0,0) = 0, and (iii) ¢(x, z) lack>o

F(x, (x, z), z) = 0. Also, show that (a, E E(s, B)" for some B > 0 and the power series E II( a' i" is a convergent power series in z. IP3i>o

Hint. This is an implicit function theorem. Write f in the form f = fo + fiy + F'gv' z"P2 fn,,r,, , where fo and fn,,n, are in E(s, A)", $ E M"(E(s, A)), and >' is the sum over all (pi i p) such that either jell = 0 and [g-,j = I or Jial j + jp2j > 2. Here, M.(R) denotes the set of all n x it matrices with entries in a ring R. Assumption (1) implies that fo(0) = 0. Assumption (2) implies that -t(0) is invertible. Therefore, I ' exists in M"(C[[x]],), and hence II4 'jI, A < +oo for some A > 0, where III-' IIB,A = sup{jj4V' f Ij, A : f E E(s, A)n, Ij f [j,.A = 11. Since IIfIIe,B 5 11A .,A for f E (C[[x]],)" if B > A, assume without any loss of generality that A = A. Then, t' IF = 90 + y" + zP2§p., p,, where g"o E E(s, A)", 9a,,a, E E(s, A)", 90(0) = 0, and 'II9n,,a2I1a,A9" i is a convergent power series ul in y" and F. Consider the equation 0 = u"+ y"+ F,'#P- z g"n, ,n, , u" = . . Then, un

there exists a unique power series 5(x, u",

_

lip ' z'na"n,

such that

fa,l+lpiI>I

dn,,a, E E(s, A)" and 0 = u'+ a'+

identically. The unique power series satisfying conditions (i), (ii), and (iii) is given by (S)

(x, z_) = 5(x, 90, zfl

Now, consider the equation &'$121 Is,A

(R) y'=u".+z-V Gn,,p,

where

Ga,,i7 =

ER". 9P% ,P2 Il .w

EXERCISES XI

369

Equation (R) has a unique solution y = E 9P'z-r'pp,,p, such that Qp,,p, E 1p1I+1P21>>-1

]R" , the entries of pP,,p, are non-negative, and the series is a convergent power series in u and z1 . Use this series as a majorant to show that defined by (S) satisfies conditions (iv) and (v).

XI-10. Assume that a covering (S,,$2,... , SN} at x = 0 is good and that N functions 01(x), 02(x),... , ON (x) satisfy the conditions: (1) 0t(x) is holomorphic in St,

(2) of(x)^_-Oasx-.0 in St, (3) 10e(x) - Ot+1(x)l < -yexp[-AIxI-kJ on St n St+1i where -y > 0, A > 0 and k > 0 are suitable numbers independent of e. Show that there exists a positive number H such that

[4t(x)I < Hexp[-AIxI-kJ

in

St.

Hint. See [Si15J.

XI-11. Assume that a covering {S1, S2, ... , SN } at x = 0 is good and that N functions 01(x), 02(x),... , y` N (x) satisfy the conditions (1) 4,(x) is holomorphic on St, (2) 4c(x) is bounded on St, (3) we have IOt(x) - 01+1(x)l < Knlxl'

(n = 1,2,...)

on St n St+1,

where K are positive numbers. Show that there exists a formal power series p = > a,,,xm E d[[xJJ such that for m=0

each t, we obtain ¢t E A(S1) and J(¢1) = p, where the notations A(S) and J are defined in Exercise XI-1.

XI-12. Assume that a covering {S1, S2.... , SN } at z = 0 is good and that n x n matrices 4i1(x),t2(x),... ,4N(x) satisfy the following conditions: (1) the entries

of tt(x) belong to A(S1 n St+1) and (2) J(tt) = I,. Show that there exists a formal power series Q = I,, + E xmQm having constants n x n matrices Qm as m=1

coefficients, and n x n matrices P1(x), P2(x), ... , PN(x) such that (i) for each e, the entries of P1(x) and P1(x)-1 belong to A(S1), (ii) J(P1) = Q (t = 1, 2,... , N),

(iii) It(x) = PI(x)-1Pt+1(x)(e= 1,2,... ,N), where the notation A(S) and J are defined in Exercise XI-1. Also, show that if the entries of (t(x) belong to A.(St), respectively, then the entries of P,(x) and P, 1(x) also belong to A,(Se), respectively. Hint. See [Si17, Theorem 6.4.1 on p. 150, its proof on pp. 152-161, and §A.2.4 on pp. 207-2081.

XI. ASYMPTOTIC EXPANSIONS

370

XI-13. Prove the formula 00

M=0

which is given in Therorem XI-2-4.

Hint. Assume that j argx1 < 2k - b, where b is a small positive number. Then, k

(r'/) v"'/ke °do

k rT tme-(t/x)ktk-Idt = xm J0

0

r (1 +

m)

T

xm - xm

Jr°O

J (t/z)k

Om/ke-`do.

Hence, N

r (i + m ) Cmxm - E cmxm

1,,k(6) _

k

M=0

/r

m-0

+00

+ k J xk

0

Om/ke °d0 (t/k)k

cmtm

e-(t/x)ktk-ldt.

m=N+1

XI-14. Let a be a positive number larger than 1. Also, let

SJ={x: a,
,b=

, and p is a positive number. Assume that v functions &1(x), 4(x), satisfy the following conditions: (i) Oj(x) is holomorphic in Sj and continuous on the closed sector

19, ={x:aj5argx
IOj(x)1 < M

in

SS

(j=1,2....,v).

Hint. This is a generalization of the Phragmen-Lindelof Theorem (Lemma XI-3-5). See ]Lin].

EXERCISES XI

371

XI-15. Let k be a positive number. A formal power series f E C[[x]] is said to be k-summable in a direction arg x = 0 if

f

E I m a g {e J : A 1/k

IT

(Ap> e

- 2k - E, 0+

7r

2k

\

+ EJ

C{[x]]1/k

for some positive numbers po and E. Show that if a formal power series f E C[[x]]

is k-summable in a direction arg x = 0, there exists one and only one function F E Al/k (po, 9 - 2k - e, 0 + 2 + EJ such that J[F] = f. Hint. Use Remark XI-3-3. For more informations concerning summability, see, for example, [Ba13], (Ram2), [Ram3], [Si17, Appendices], and [Si19]. r e_t-s XI-16. Consider the integral f (x) = f - x dl;, where -1 is a line segment 0 <

x < 1 in the sectorial domain D 1, - 4 27r + 7r §XI-5, show that

\

!.

Using the argument given in

/

(i) f(x) admits an ``asymptotic expansion J[f] E C[[x])112 as x -. 0 in

DI 1,-427x+ 4), (ii) J[f] is 2-summable in the direction argx = 0 if 0 < 0 < 27r.

CHAPTER XII

ASYMPTOTIC SOLUTIONS IN A PARAMETER In this chapter, we explain asymptotic solutions of a system of differential equations E°fy = f (x, y, f) as a -i 0. In §§XII-1, XII-2, and XII-3, existence of such asymptotic solutions in the sense of Poincar6 is proved in detill. In §XII-4, this result

is used to prove a block-diagonalization theorem of a linear system E° d = A(x, e)y.

In §XII-5, we explain similar results in the Gevrey asymptotics. In §XII-6, we explain how much we can simplify a linear system by means of a linear transformation with a coefficient matrix whose entries are convergent power series in the parameter. This result is given in [Hs1] and similar to a theorem due to G. D. Birkhoff [Bi] concerning singularity with respect to the independent variable x (cf. Theorem XII-6-1 and (Si17, Chapter III]). The materials in §§XII-1- XII-5 are also found in [Wasl], [Si3], [Si7], and [Si22].

XII-1. An existence theorem In §§XII-1, XII-2, and XII-3, we consider a system of differential equations dv

(XII.1.1)

E° V -

=

f3 (x, V 1 , V2, ... , Vn, E)

(j = 1, 2, ... n),

where a is a positive integer and f J (x, V1, v2, ... , vn, f) are holomorphic with respect to complex variables (x, v1, v2, ... , vn, E) in a domain

(XII.1.2)

Ix[ < do,

0< IEI < Po,

I arg EI < ao,

Iv. I < 'ro

(j = 1, 2, ... , n),

6o, po, ao, and 'Yo being positive constants. Set

f) (XII.1.3)

f j (x, V,

()

n

a,h(x, e)Vh + 1 fi, (x, E)iJ

= fJO(x,

,

Icl?2

h=1

where i I E Cn with the entries (V1, V2, ... , vn).

We look at (XII.1.1) under the following three assumptions.

Assumption I. Each function f j (x, v, c) (j = 1, 2,... , n) has an asymptotic expansion (XII.1.4)

fi (x, v, E) "'00E f

v)E"

L=o

in the sense of Poincare as c (XII.1.5)

0 in the sector I arg EI < ao,

0 < IEI < po, 372

373

1. AN EXISTENCE THEOREM

where coefficients f j,, (x, v) are holomorphic in the domain

W < 6o,

(XII.1.6)

IV1 < 7o.

Furthermore, we assume that (XII.1.7)

fjo(x,) = 0

(j = 1, 2, ... , n) for 1x1 < do.

Observation XII-1-1. Under Assumption I, fjo(x, e), a jh(x, e), and f jp(x, e) admit asymptotic expansions 00

00

(XI1.1.8)

fjo(x, () E f o (x)e", =1

fjp(x, e)

' E fJp.(x)f" v=0

and

(XII.1.9)

(j, h = 1, 2, ... , n)

aj h(x, e) '- > --o

as e - 0 in sector (XII.1.5) with coefficients holomorphic in the domain (XII.1.10)

1x(< 60-

Let A(x, e) be the n x n matrix whose (j, k)-entry is a.,&, e), respectively (i.e., A(x, e) = (ajk(x, e)). Then, A(x, c) admits an asymptotic expansion (XII.1.11)

A(x, e) ^_- E e"A,,(x) =o

as a -+ 0 in sector (XII.1.5), where the entries of coefficient matrices (ajk,,(x)) are holomorphic in domain (XII.1.10). The following second assumption is technical and we do not lose any generality with it.

Assumption II. The matrix Ao(0) has the following S-N decomposition Ao(0) = diag

[µh,02,... ,p1 + N,

where Al, µ2f ... , An are eigenvalues of Ao(0) and .,V is a lower-triangular nilpotent matrix.

Note that [Ni can be made as small as we wish (cf. Lemma VII-3-3). The following third assumption plays a key roll.

Assumption III. The matrix Ao(0) is invertible, i.e., t
(j = 1, 2,. .. , n).

In §§XII-2 and XII-3, we shall prove the following theorem.

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

374

Theorem XII-1-2. Under Assumptions I, II, and III, system (X11.1.1) has a solution (XII.1.12)

(j = 1,2,... ,n)

vJ = pj (x, c)

such that (r) pi(x,e) are holomorphie in a domain jxl < b,

(XII.1.13)

0 < lei < p,

l arg el < a,

where b, p, and a are suitable positive constants such that 0 < 6 < bo, 0 <

p
(ii) p, (x, e) admit asymptotic expansions 00

(j = 1,2,... n)

p3(x,e) -

(XII.1.14)

as e - 0 in the sector

urge) < a,

(XII.1.15)

0 < Iej < p,

where coefficients

of (XII.1.14) are holomorphic with respect to x in

the domain {x : lxi < 6}.

XII-2. Basic estimates In order to prove Theorem XII-1-2, let us change system (XII-1-1) to a system of integral equations.

Observation XII-2-1. Expansion (XII.1.14) of the solution p3(x,e) 00

(XII.2.1)

vJ = Epi,(x)e"

(j = 1,2,... n)

1-1

must be a formal solution of system (XII.1.1). The existence of such a formal solution (XII.2.1) of system (XII.1.1) follows immediately from Assumptions I and III. The proof of this fact is left to the reader as an exercise.

Observation XII-2-2. For each j = 1, 2.... , n, using Theorem XI-1-14, let us construct a function g1(x,e) such that (i) q, is holomorphic in a domain (XI1.2.2)

lxi < 6',

0 < jej < p',

l arg el < a',

where 0<6'<6o,0
2. BASIC ESTIMATES

375

(ii) qj and !j admit asymptotic expansions d-z

(XII . 2 . 3)

gj (x ,E )

> p,v (x) "

an d

E

d9,

,E)

_

=i

E dp,, x) " E

V=1

as f -. 0 in the sector (XII.2.4)

I argel < o .

0 < IEI < p',

Consider the change of variables (j = 1,2,...,n).

v, = u, + q,(x,E)

Denote (ql, q2, ... , qn) and (U I, U2,. .. , un) by q" and u, respectively. Then, ii satisfies the system of differential equations (XII.2.5)

E°

dd

(j = 1,2,... ,n),

xj = gj(x,19, c)

where E° dq,(x,E)

93 (x,u,E) = fj(x,U + q' f) -

(j = 1, 2,. .. , n).

dx

Set n

(XII.2.6)

00

bjk(x,E)uk + E bJp(x,E)u-P

g)(x,u,E) = g2o(x,E) +

IpI?2

k=1

(j = 1, 2, ... , n).

In particular, (XII.2.7)

g,o(x,E) = fi(x,9,E)

-E°dq)'E) ^ 0

(j = 1,2,... n)

and

b,k(x, E) - a,k(x, c) = O(E)

k = 1, 2,... , n)

as c - 0 in sector (XII.2.4). Therefore, (XII.2.8)

bik(x, E) = ajko(x) + O(e)

(j, k = 1, 2,... , n)

asE-+0insector (XII.2.4). Set (XII.2.9)

g, (x, ",E) = p, u, +R.(x,u,E)

(j = 1,2,... ,n).

Then, for sufficiently small positive numbers 6, p, and -y, there exists a positive constant c, independent of j, such that for every positive integer N, the estimates (XII.2.10)

IR7(x,u,E)I

CIUI+BNIEIN

(j=1,2,...,n)

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

376

and

eIu - u I

(XII.2.11)

(j = 1, 2,... , n)

hold whenever (x, u", f) and (x, u~', e) are in the domain (XII.2.12)

largfl
0
lxI <8,

Here, BN is a positive constant depending on N and 1191. From (XII.2.5) and (XII.2.9), it follows that (XII.2.13)

E°

uj

= µ,u, + R.(x,u,E)

(j = 1,2,... ,n).

Change system (XII.2.13) to the system of integral equations

(XII.2.14)

I

u, =

exp of( (t - x) R, (t, u, E)dt

E

where the paths of integration must be chosen carefully so that uniformly convergent successive approximations can be defined.

Hereafter in this section, we explain how to choose paths of integration on the right-hand side of (XII.2.14).

Observation XII-2-3. Set

(j=1,2,...,n)

w,=argµ,

(XII.2.15)

and suppose that - 27r < wj <

(XII.2.16)

27r

(j = 1, 2,... , n).

Then, there exists a positive number a less than a such that

-2rr
21

7r

and

w, - e

-2ir

,n).

Without loss of generality, suppose that n real numbers w, are divided into the following two groups: (I)


(j =1,2,...,m')

and (II)

-27r
(3=m'+1,...,n).

377

2. BASIC ESTIMATES

Choose two positive numbers a and /3 sufficiently small such that

-2ir+aa+3
7 r - (Qa+Q)

(.7 =

1

m'),

(j=m'+1,...,n).

2"+oa+13
Set

x(1) = be-i°,

x(3) = -x(1),

x(2) = ix(1) tan [3,

and x(4)

_ -x(2),

where b is a sufficiently small positive number. Then, a rhombus is defined by its

four vertices x( j) (j = 1,2,3,4) (cf. Figure 1). Note that the angle at x(1) is 20.

FIGURE 1.

Denote the interior of this rhombus by D(5). It is noteworthy that the domain D(b) contains a small open neighborhood of x = 0 and is contained in the domain {x : IxI < b}. The basic estimates for the proof of Theorem XII-1-2 are given by the following lemma.

Lemma XII-2-4. For each j = 1.... , n, consider the function (XII.2.17)

U3(x,E) =

=exp

J z,

[

- µi(t_

dt

L

J

where the path of integration is the straight line 77! and ( x(1)

xj=51

x(3)

(j = 1, 2,... , m'), (j = m' + 1, ... , m).

Then, there exists a positive number c such that (XII.2.18)

I Ul (x, E)1 :5 c1cl, lexp

'1l L

(x - xl) E°

J

for (XII.2.19)

x E D(8),

I argel
IEI > 0.

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

378

Proof We prove this lemma for the cases j = 1, 2,... , m'. The cases j = m' + 1,... , m

can be treated in a similar manner. Set f = It - x(l)I and 0 = arg(t - x(')) for t E D(b) and set w = arg e. Then, (XII.2.20) 11

Ix- (l, l exp

I U j (x, e) I s

[-19'J'

£ cos(w, + 0 - aw)] d {

(j = 1, 2, ... m').

From the definition of D(8), it follows that 7r - e - f3 < 0 < 7r - e +,6 for x E D(8). Thus, if a is in the sector I arg eI < a, the inequalities 3 2a
hold f o r x E D(b) and j = 1, 2, ... , m'. Therefore, there exists a positive constant c' such that

-cos(wj +0-aw)>c

(XII.2.21)

(j=1,2,...,m')

for (x, e) in (XII.2.19). By virtue of (XII.2.20) and (XII.2.21), we obtain IUj(x,E)I <

µ,(

leXp [

xj)

(j = 1,2,... ,m')

for (x, e) in (XII.2.19). Thus, Lemma XII-2-4 is proved. D In the next section, we shall consider system (XII-2-14) of integral equations assuming that x E 1)(6) and the paths of integration are chosen in the same way as in Lemma XII-2-4.

XII-3. Proof of Theorem XII-1-2 Let us construct a solution u, _ 0, (x, e) of (XII.2.14) so that Oj(x,e)

(XII.3.1)

(j = 1,2,... n)

0

as e-'0in (XII.1.15). Now, by virtue of Assumption II, three positive quantities 6, p, and (Nj can be chosen so small that c in (Xl.2.10) and (XII.2.11) satisfies the condition (XII.3.2)

cc < 1,

where c is the constant given in (XII.2.18). Define successive approximations of a solution of (XII.2.14) in the following way:

I (XII.3.3)

u°) (x, e)

Uh(xe) _

0,

i

(Z

1

exp

[__i.(t - x)} Ri (t, u"("-1)(t, e), e)dt

(j = 1,2,... ,n; h=1,2,...)1

3. PROOF OF THEOREM XII-1-2

379

. For a given where x E D(b) and the integration is taken over the straight line positive integer N, it will be shown that (i) for each j, the sequence I (h) (x, E) : h = 0,1, ... } is well defined, (ii) for the given integer N, there exist positive constants KN and PN (0 < PN p) such that (XII.3.4)

(j=1.2,...,n; h=0,1,...),

Iu(h)(x,E)I
uniformly for (x, e) in the domain

x E D(b),

(XII.3.5)

0 < IEl < pN,

I argel < a',

(iii) the sequence {u(') (x, f) : h = 0,1, ...) converges uniformly to (x, E) _ (01(x, E), ¢2(x, E), ... , 0n (x, E)) in (XII.3.5), where iZ (x, f) is the C"-valued function with the entries (u(h) (x, E), ... , u,(th) (x, E)).

The limit function (x, e) is independent of N since the successive approximations are independent of N. If (i), (ii), and (iii) are proved, it follows that (XII.3.6)

0l(x,E) = llim u(jh)(x,E) ^-0

(j = 1,2,... ,n)

as e - 0 in the sector S = E : 0 < lei < sup(pN), I argel < a' }, and the functions N

inll

the domain D(b) x S and satisfy (XII.2.14). Setting pf(x,E) = j(x,E) +g2(x,E)(j = 1,... ,n), we obtain a solution v. = p., (x,E)(j = (x, c) are holomorphic

1,2,... , n) of (XII.1.1) which satisfies all of the requirements of Theorem XII-1-2. Thus, the proof of Theorem XII-1-2 will be completed. To show (i) and (ii), choose two constants KN and pN so that KN >

i

BNOC and

PNNKN < ry. This is possible since condition (XII.3.2) is satisfied. Now, assuming

that (i) and (ii) are true for ugh-1)(x,E) in (XII.3.5), let us prove that also satisfy conditions (i) and (ii). First from (XII.3.3), it follows that ..(h)i_ _' _ I _-._ I uJ I_

_

,l

(XII.3.7) x

j:exp {

(t

-x.,)]Ri(tu(h-1)E)dt (J = 1,2,....n).

Then, from Lemma XII-2-4, (XII.2.10), (XII.3.4), and the inductive assumption, we conclude that E)I < C{CKN + BN}IEI N < KNIEIN

for (x, e) in (XII.3.5). Thus, (i) and (ii) are true for ujh) (x, E) (j = 1, 2, ... , n).

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

380

To show (iii), from (XII.2.11), (XII.3.3), and Lemma XII-2-4, we obtain Iu-(h+l)(x,E)-u-(h)(x,E)I :5c8 sup Iiz(h)(t,E)-tL(h-1)(t,E)I tED(6)

in (XII.3.5) for h = 1 , 2, 3, .... Hence, the sequence (u -1 h ) (x, E) : h = 0,1, 2, ... } is convergent to (x, f) uniformly in (XII.3.5). Furthermore, uj = Oj (x, e) (j = 1, 2,... , n) satisfy system (XII.2.14). Thus, the proof of Theorem XII-1-2 is completed.

0

XII-4. A block-diagonalization theorem Consider a system of linear differential equations E° fy = A(x, E)y,

(XII.4.1)

where a is a positive integer, y E C", and A(x, c) is an n x n matrix. The entries of A(x, E) are holomorphic with respect to a complex variable x and a complex parameter c in a domain (XII.4.2)

lxi < ao,

0 < IEI < po,

I argEl < no,

where bo, po, and ao are positive numbers. Assume that the matrix A(x, e) admits a uniform asymptotic expansion in the sense of Poincar6, 00

(XII.4.3)

A(x, e) E E"A, (x), V=o

in domain (XII.4.2) as c --# 0 in the sector (XII.4.4)

0< IEI < po,

where the entries of coefficients domain (XII.4.5)

I

ao,

are holomorphic with respect to x in the Ixl < do.

Suppose that Ao(O) has a distinct eigenvalues A1, A2i ... , At with multiplicities n1, n2i ... , nt, respectively (n1 + n2 + - - + ne = n). Without loss of generality, assume that Ao(0) is in a block-diagonal form (XII.4.6)

Ao(0) = diag [Al, A21 ... , At]

,

where Aj are n, x n, matrices in the form (XII.4.7)

Aj =A,I",+N, (j=1,2,...,t).

Here, I,, is the nj x n, identity matrix and Yj is an n, x nj lower-triangular nilpotent matrix. The main result of this section is the following theorem.

4. A BLOCK-DIAGONALIZATION THEOREM

381

Theorem XII-4-1 ([Si7]).

Under assumptions (X11.4.8) and (X11.4.6), then exists an n x n matrix P(x, e) such that (i) the entries of P(x, e) are holomorphic with respect to (x, e) in a domain (XII.4.8)

Ix j < 6,

0< kkI < p,

l argeI < a,

where 6, p, and a are positive numbers such that 0 < 5 < 60, 0 < p < po and

0
P(x, e)

(XII.4.9)

etPP(x)

(P0(0) = I,,)

V=0

in domain (XII.4.8) as e - 0 in the sector 0< IkI < p,

(XII.4.10)

argel < a,

where the entries of coefficients P,(x) are holomorphic with respect to x in the domain (XII.4.11)

fix, < 6,

(iii) the transformation (XII.4.12)

y" = P(x, e)zi

reduces system (X11.4.1) to a system (XII.4.13)

e° d = B(x, e)z

with the coefficient matrix B(x, e) in a block-diagonal form (XII.4.14)

B(x, e) = diag[Bj (x, e), B2(x, e),

... , B,(x, e)],

where B , (x, e) is an n, x n, matrix (j = 1, 2, ... , e), (iv) the matrix B, (x, e) admits a uniform asymptotic expansion (XII.4.15)

ficients

Bj (x, e) ^- 00 1: e in sector (XII.4.10), where the entries of coefare holomorphic with respect to x in domain (X11.4.11).

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

382

Remark XII-4-2. (a) Set diag [B1v(x), B2 ,(x), ... , Bl,(x)] -

(XII.4.16)

Then, the coefficient matrix B(x, e) of system (XII.4.13) admits a uniform

0

asymptotic expansion B(x, e)

'=a

(b) When a fundamental matrix solution Z(x, e) of (XII.4.13) is known, a fundamental matrix solution Y(x, e) of (XII.4.1) is given by Y(x, e) = P(x, e)Z(x, e). (c) In the case when the matrix Ao(O) has n distinct eigenvalues, by Theorem XII-4-1, we can diagonalize system (XII.4.1). (d) In the case when eigenvalues of the matrix Ao(O) are not completely distinct, the point x = 0 is, in general, a so-called transition point. In order to study behavior of solutions in the neighborhood of a transition point, we need a much deeper analysis of solutions of system (XII.4.1). For these informations, see, for example, [Wash], [Was2], and [Si12]. Proof of Theorem XII-4-1.

We prove this theorem in two steps. The proof is similar to that of Theorem VII-3-1.

Step 1. We show that there exist a positive number 6 (< oo) and an n x n matrix Po(x) such that (i) the entries of Po(x) and Po(x)-1 are holomorphic in the domain {x : JxI < b} and Po(0) = In, (ii) the matrix Co(x) = Po(x)-1Ao(x)Po(x) is in a block-diagonal form Co(x) = diag [Col)(x),Co2)(x),... ,C(c)(x)]

(XIL4.17)

where Ca) (x) is an n, x nj matrix such that

Co)(0) = Aj=a,ln,+Arj

(J=1,2,...,e).

In fact, two matrices Po(x) and Co(x) must be determined by the equation Ao(x)Po(x) - Po(x)Co(x) = 0.

(XII.4.18) Set

4(11)(x)

Ao(x)

Ao2 1) (x

Aat1)(x) nd

'4(12)(x)

4

) (x)

...

40(2)

AA(x) ... o2)

Aou) (x)

4(13)(x)

Aotz}(x)

Ao)(x)

Poll)(x)

Po12)(x)

Po's)(x)

Pa21)(x)

P

Po2)(x)

Ao r)(x) Pol() (x)

o

Po(x) =

)(x)

Poti)(x)

P ' (x)

Po2'(x) ,

Po )(x)

Poti)

(x) 1

383

4. A BLOCK-DIAGONALIZATION THEOREM

k) where A4(jk) (x) and PO (x) are nj x nk matrices. Furthermore, set PO l) (x) _

In, (j = 1,2,... , t). Then, (XIL4.19)

Co ) (x) = Ao(jj) (x) + 1` 4h) (x) p(hj)(x) (j = 1, 2,... , f) h#3

and

Ao(jj)(x)po (XII.4.20)

k)(x)

- po k)(x)Cc(k)(x)

+

+AD(jk)(x)

0

=

(j i4 k)

h#j,k

we obtain

from ( XII.4.18). Combining (XII.4.19) 41) (x)p0 k) (X)

-

p(1k)

(x)("Okk)(x)

4h)

+

(x)pOhk)(x)) hjAk

(XII.4.21)

vh)(x) p(hk)(x) +

0

"(j # k).

h#j,k

Upon applying the implicit function theorem to (XII.4.21), matrices POk)(x) (j L k) can be constructed. Then, Co(x) is given by (XII.4.19) and (XII.4.17).

Step 2. Now, assume without loss of generality that Ao(x) is in a block-diagonal form

Ao(x) = diag [4)(x),4)(x),... ,Afl[)(x)] .

(XII.4.22)

To prove Theorem XII-4-1, it suffices to solve the differential equation (XII.4.23)

Ea

dP(x, e)

= A(x, E)P(x, E) - P(x, E)B(x, E),

where (XII.4.24)

A(x, E) = Ao(x) + EA(x, E), P(x, E) = In + EP(x, E) and

B(x, E) = Ao(x) + EB(x, e).

Set

A(x, c) _

A(11)(x,E)

A(12)(x,E)

...

A(21)(,, )

A(22) (x, E)

...

A([I) (x, E)

A(M) (x, E)

P(11)(x,E) P(21) (x, E)

P(12)(x,E) P(22) (x, E)

... A(t[) (x, E) ... P(I[)(x'E)

P([1)(x E)

p(t2)(x, E)

A(11)(x,E) A(21) ('T' E)

...

P(2[) (x, E)

...

PIP (x,

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

384 and

B(x, e) = diag [b(') (X, E), b(2) (X, E), . . . , B(') (x, E)]

,

where A(jk) (x) and P(Jk) (x) are nj x nk matrices and BJ (x, c) is an nj x nJ matrix

(j, k = 1,... , e). Furthermore, set

(j = 1,2,...

P(JJ)(x,E) = 0

(XII.4.25)

Then, equation (XII.4.23) becomes

l

B(.)(x, f) = A(Jj)(x,E) + Ey: A(Jh)(x,E)P(hJ)(x,e)

(XII.4.26)

h=1 (j = 1,2,...,t)

and (XII.4.27) EodP(jk)(x,E)

+EE

= A(J)(x)P(Jk)(x,E)

-

P(Jk)(x,E)A(k)(x) + A(jk)(x,E)

A(jh)(x f)P(hk)(x,E)

-

P(jk)(x,f)P(k)(x,E)

(i

k).

1hjAk

Combining (XII.4.26) and (XII.4.27), we obtain Eo

dP(J c) (x, E)

4

J)

x P(Jk) x E - P(Jk) x, e A(k) x + A(Jk) x, E I

+E

L.:A(Jh)(x, E)P(hk)(x, E)

- E p(Jk)(x, E)A(kk) (x, E)

h=1

- (2P(Jk) (x, c)

I

A(kh) (x, E)P(hk) (x, E)

(j0k).

h=1

Replacing P(jk)(x,E) by an nJ x nk matrix X(Jk), we derive a system of nonlinear differential equations E

o dX (2k)

dx

= A (J) ( x) X (jk)

- X (jk) A(k)x+(X)A(Jk) x( E) ,

I

(XII.4.28)

+ E E A(Jh)(x, C)X(hk) - EX (Jk)A(kk)(x, f) h=1 I

- E2X(jk) FA(kh)(x,e)X(hk)

(j0k)

h=1

Consider the entries of X (jk) (j, k = 1, 2, ... , e; j # k) altogether to form a system of nonlinear differential equations. Upon applying Theorem XII-1-2, we

5. GEVREY ASYMPTOTIC SOLUTIONS IN A PARAMETER

385

construct an analytic solution of this nonlinear system in a domain (XII.4.8) admitting an asymptotic expansion in e as described in Theorem XII-4-1. This completes

the proof of Theorem Xll-4-1. 0

XII-5. Gevrey asymptotic solutions in a parameter In this section, using Theorems XI-2-3 and the proof of Theorem XII-1-2, we construct Gevrey asymptotic solutions of a system of differential equations (XII.5.1)

Ea

dx

= Ax, y, E),

where x is a complex variable, y E Cn, E is a complex parameter, a is a positive integer, and Ax, if, e) is a Ca-valued function of (x, y', f). Define three domains by A(So) = {x E C : IxI < bo}. (XII.5.2)

S(ro, no) = {E E

, arg E

< ao, 0 < H < ro}.

Also, define two matrices A(x, E) and Ao(x) by

(XII.5.3)

A(x, e)

89 Oyn

8y1(x,E)

(x, 0, f)

and (XII.5.4)

Ao(x) = limo A(x,e),

respectively. We first prove the following lemma.

Lemma XII-5-1. Assume that (i) Ax, y", e) is holomorphic with respect to (x, f, c) in a domain A(bo) x Q(po) x S(ro, no), where 6o, po, ro, and no are positive numbers, (ii) f (x, y", c) is bounded on A(60) x i2(po) x S(ro, no), (iii) the matrix Ao(x) defined by (XII. 5.4) exists as e - 0 in S(ro, no) uniformly in x E A(6o) and Ao(0) is invertible,

(iv) f (x, 0, e) is flat of Gevrey order -r as e -+ 0 in S(ro, no) uniformly in x E A(60), where r is a non-negative number. Then, there exist three positive numbers b, r, and or such that system (XIL5.1) has a solution O(x, e) which is holomorphic in (x, t) E A(6) x S(r, a) and that (x, c) is flat of Gevrey order -r as e -' 0 in S(r,a) uniformly in x E L1(6).

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

386

Proof.

Note first that (XII.5.5)

I exp1-

k1I = expJ-cJfJ-'cos(k(arge))]

for any positive numbers c and k. Note next that assumptions (i) and (iv) imply tha in the case when r = 0, Ax, 0, e) is identically equal to 0 for (t, f) E A(b) x S(r, a). Hence, in this case, 0 is a solution of (XII.5.1). In the case when r > 0, it holds

that (XII.5.6)

If(x,o,e)I < h

exp[-2cIEl-k1

for some positive numbers K and c if (x, e) E A(b) x S(r, a) for sufficiently small positive numbers 6, r, and a and if k

(XII.5.7)

(cf. Theorem X1-3-2). Therefore, (XII.5.5) and (XII.5.6) imply that (XII.S.8)

I exp[ce-k] I If (x, O, E)I = if (x, o, e)I expf clkl-k cos(k(arg e))]


for (x, e) E A(6) x S(r, a). Note also that (XII.5.9)

cos(k(arge)) > cos(ka) > 0

for

f E S(r,a) if

ka < 2.

Let us change l/ in (XII.5.1) by

Ii = exp(-ce-'16.

(XII.5.10)

Then, (XII.5.1) is reduced to (XII.5.11)

el" = exp[m-kJf(x,eXP[-ce-k]u,e).

Set

f (x+ FJ, e) = f (x, 0, e) + A(x, e)il + E yPfp(x, e). JpI>2

Then, exp[cE-k1f(x, eXp[-cE-k]u, e)

= eXPfce-kJf (x, 0, e) + A(x, e)U +

explc(1 - lpl)E-k]ul fp(x, e)' ip1>2

Using a method similar to the proof of Theorem XII-1-2, we construct a bounded solution u = tlr(x, e) of (XII.5.11). Therefore, system (XII.5.1) has a solution of

the form y" _ ¢(x,e) = expl-ce-k]y,(x,e). This completes the proof of Lemma XII.5.1.

The main result of this section is the following theorem, which was originally conjectured by J: P. Ramis.

5. GEVREY ASYMPTOTIC SOLUTIONS IN A PARAMETER

387

Theorem XII-5-2. Assume that (i) f (x, y', e) is holomorphic in (x, y, e) on a domain 0(6o) x f2(po) x S(ro, ao), where bo, po, ro, and ao are positive numbers, (ii) f (x, y, e) admits an asymptotic expansion F(x, y, e) of Cevmy orders as a --+ 0 in S(ro, ao) uniformly in (x, y-) E A(bo) x f2(po), where s is a non-negative number,

(iii) the matrix Ao(x) given by (XIL5.4) is invertible on 0(bo), (iv) it holds that

on 0(bo).

lim f (x, 0, e) = 0

(XII.5.12) Then,

(1) (XIL5.1) has a unique formal solution

Plx, e) _

(X1.5.13)

c-

eep'e(x)

with coefficients p"t(x) which are holomorphic on A(bo), (2) there exist three positive numbers 6, r, and a such that (XII. 5.1) has an actual solution ¢(x, e) which is holomorphic in (x, e) E A(5) x S(r, a) and that ¢(x, e) admits the formal solution P(x, e) as its asymptotic expansion of Gevrey order max {

1, s } as e - 0 in S(r, a) uniformly in x E L1(b). a

J11

Proof.

If a positive number & is sufficiently small, for every real number 6, there exists

a C"-valued function fe(x,y,e) such that (a) fe(x, y', e) is holomorphic in (x, y", e) on a domain A(60) x f2(po) x Se(ro, &), where (XIL5.14)

Se(ro,&) = {e: jarge - 61 < &, 0 < fej < ro},

(b) fe(x, y, e) admits an asymptotic expansion F(x, y e) of Gevrey order s as e - 0 in Se(ro, &) uniformly in (x, y) E 0(bo) x f2(po), where s is the nonnegative number given in Theorem XII.5.2. Such a function fe(x, y, e) exists if & is sufficiently small (cf. Corollary XI-2-5). In particular, set (XII.5.15)

fo(x, b, e) = f (x, v, e)

Let y = & (x, e) be a solution of the system (XII.5.16)

e°

= fe(x, y, e)

such that &(x, e) is holomorphic and bounded in (x, e) E i (b1) x Seri, al) for suitable positive numbers 61, r1, and al. Using Theorem XII-1-2, it can be shown that such a solution of (XII.5.16) exists.

388

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

Suppose that So,(ri,al) nSo,(ri,al) 0 0. Set

/i(x, E) = 4 (x, E) - 4 (x, E)

(XII.5.17)

for (x, E) E A(5j) X {So,(r1,a1)nSo.(rl,a,)}. Then, obi satisfies the following linear system

E°L = B(x, E)/+ 6(x, E),

(XII.5.18) where (XII.5.19)

B(x,E)W(x,E) =

fet(x,iez(x,E),E),

6(x,0 =

Note that (XII.5.20)

B(x, c) _

JlOfJ (x,tiet(x,E) + (1 -

and that, if s > 0, then jb(x,E)I
(XII.5.21)

for (x,E) E 0(6) x {Se,(rl,a,)nS02(rl,a,)}, where k

(XII.5.22)

and u and v are suitable positive numbers (cf. Theorem XI-3-2). From (XII.5.20), it follows that

lim B(x, E) = Ao(x),

(XII.5.23)

where the matrix Ao(x) is given by (XII.5.4). If s = 0, the power series f is convergent in c (cf. Exercise XI-6). Hence, b(x, E) = 0. If

(XII.5.24)

Se(r2,a2) C Se,(ri,a'i)nSo,(ri,ai)

and if positive numbers 62, r2, and a2 are sufficiently small, applying Lemma XII5-1 to (XII.5.18), the functions j'(x, f) can be written in the form (XII.5.25)

i(x, E) = I (x, E)y'(E) + w(x, E),

where

(XII526) ..

1.-.f

.)l

1=0

if

s=0,

5. GEVREY ASYMPTOTIC SOLUTIONS IN A PARAMETER

389

for (x, e) E A(S2) x Se (r2, a2), K and c are suitable positive numbers, t(x, e) is a fundamental matrix solution of the homogeneous system to dV = B(x, E)v,

(XII.5.27)

and I(e) is a C"-valued function of a independent of x. Assuming that Ao(0) has distinct eigenvalues A1,.12, ... , Al with multiplicities + ne = n) and that AO(O) is in a blocknl, n2i ... , ne, respectively (n1 + n2 + diagonal form (XII.4.6) with (XII.4.7), where N, (j = 1, 2,... , l) are n, x nj nilpotent matrices, respectively, apply Theorem XII-4-1 to block-diagonalize (XII.5.27). More precisely speaking, there exist three positive numbers 53, r3, 03, and an n x n matrix P(x, e) which is holomorphic in (x, e) E A(53) x So (r3, a3) such that (i) P(x, e) admits an asymptotic expansion (XII.4.9) in the sense of Poincart as e -+ 0 in S(r3,a3) uniformly in x E A(53), (ii) the transformation V = P(x, E)z

(XII.5.28)

reduces (XII.5.27) to (XII.5.29)

E°

di'

T = E(z, E) L,

where E(x, e) is in the block-diagonal form E(x, e) = diag [El (z, e), E2(z, c),. . . , Et(x, E)J ,

(XII.5.30)

with an n1 x n, matrix E J (x, c) for each j = 1, 2, ... , t, and E(x, e) admits

in the sense of Poincar6 as t -+ 0 in

an asymptotic expansion

SB(rs,a3) uniformly in x E A(63), where Eo(0) = Ao(0). Let us construct a rhombus 7)(64) with vertices x{'), x(2), xi3i, and xt4i as it is defined in §XII-2, in such a way that (a) 7)(54) C a(53),

(b) it holds that

(Ai(x_r(I))\ J

E°

(XII.5.31)

< _i4> HIx--x""I

E° (A,(z_z(3))\ <

IEI°

i4 A,1 Ix - x3l IEI°

for 3=1,...,m',

for j=m'+1,...,1

on the domain V(54) x So(r4,Q4), where 64i r4, and a4 are suitable positive numbers. El

Write fin the form z" _

I

where z"J E C"' (j = 1, 2, ... , f). Then, (XII.5.29)

ztJ can be written in the form (XII.5.32)

E° L =

(j = 1,2... ,f).

390

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

For each j, let 4ij(x, e) be a fundamental matrix solution of cv Then, the general solution of (XII.5.29) is given by (XII.5.33)

= Ej(x, e)z j.

%(x,E) _

where

xl

(XII.5.34)

for for

x3

j = 1,... , m',

j = m+ I,... ,

1.

In this way, we can prove that t 5(x, e) - w'(x, e) is flat of Gevrey order 1 as e - O in S e, (r1, a i) n Sez (ri , a 1) uniformly for x E A(S) if a positive number 6 is sufficiently small (cf. [Si6; Proof of Lemma 1 on pp. 377-379]). Therefore, by using Theorem X1-2-3, we can complete the proof of Theorem XII-5-2. Materials of this section are also found in JSi22J.

XII-6. Analytic simplification in a parameter For a system of linear differential equations of the form

dx = xkAl

(E)

ly,

where k is an integer, y E C', and A(t) is an n x n matrix whose entries are holomorphic in a domain (D)

{ t : it( <

(t = x; Ro : positive constant),

},

G. D. Birkhoff proved the following theorem.

Theorem XII-6-1 ((Bil). There exists an n x n matrix P(t) such that the entries of P(t) and P(t)-' are holomorphic in domain (D) and that the transformation y=

(T)

PMg

reduces system (E) to a system of the form

= xkB(z}v,

(Eo)

where B(t) is an n x n matrix whose entries are polynomials in t. Moreover, the

matrix P(t) can be chosen so that x = 0 is, at worst, a regular singular point of (E).

Observe that if we set t = x-3, (E) becomes {E1)

dg dt

=

_t-k-2

A(t)9.

6. ANALYTIC SIMPLIFICATION IN A PARAMETER

391

If k < -2, (E1) has the coefficient holomorphic in (D). Hence, there is a fundamental matrix solution V(t) of (E1) whose entries are holomorphic in (D). Then, the transformation v = V (1) i reduces (E) to the system

f

= 6. Therefore, the

main claim of Theorem XII-6-1 concerns the case when k > -1. In this theorem of G. D. Birkhoff, the entries of the matrix B(t) are polynomials in t. However, even though we can choose P(t) so that x = 0 is, at worst, a regular singular point of (Eo), the degree of B(t) with respect to t may be very large. In order that the

degree of B(t) with respect to t is at most k + 1 so that x = 0 is a singularity of the first kind of system (Eo), we must impose a certain condition on A(t) (cf. Exercises XII-8, XII-9, and XII-10). For interesting discussions on this matter, see, for example, [u2], [JLP], [Ball], and [Ba12j. A complete proof of Theorem XII-6-1 is found, for example, in [Si17, Chapter 3]. In this section, we prove a result similar to Theorem XII-6-1 for a system of linear differential equations f '!L

(XII.6.1)

= A(x,e)y,

dx

under the assumption that a is a positive integer, y" E C", and A(x, e) is an n x n matrix whose entries are holomorphic with respect to complex variables (x, e) in a domain (XII.6.2)

X E Do,

[e[ < 60,

where Do is a domain in the x-plane containing x = 0, and 6o is a positive constant. Let 00

A(x,e) = E ekAk(x)

(XII.6.3)

k=0

be the expansion of A(x, e) in powers of e, where the entries of coefficients Ak(x) are 00

holomorphic in Do. We assume that the series >26o (Ak(x)I is convergent uniformly k=0

in Do. The main result of this section is the following theorem, which was originally proved in [Hsl].

Theorem XII-6-2 ([Hsl]). For each non-negative integer m, there is an n x n matrix P(x, e) satisfying the following conditions: (i) the entries of P(x, e) are holomorphic in (x, e) in a domain (XII.6.4)

x E D1i

jeJ < 60,

where V1 is a subdomain of Do containing x = 0, (ii) P(x, 0) = In for X E V1 and P(0, e) = In for je[ < 6o, (iii) the system (XII.6.1) is reduced to a system of the form (XII.6.5)

e°

dii

m

o-1

k=0

k--O

_{kA(X) E + em+1 r` c'Bk(: W

)

u

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

392

by the transformation P(x, e)u,

(XII.6.6)

where Bk(x) (k = 1, 2, ... , or - 1) are n x n matrices whose entries are holomorphic for x E D1.

Remark XII-6-3. In case when a = 1, (XII.6.5) becomes dil ed

En

CkAk(x)+em+IB0(x)I!

.

k=0

In particular, if a = I and m = 0, (XII.6.5) has the form e

dil

= {Ao(x) + eB0(x)} u.

It should be noticed that in Theorem XII-6-2, if we put m = 0, then the right-hand side of (XII.6.5) is a polynomial in a of degree at most a without any restrictions on A(x, e). We prove Theorem XII-6-2 by a direct method based on the theory of ordinary differential equations in a Banach space. (See also [Si8j.) Proof of Theorem XII-6-2.

The proof is given in three steps.

Step 1. Put (XII.6.7)

a

m+o

k=0

k--0

P(x, e) = I + Em+1 [-` ekPk(x) and B(x, e) = E EkBk(x),

where

(XII.6.8)

Bk(x) _

(k = 0,1, ... , m),

J A&. (x)

Bk-m-1(x)

(k = m+ 1,m+2,... ,m+a).

From (XII.6.1), (XII.6.5), and (XII.6.6), it follows that the matrices P(x,e) and B(x, e) must satisfy the equation (XII.6.9)

e

QdP

= A(.x, e)P - PB(x, e).

From (XII.6.3), (XII.6.7), (XII.6.8), and (XII.6.9), we obtain k

(XII.6.10)

Am+1+k(x) - Bm+1+k(x)+ E{Ak-h(x)Ph(x) - Ph(x)Bk-h(x)} h=O

(k=0,1,...,a-1)

6. ANALYTIC SIMPLIFICATION IN A PARAMETER

393

and

dPk(x) dx

o+k

Ad+k-h(x)Ph(x)

=A m +1+ o +k(x) + h=o

(XII.6.11)

o+k (k = 0,1,2,...),

E Ph(x)Bo+k-h(x)

h=k-m where

Ph(x) = 0

(XIL6.12)

if

h < 0.

It should be noted that the formal power series P and B that satisfy the equation (XII.6.9) are not convergent in general. In order to construct P as a convergent power series in e, we must choose a suitable B. To do this, first solve equation (XII.6.10) for B,,,+1+k(x) to derive

Bm+1+k(x) = Am+I+k(x)+Hm+1+k(x;P0,P1,... Pk)

(XII.6.13)

(k=0,1, ..,Q-1),

where H, are defined by (XII.6.14)

H , = 0,

(k = 0,1, ... , m), k

k

Hm+I+k(x; Po, PI, ... , Pk) = E{Ak-h(x)Ph - PhAk-h(x)) h=0

- E PhHk-h, h=0

(k = 0,1,...,a - 1). Denote by P an infinite-dimensional vector {Pk : k = 0, 1, 2,... }. Then, by substituting (XII.6.13) into (XII.6.11), we obtain dPk(x)

(XII.6.15)

dx

= fk(x; P)

(k = 0,1, 2, ... ),

where o+k

o+k

fk(x; P) = Am+1+o+k(x) + E Ao+k-h(x)Ph - 1: PhA,,+k-h(x) h=0

(XII.6.16)

h=k-m

a+k

-

E PhHo+k-h(x;P)

(k=0,1,2,...).

h=k-m

Denote by -F(x; P) the infinite-dimensional vector { fk(x; P) : k = 0,1, 2,... }. Then, equation (XII.6.15) can be written in the form (XII.6.17)

dP = F(x; P). dx

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

394

Solve this differential equation in a suitable Banach space with the initial condition P(0) = 0. Actually, we solve the integral equation

P(x) = f0F(;P())de.

(XII.6.18)

This is equivalent to the system (XII.6.19)

Pk(x) =

ffk(;P())de

(k = 0,1, 2, ... ).

If P is determined, then the matrices Bk are determined by (XII.6.13) and (XI1.6.14).

Step 2. We still assume that A(x, c) is holomorphic in domain (XII.6.2). Denote by B the set of all infinite-dimensional vectors P = {Pk : k = 0, 1, 2.... } such that (i) Pk are n x n matrices of complex entries, 00

(ii) j:boilPkll < oo, where IlPkll is the sum of the absolute values of entries k=0 of Pk.

For each P, define a norm 11P11 by 00

(XII.6.20)

IIPII = Ebo IlPkll k=0

Then, we can regard B as a Banach space over the field of complex numbers. Now, we can establish the following lemma.

Lemma XII-6-4. Let F(x; P) be the infinite-dimensional vector whose entries fk(x,P) are given by (XII.6.16). Then, for each positive numbers R, there exist two positive numbers G(R) and K(R) such that (XII.6.21)

for IIPII <- R

flF(x; P) II 5 G(R)

and

(XII.6.22)

II.(x; P) - F(x; P)ll 5 K(R)IIP - PII for IIPII 5 R, IIPI{ 5 R.

In order to prove this lemma, consider a formal power series of e which is defined by

.r(x,P,E) _

(XII.6.23)

Ekfk(x;P)k

Then, using (XII.6.16), we obtain (XII.6.24) 00

.F(x, P, e) _ >2 EkA,,,+1+o+k(x) + k=0

a-1

- k=0 >

l

\ k=0 EkAk(x)/

(e'Pk) k=0

k

- Ek=OEk hE Ak-h Ph 0-1

1

M+V

C*

k=o Ekpk/

k

ek E Ph[Ak-h(x) + Hk-h(x;P)1 h=O

Hence, Lemma XII-6-4 follows immediately.

l }.

r

k o Ek [Ak(x) +

Ik(x; PA)

395

EXERCISES XII

Step 3. We construct the matrix P(x, e), solving the integral equation (XII.6.18) by the method of successive approximations similar to that given in Chapter I. By virtue of Lemma XII-6-4, we can construct a solution P(x) in a subdomain Dl of Do containing x = 0 in its interior. Since (XII.6.18) is equivalent to differential equation (XII.6.15) with the initial condition P(0) = 0, the solution P(x) gives the desired P(x, e). The matrix B(x, c) is given by (XII.6.13) and (XII.6.14). 0

EXERCISES XII 00

XII-1. Find a formal power series solution y = F, e'd,,,(x) of the system of m=0

differential equations CO

LY = Ay" + > Embm(x), m=0

where y E C", A is an invertible constant n x n matrix, and 5,n (x) and bm(x) are C"-valued functions whose entries are holomorphic in x in a neighborhood of x = 0. XII-2. Using Theorem MI-4-1, diagonalize the system dy = e dx

0 11-X

1+x1 ex

y'

where

[1.

XII-3. Using Theorem XII-4.1, find two linearly independent formal solutions of each of the following two differential equations which do not involve any fractional powers of e. E2d2

(1)

Y

2 + y = eq(x)y,

(2) a2 + y = eq(x)y,

where q(x) is holomorphic in x for small jxj.

Hint. If we set '62 = e, differential equation (2) has two linearly independent solutions e4=/00(x, 0) and a-1/190(x, -/3). The two solutions

= Q reiz/°4(x, Q) -

-Q),

do not involve any fractional powers of e.

XII-4. Let x be a complex independent variable, y" E C", z" E C, e be a complex parameter, A(x, y, z, e) be an n x n matrix whose entries are holomorphic with respect to (x, y, z, e) in a domain Do = {(x, y, X, e) : jxj < ro, jyi < Pi, Izl < p2, 0 < jej < ao, j arg ej < flo}, f (x, y, z, e) be a C-valued function whose entries are holomorphic with respect to (x, y, 1, e) in Do, and #(x, ,F, e) be a C"-valued function whose entries are holomorphic with respect to (x, z, e) in the domain Uo = ((x, zl, e) : jxj < ro, jzj < p2,0 < jej < ao, j argej < /3o}. Assume that the entries of the matrix

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

396

A(x, if, z, c), the functions f (x, y, z, e), and §(x, .F, e) admit asymptotic expansions as f - 0 in the sector So = {E : 0 < IEI < ao, I arg EI < /30} uniformly in (x, y, z) in the domain Ao = {(x, y, z) : IxI < ro, Iv'I < pl, I1 < p2} or (x, z-) in Vo = {(x, z) : I xI < ro, Izl < p2}. The coefficients of those expansions are holomorphic with respect to (x, y", I) in Do or (x, z-) in V0. Assume also that det A(0, 0, 0, 0) 0 0. Show that the system f dy = A(x, y, z, E)y + E9(x, Z, f),

= Ef (x, g, Z, f)

has one and only one solution (y, z) = (¢(x, f),

e)) satisfying the following

conditions:

(a) the entries of functions O(x, E) and '(x, f) are holomorphic with respect to (x, e) in a domain {(x, f) I xI < r, 0 < lei < a, I arg EI < /3} for some positive numbers r, a, and 0 such that r < ro, a < ao and /3 < /io, (b) the entries of functions (x, c) and i/ (x, c) admit asymptotic expansions in :

powers of f as c - 0 in the sector if : 0 < lei < a, I argel < (3} uniformly in x in the disk (x : Ixi < r), where coefficients of these expansions are holomorphic with respect to x in the disk {x: Ixi < r}, (c) 0(0,c) = O for f E {E : 0 < IEI < a, I arg EI < ,0}.

Hint. Use a method similar to that of §§X11-2 and XII-3.

XII-5. Find the following limits:

j

f (t) sine I

r / c-0 0

(1) lim

at

1

f (t) sin 1

\E

dt and (2)

lim

I dt, where a is a nonzero real number and f (t) is continuous

and continuously differentiable on the interval 0 < t < 1.

XII-6. Discuss the behavior of real-valued solutions of the system

Edg = rEE

-1 +Ex1 y as e -40,

y"_ [y2J

where

XII-7. Discuss the behavior of real-valued solutions of the following two differential

equations as f

O+: (1) E2

d3 y

+ 2 + xty = O,

(2) E3

1

(1

x)y = O.

XII-8. Assume that (i) the entries of a C"-valued function Ax, y", e) are holomorphic with respect to (x, y", e) in a domain A1(Eo) X II(po) X O2(ro), where b0, po, and ro are positive

numbers and

0l(6o)={xE(C:IxI
(ii) the matrix Ao(x) =

Of

(iii) f(x, 0, 0) = 0 on A(bo), (iv) a is a positive integer.

-

(x, 0, 0) is invertible on 0(bo),

EXERCISES XII

397

Denote by S(r, a, 0) the sector (EEC : 0 < IEI < r, I arg e - 01 < a). Show that

y (1) the system (S) c'±dx = f (x, y, E) has a unique formal solution p(x, E) _ +00

with coefficients p"l(x), which are holomorphic in A(50), c=1

(2) for any real number 0, there exist three positive numbers b, r, and a such that (S) has an actual solution ¢(x, e), which is holomorphic in (x, E) E 0(b) x S(r, a, 0), and that ¢(x, c) has the formal solution p(x, e) as its asymptotic expansion of Gevrey order

1 as c -. 0 in S(r, a, 0) uniformly in x E 0(b) or

Hint. This is a special case of Theorem XII-5-2. +M

XII-9. Assume that p(x,E) _ E Emp',,,(x) is a formal solution of a system (S) ca

dy

M=1

= f (x, 17, f), where a is a positive integer, y" E C, Ax, y", e) is a C"-valued

dx function whose entries are holomorphic in a neighborhood of (x, y, f) = (0, 0, 0), and the entries of pm(x) (m = 0.1, ...) are holomorphic in a disk Ixi < ro, where ro is a positive ember. Assume also that p(x, E) E {G[[E]]3}" uniformly for Ixi < ro

and that 0 < s < 1. Show that if S is an open sector in the E-plane with vertex at e = 0 and whose opening is smaller than sir, there exists two positive numbers ri and r2 and a solution $(x, e) of (S) such that the entries of q,(x, e) are holomorphic in (x, E) for Ixt < r1, IEI < r2,( E S, and that (x, E) admits the formal solution p(x, c) as the asymptotic expansion of Gevrey order s as e -+ 0 in S n {I-El < r2} uniformly for IxI < r1. Note. No additional conditions on the linear part of Ax, y", c) are assumed. +00

XJI-10. Assume that p(x, c) = F E"`p""m(x) is a formal solution of a system m--O

e

= f (z, y, E), where y E C", f (x, y, e) is a C"-valued function whose entries are

holomorphic in a neighborhood of (x, y, c) = (0, 6, 0), p".. E C[(x]]" (m = 0,1, ... ), p"o(0) = 6, and p(O, e) is convergent. Show that p(x, e) is convegent for ]x) < r and IEI < p for some positive numbers r and p.

Hint. Regard p(ET, e) as the solution of the initial-value problem y = d 00) = P(0. E.

f (.ET, y, E),

XII-11. Let t and e be complex variables, y" E G", and the entries of a G"valued function f(t, y, e) be holomorphic with respect to (t, y. E) in a domain 1)o = E)

: ICI < do, -oo < tt < oo, Jyj < po, 0 < IEI < ro, I arg E) < ao). As00

some that f(t, y, E)

E E'f,,,(t, y-) as e -+ 0 in the sector So = {E : 0 < IEI < m=0

ro, I arg EI < ao } uniformly with respect to (t, y-) in the domain Do = {(t, y) : J 3'tI < do, -oo < Iftt < +oc, Iyl < po}, where the coefficients fm(t,y-) are holomorphic in

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

398

Ao. Assume also that f(t, y, e) and the coefficients fm(t, y) are periodic in t of dy = 27riy + e[y + e f (t, y, e)] has a period 1. Show that the differential equation dt e) of period 1 such that periodic solution U = (a) the entries of (t, e) are holomorphic with respect to (t, e) in a domain D = { (t, e) : I `3'tJ < d, -oo < Rt < +oo, 0 < JeJ < r, I arg el < a} for some (d, r, a)

such that0
a}, where coefficients pm(t) are holomorphic and periodic of period 1 in the domain D' = It : J:33tI < d, -oo < Rt < +oo}.

Hint. Step 1. Construct the solution +li(t, c", e) to the initial-value problem (IP)

dt = 27riy+ e[y+ ef (t, y, e)],

9(0) = c.

Substep 1. First, construct a formal solution +00

+G(t, , e) _ E em'+Gm(t, l = e2x1cd + ete2attC + O(e2). M=0

The coefficients 1& (t, a) are holomorphic in a domain S2o = {(t, cl : 13'tI < do, -oo < Rt < +oo, Icl < ryo),

where -yo is a positive number. To prove this, set y = e2i't(c + CZ-) to change (IP) to

(IP')

d = 6 + e[z + e-2i't f (t, e2"`t(c + ez_), e)J,

z(0) = 0.

The function f (t, e2x't (c" + ez-),e) admits an asymptotic expansion in powers of e (cf. Proof of Theorem XI-1-8) whose coefficients are polynomials in the entries of ci can be calculated successively. the vector z. Coefficients

Substep 2. Show that iP(t, c", e) V;(t, c, e) as e -. 0 in a sector S = {e : 0 < je1 < r, I argel < a} uniformly in a domain 1Y = {(t,c) : J)tj < d, IRtI < T, Icl < ry}, where T is an arbitrary positive number and -y > 0 depends on T. In this argument, the Gronwall's inequality (Lemma 1-1-5) is useful. Step 2. Solve the equation 1 (1, cE, e) = E. This equation has the form

c" = eh(c, e),

(E)

where h(c", e) admits an asymptotic expansion as a -' 0 in the sector S uniformly for Icd < y. To solve this, we can use successive approximations as follows:

4W = 0,

Ch(e) = e46y_1(e),e).

Then, the approximations 64(e) converge to a limit c(e). Using Theorem XI-1-12, we can conclude that ee) admits an asymptotic expansion in powers of e.

399

EXERCISES XII

Step 3. The particular solution ¢(t, e) = j(t, cue), e) is the periodic solution satisfying all of the requirements. XII-12. Consider a system of differential equations dy = xka(x,e)y,

(s)

dx

where x is a complex independent variable, a is a complex parameter, k is a nonnegative integer, and y is an unknown element in a Banach algebra U over the field C of complex numbers with a unit element I . Assume that +00

a(x, e) _ E x-ma,.,(e),

(a)

m=o

where am(e) E U and these quantities are holomorphic and bounded with respect toe in a sector S = {e : 0 < je' < bo, I argel < bl}, and the series (a) is convergent in norm for lx[ > Ra uniformly for e in S. Also, assume that a(x, e) admits an asymptotic expansion in powers of a uniformly for x in jxl > Ro as e -. 0 in S. Show that if a positive integer N is sufficiently large, there exist elements p(x, e), bo(e ), ... , bM (e) of U such that (i) p(x, e) is holomorphic and bounded in S and large [x[, (ii) p(x, e) admits an asymptotic expansion in powers of e uniformly for large [xi

a positive integer and the quantities bo(e), ..., bkf(e) are holomorphic in S and admit asymptotic expansions in powers of e as a -+ 0 in S,

m>k+l,

(iv)

(v) the transformation y = [I + x-(N+1)p(x, e)Ju changes (s) to

(S')

du dx

N

xk

E x-mam(e) + x- (N+1)

M

E x-mb,,,(e)I

U.

Comment and Hint. See [Sill; in particular Theorem 2 on p. 157]. The point x = 0 is not necessarily a regular singular point of (s') as in Theorem XII-6-1. Step 1. The main idea is, assuming that N > k, to solve equations of the following forms:

(N + 1) x -(k+l)Q(x) (1)

-

x -k d((x) + a (x )Q( x )

+ x-(N+1) [a(x)Q(x)

-

Q( x ) a ( x )

-

B(x )

- Q(x)B(x)] = F(x)

and

(II)

(N + 1)x-(k+l)Q(x) - x-kdQ(x) + a(x)Q( x )

- Q(x ) a(x ) - B( x )

+ x-(N+1) [a(x)Q(x) - Q(x)$(x) - 'Y(x)B(x)) = F(x),

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

400

where

(1) Q(x) is an unknown quantity which should be a convergent power series in x-1,

(2) B(x) is an unknown quantity which should be a polynomial in x-1 of degree k, (3) a(x) is a given polynomial in x-1 of degree N, (4) a(x) is a given convergent power series in x-1, (5) 0 is a given polynomial in x-1 of degree k, (6) -r(x) is a given convergent power series in x-1, (7) F(x) is a given convergent power series in x-1. To solve these equations, first express B in terms of Q. In fact, setting too

N

k

Q(x) _ E x-'Q,,,. B(x) _ F, m=0

x-mBm,

a(x) _ E X -0m,

m=0

m=0

we obtain M

Bm = 1:fam-hQh - Qham-hl,

m = 0,1,..., k.

h=1

Then, we can write (I) and (II) in the form 1

(m>0), = where 9,,, is either quadratic or linear in Q (n > 0). Step 2. If N > 0 is sufficiently large, we can solve (III) by defining a norm for a QM

(III)

+oo

+oo

convergent power series P(x) = E x-'P,,, by 11P11 = E p'11 P,,, 11, where p is a

m0

m=0

sufficiently small positive number.

Step 3. Using Steps 1 and 2, we can construct a formal power series q(x, e) _ +00

E e'ge(x) such that the coefficients qt(x) are holomorphic and bounded in a disk t=0

A = {x : Ix] > R > 0} and that the formal transformation y = [I+x-(N+1)4(x, e)Ju changes (s) to (s'). Step 4. Find a function q(x, e) such that q is holomorphic and bounded in 0 x S and that q(x, e) q(x, e) as e --+ 0 in S uniformly for x E A. Then, transformation Y= [I + x-(N+1)q(x, e)Iv changes (s) to (s") dv

k

=

dx

xk

F

x-mam(E) + x- (N+1) E

M=

m=0

+oo

x-mbm(E) + E x-mbm(0) 1 u, m=k+1

where bm(E)

^_-

b,1 (E)

as

E

and

x-mam(E) ^- 0 m=k+1

as

0 in S uniformly for x E A.

0 in S

401

EXERCISES XII

Step 5. Using again Steps 1 and 2, find a function r(x, e) such that (a) r(x, e) is holomorphic and bounded for jxj > R' > 0 and e E S, (b) r(x, e) = 0 as a - 0 in S uniformly for jx[ > R', (c) for a sufficiently large M' > 0, the transformation v = [I +x-(h1'+1)r(x,e)jw changes (s") to

= xk

dw

M

N

` x-mam(f) + x -(N+1) E ? bm(e) w, Lm=o M-

whereb,,,(e)-0ase-0 inSform>k+1. XII-13. Let A(t) be a 2 x 2 matrix whose entries are holomorphic in a disk It it[ < po} such that the matrix (M)

P

()' {kA(!)P(1)

-P

:

(-X1)1

is not triangular for any 2 x 2 matrix P(t) such that the entries of P(t) and P(t)-1 are holomorphic in the disk D = It : jtj < po}. Show that there exists such a 2 x 2 ) matrix P(x) for which matrix (M) has the form xk B Gl with a 2 x 2 matrix B(t) wh ose entries are polynomials in t and whose degree in t is at most k + 1.

Hint. See [JLP]. This result can be generalized to the case when A(t) is a 3 x 3 matrix (cf. [Ball) and [Bal2J).

XII-14. Let P(t) be a 2 x 2 matrix such that the entries of P(t) and holomorphic in a disk V = it : jtj < po} and that the transformation y" = P

P(t)-1

are

(!)

u"

X

changes the system dil dx

[ l +x-1 I.

0

x-3

1 1-x'1Jy'

y =

[ i12,

to

dil

ds

Blx/u,

u=

u1 112

with a 2x 2 matrix B(t) whose entries are polynomials in t. Show that the degree of the polynomial B(t) is not less than 2. Also, show that there exists P(t) such that the degree of the polynomial B(t) is equal to 2.

Hint. To prove that there exists P(t) such that the degree of the polynomial B(t) -17+ at is equal to 2, apply Theorem V-5-1 to the system t d'r = 1 +tbt ] v with t .ii L suitable constants a, Q, y, and b. To show that the degree of the polynomial B(t) is not less than 2, assuming that the degree of the polynomial B(t) is less than 2, derive a contradiction from the following fact:

402

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

The transformation y = 0

0

dy

rl+x'1

dx

l x-3

dil dx =

[lx-I

u changes the system 0

1

xtJ',

to

1-x-11

u,

u = IuaJ.

XII-15. Let A(t) be a 2 x 2 triangular matrix whose entries are holomorphic in a disk it : Its < po}. Show that there exists a 2 x 2 matrix Q(x) such that (i) the entries of Q(x) and Q(x)-I are holomorphic for jxi > o and meromorphic at x = oo, (ii) the transformation y" = Q(x)t changes the system dx = xkA

\ x/

y to

_

xkB 1 f u" with a 2 x 2 matrix B(t) whose entries are polynomials in t and whose degree in t is at most k + 1.

Hint. See IJLP]. This result can be generalized to the case when A(t) is a 3 x 3 matrix (cf. [Ball] and [Bal2j).

CHAPTER XIII

SINGULARITIES OF THE SECOND KIND

In this chapter, we explain the structure of asymptotic solutions of a system of differential equations at a singular point of the second kind. In §§XIII-1, XIII-2, and XIII-3, a basic existence theorem of asymptotic solutions in the sense of Poincar6 is proved in detail. In §XII-4, this result is used to prove a block-diagonalization

theorem of a linear system. The materials in §§XIII-I-XIII-4 are also found in [Si7J. The main topic of §XIII-5 is the equivalence between a system of linear differential equations and an n-th-order linear differential equation. The equivalence is based on the existence of a cyclic vector for a linear differential operator. The existence of cyclic vectors was originally proved in [De1J. In §XIII-6, we explain a basic theorem concerning the structure of solutions of a linear system at a singular point of the second kind. This theorem was proved independently in [Huk4] and (Tull. In §XIII-7, the Newton polygon of a linear differential operator is defined. This polygon is useful when we calculate formal solutions of an n-tb-order linear differential equation (cf. [Stj). In §XIII-8, we explain asymptotic solutions in the

Gevrey asymptotics. To understand materials in §XIII-8, the expository paper [Ram3J is very helpful. In §§XIII-I-XIII-4, the singularity is at x = oo, but from §XIII-5 through §XIII-8, the singularity is at x = 0. Any singularity at x = oo is changed to a singularity of the same kind at = 0 by the transformation x = I

XIII-1. An existence theorem In §§XIII-1, XUI-2, and XIII-3, we consider a system of differential equations (XIII.1.1)

= x'f,(x,L'i,v2,... ,un)

(.7 = 1,2,... n),

where r is a non-negative integer and ff (x, v1, t,2,... , vn) are holomorphic with respect to complex variables (x, v1, v2, .... vn) in a domain (XIII.1.2)

JxJ>No,

JargxI
Jv21<6o

(j=1,2,...,n),

No, oo, and 6a being positive constants. Set n

(XIII.1.3)

f2 (x,

ffo(x) + E ain(x)va + h=1

f, ,(x)v Ip1?2

where v" E C" with the entries (vl, V2.... , vn). We look at (XIII.1.1) under the following three assumptions. 403

+

404

XIII. SINGULARITIES OF THE SECOND KIND

Assumption I. Each function f j (x, V-) (j = 1, 2,... , n) admits a uniform asymptotic expansion

fi(x,v1 =00>fjk(la

(XIII.1.4)

k

k=0

in the sense of Poincare as x - oo in the sector (XIII.1.5)

Jx j > No,

J argxj < cto,

where coefficients fjk(v-) are holomorphic in the domain (XIII.1.6)

jvl < bo.

Furthermore. we assume that

fio(d) =

(XIII.1.7)

0

(j = 1, 2,... , n).

Observation XIII-1-1. Under Assumption I, fo(x), ajh(x), and fl,(x) admit asymptotic expansions oe

00

f70kx-k,

f}0(x)

fjp(x) ti

1: fjpkx-k, k=0

k=1

ajhkx-

a,h(x) ti00E k=0

(j, h = 1, 2,... , n)

as x oo in sector (XIII.1.5) with coefficients in C. Let A(x) be then x it matrix whose (j, k)-entry is ajk(x), respectively (i.e., A(x) = (a,h(x)). Then, A(x) admits an asymptotic expansion 00

(XIII.1.10)

A(x) '=- E x-kAk k=0

as x - oo in sector (XIII.1.5), where Ak = (ajhk). The following assumption is technical and we do not lose any generality with it.

Assumption II. The matrix A0 has the following S-N decomposition: (XIII.1.11)

Ao = diag [µ1,µ2, ... 44n] + N,

where µ1,µ,2, ... , µ are eigenvalues of Aa and N is a lower-triangular nilpotent matrix.

Note that WI can be made as small as we wish (cf. Lemma VII-3-3). The following assumption plays a key roll.

405

1. AN EXISTENCE THEOREM

Assumption III. The matrix A0 is invertible, i.e., p,54 0

(XIII.1.12)

(j=1,2,...,n).

Set

w = arg x,

(XIII.1.13)

1 ,2 , .n ) .

w , = arg p3

and denote by D, (N,'y,q) the domain defined by D3 (N, y, q)

(XIII.1.14)

l

x:

Ixi > N, jwj < ao,

(2q_)r+Y
,w

q is an integer, N is a sufficiently large positive constant, and -y is a sufficiently

small positive constant. For each j, there exists at least one integer q, such that the real half-line defined by x > N is contained in the interior of V., (N, y, qj). Set n

(XIIL1.15)

7)(N,7) = n V, (N,y,qj). =1

In §§XIII-2 and XIII-3, we shall prove the following theorem.

Theorem XIII-1-2. If N-1 and y are sufficiently small positive numbers, then under Assumptions 1, II, and III, system (XIII. 1.1) has a solution (XHI.1.16)

U = 1,2,... , m),

vj = pj (x)

such that (i) p,(x) are holomorphic in D(N,-y), (ii) p, (x) admit asymptotic expansions 00

(XIII.1.17)

Pi (z)

>2 P,kx-k

(j = 1, 2, ... , m)

k=1

as x - oo in 7)(N,7), where p,k E C. To illustrate Theorem XIII-1-2, we prove the following corollary.

Corollary XIII-1-3. Let A(x) be an n x n matrix whose entries are holomorphic and bounded in a domain Ao = {x : lxi > Ro} and let f (x) be a C'-valued function whose entries are holomorphic and bounded in the domain Do. Also, let A1,A2, ... An be eigenvalues of A(oo). Assume that A(oo) is invertible. Assume also that none of the quantities Aje1ka (j = 1, 2,... , n) are real and negative for a real number 9 and a positive integer k. Then, there exist a positive number e and a solution y"= J(x) of the system d-.yi =

xk-1A(x)y+x-1 f(x) such that the entries of

XIII. SINGULARITIES OF THE SECOND KIND

406

O(x) are holomorphic and admit asymptotic expansions in powers of x-1 as x -+ 00 in the sector S = {x : IxI > Ro, I argx - 01 < Zk + e}. Proof.

The main claim of this corollary is that the asymptotic expansion of the solution is valid in a sector I argx-8 < 2k +e whose opening is greater than k So, we look at the sector D(N, y) of Theorem XIII-1-2. In the given case, ao = +oo, r = k -1, and µ, = Aj (j = 1,... , n). The assumptions given in the corollary imply that

w, + k6 -A (2p + 1)7r,

where wj = arg Aj (j = 1, ... , n),

for any integer p. Therefore, for each 3, there exists an integer q j such that

either - it < w, - 2qjr, + k8 < 0

or

0 < w, - 2gjir + k8 < rr.

Therefore, either

- 3r.

< wj - 2q, 7r + k8 - 2 < w) - 2q,rr + kO +

2

<2

or

2


32

T.

2nk

This proves that a sector S ={ x : x > &,, I argx - 81 < + e is in D(N, y) for a sufficiently large Ro > 0 and a sufficiently small e > 0 if we use xe-i8 as the independent variable instead of x.

XIII-2. Basic estimates In order to prove Theorem XIII-1-2, let us change system (XIII-1-1) to a system of integral equations.

Observation XIII-2-1. Expansion (XIII.I.17) of the solution pj (x) 00

(XIII.2.1)

Epjkx_k

uJ =

(j = 1,2,... n)

k=1

must be a formal solution of system (XIII.1.1). The existence of such a formal solution (XIII.2.1) of system (XIII.1.1) follows immediately from Assumptions I and III. The proof of this fact is left to the reader as an exercise.

2. BASIC ESTIMATES

407

Observation XIII-2-2. For each j = 1,2,... , n, using Theorem XI-1-14, let us construct a function z3(x) such that (i) z j (x) is holomorphic in a sector (XIII.2.2)

I argxl < ao,

In l > No,

where No is a positive number not smaller than No,

(ii) zj (x) and

(XIII.2.3)

dz) admit asymptotic expansions z,(x)

L.f,kx-k

dz)

and

k=1

E(-k)P,kx-k-1

k=1

as x - oo in sector (XIII.2.2), respectively. Consider the change of variables

v, = u, +z,(x)

(XIII.2.4)

(j = 1,2,... ,n).

Denote (zl, z2, ... , z,a) and (u1, U2,. , isfies the system of differential equations -

du,

(XIII.2.5)

-

= x'g,(x,U-)

by i and u, respectively. Then, u' sat-

(.7 = 1,2,... ,n),

where dx Set

m (XIII.2.6)

g,(x, u") = go(x) + >2 b,k(x)uk +00E b,p(x)iI k=1

(j = 1, 2,... , n).

jp1>2

In particular, (XIII.2.7)

92o(x) = f, (x, a) - x-' d

dxx)

0

(j = 1, 2,... , n)

and

b3k(x)-ajk(x)=O(IxI-')

(j,k=1,2,...,n)

as x -+ oo in sector (XIII.2.2). Thus,

()III.2.8)

b,k(x) = a,k(oc) +o(IxI-1)

as x -' oo in sector (XIII.2.2).

(j, k = 1,2,... , n)

408

XIII. SINGULARITIES OF THE SECOND KIND

Observation XIII-2-3. Set

(j = 1,2,... ,n).

g1(x,u") = u, uj +R1(x,u')

(XIII.2.9)

Then, by virtue of Assumption III, (XIII.2.7), and (XIII.2.8), for sufficiently small positive numbers No 1 and 5, there exists a positive constant c, independent of j, such that for any positive integer h, the estimates (XIII.2.10)

+bhlxl-(n+1)

lR3(x,u)I ( clui

(j = 1,2,... n)

and (XIII.2.11)

IR,(x,u) - R, (x,1U') I < clu"- u l

(j = 1,2,... ,n)

hold whenever (x, u") and (x, u"') are in the domain (XIII.2.12)

lxl > No,

luil <6 (j =1,2,... ,n).

Iargxl
Here, by, is a positive constant depending on h. Furthermore, by virtue of Assump-

tion II, it can be assumed without loss of generality that the constant c satisfies the condition

c< r+1 , H

(XIII.2.13)

where H is a positive constant to be specified later. From (XIII.2.5) and (XIII.2.9), it follows that d

(XIII.2.14)

ds = X'" [µuj + R, (x, u)]

Change system (XII/I.2.14) to a system of integral equations (XIII.2.15)

t

u, = J

trR, (t, u) exp Ir -

1(tr+l

_ xr+1)1 dt (j = 1, 2,... , n), J

where the paths of integration L3 start from x. The paths of integration must be chosen carefully so that uniformly convergent successive approximations can be defined in such a way that the limit is a solution uf(x) of (XIII.2.15) which satisfies the conditions (i) u , (x) (j = 1, 2, ... , n) are holomorphic in D(N,'y) and (ii) u., (x) ^ _ - 0 (j = 1, 2, ... , n) as x - oo in D(N, ry) for suitable positive numbers N and ry.

Hereafter in this section, we explain how to choose paths of integration on the right-hand side of (XIII.2.15).

Observation XIII-2-4. Since for each j, the domain Dj (N, y, q,) contains the real half-line defined by x > N in its interior, their common part D(N, ry) is given by the inequalities (XIII.2.16)

l xI > N,

-t < argx < e',

409

2. BASIC ESTIMATES

where t and f are positive constants. It is noteworthy that if x E D(N, y), then x satisfies the inequalities (XIII.2.17)

argxI
-

2r+y
(j=1,2,...,n),

provided that w1 = arg u, are chosen suitably. Therefore,

r


2

(XIII.2.18)


3

2r-y

Moreover, since D(N,1) is the common part of Dj (N, y, qj), the equalities

-Zr+y =w3 -(r+ 1)f

(XIII.2.19)

and 3

(XIII.2.20)

r-y=Wh+(r+1)f

hold for some j and h. This implies that t and f depend on y. Thus, the quantity y can be chosen so that Wa - (r+ 1)f 36

(XIII.2.21)

f2r

and

wj +(r+1)f 54± r

(XIII.2.22)

for all j.

Observation XIII-2-5. Define the paths of integration in each of the following two cases.

Case 1. Consider first the case when

(r+1)(f+f)
(XIII.2.23)

In this case, the set of indices J = {j : j = 1, 2, ... , n} is divided into four groups: (XIII.2.24)

(XIII.2.25)

G1 =

G2

jj

:

2r
{j: -2tr
2. BASIC ESTIMATES

411

For every point E D(N), the lines Lj4 in D(N) (except possibly for their starting points) are defined by

I s = fo + t expji arg( - £o)], s = t + texp[i(r + 1)t'], (XIII.2.36)

0 < t < I - toI for j E G1 u G2,

s=C+texp[-i(r+1)t],

0 < t < oo

for j E G3,

0
for jEG4.

Note that from (XIII.2.28) and (XIII.2.29), it follows that

Ir+ 2

2


2

(j E G1),

2

J-2r+ 2
2

(jEG2)

Case 2. For the case when (XIII.2.37)

(r + 1)(t + t') > r,

the set of indices J is divided into three groups:

-2r+ry'
-2-.+y
62= j: -2r+ry'
2r+-y'
r-ry'},

-2r +'y'
2r+ where -y' is a sufficiently small positive constant such that r - 2^y' > 0. Note that (r + 1) (t + t') > r - y'. These inequalities imply that 2 r - 2 ry' > 0 and

-(r + 1)t < -(r + 1)t - 27' + 17r < (r + 1)t' + 27' - 2r < (r + 1)t'. In D(N), let

1 = Nr+1 exp i{ (r + 1)t' + ifl

12'Y'

- Zr}J

,

C2Nr+lexp i -(r+1)t-+2r} 1,Y

,

413

2. BASIC ESTIMATES

For every t E D(N), the paths of integration in t(N) are defined in the following manner: (a) For j E G1, from (XIII.2.37) and (XIII.2.38.1), it follows that -27r + 27' <

-(r+1)t-21+rand w, -(r+l)t-2ry'+r
w, hence,

-2r+2ry'
(0 < t < cc),

s = + texp(i(O({))

where 0({) is a real-valued and continuous function off such that

-(r+1)(- 2y'+r

(r+1)t"

in D(N). This implies that - 2 + 2 < wJ + 8(t;) < 2 - 2 . Precisely speaking, the function

is defined in the following way. Note that (XIII.2.37) implies

-(r+1)t- 12 7 +

2

lry`. <(r+ 1)e-2lr<(r+1)(- ir+ 2 2

Let

6 =Nr+1expli{(r+1)f -2r}J. Then, o is on arc (A-3). Let t;' be a point on arc (A-3) such that argt2 < argt' < arg {o. Then, the tangent T4, of the circle j = Nr+1 at the point t:' is given t < +oo), where 0(l;') is continuous in {' and by s = £' + (r + 1)e'. Moreover, the part of T. in D(N) is given -(r + 1)t - 2 + r < by

(XIII.2.40)

s

+ t exp(iQ(t:'))

(0 < t < +oo).

Note that If a point f in D(N) is on such a tangent (XIII.2.40), set 9(e) = the tangent of the circle I{1 = Nr+' at to is parallel to the line (A-1). If a point is in a domain between these two lines, then set O(ff) = (r + 1)Q'. For all points other than those given above, set 9(e) = -(r + 1)t - try' + r. (b) For j E G2, (XIII.2.37) and (XIII.2.38.2) imply that

-2r+y'
-(r+1)e
r

414

XIII. SINGULARITIES OF THE SECOND KIND

in )(N). This implies that - 2 + 2 < wj + O(l:) < 2 - 2 The function O(C) is defined in a way similar to the previous case. (c) For j E C3, from (XIII.2.38.3), it follows that

-17r+ 2

I 2

< wj - (r+ 1)e- 'y'+1r < 2

lry'-7r<

2

Let

17r 2

2

- 32

7- 1>'

[i{(r+1)t'+.i'_ 32rr1l

o = N''+1 exp

.

In the case when (r + 1)(t + e') >_ 2a - ry', the inequality (XIII.2.41)

arglo ? argf2,

holds, whereas in the case when (r + 1)(t + e') < 2a - ', it holds that arg o < arg t;2

(XIII.2.42)

In the case when (XIII.2.41) holds, the lines Lj( are defined by (XIII.2.39) with

-(r+1)e- 2y'+a

(r+1)e'+ 2ry'-7r

in D(N). This implies that - 2 + 2 < wj + B({) < 2 - 2 . Since to is on arc (A-3), O(l;) is defined as given above. In the case when (XIII.2.42) holds, the lines Ljf are defined by (XIII.2.39) with

(r+1)e'+ try'-7r <9({) in D(N). This implies that - 2 +

32

-(r+l)e- Zry'+7r Y.

< wj + 9({) < 2 - 2

In this case, o is

not on arc (A-3), but 8(e) is still defined in a way similar to the previous cases. For all of the cases considered above, we prove the following lemma.

Lemma XIII-2-6. There exists a positive constant H independent of h such that the inequalities

(XIII.2.43)

I

Isi-11 I exp[

-

r+ 1

a11 Ids)


1r

l

]

(j = 1,2,... n) hold in a domain D(Nh) for any positive number h, where Nh is a sufficiently large positive constant depending only on h.

2. BASIC ESTIMATES

415

Proof First let j E G1 U G2. Then, exp [

r+1ji -

r+l{to+texP(iarg(t-to))}JI

[-

r + 1 to] I I exp I - r +tl exp(i arg(t - Wd

- I exp [ exp

r "J 1 to]

L

for

exp I

I

exp

I +I1 cos(wj

+ g(t - to))]

E D(N). By virtue of (XIII.2.28), (XIII.2.29), and (XIII.2.35), it holds that

27r+2Y'
(9EG1),

-rr2 + try' < W, + arg( - W) < -Zrr - 27 Therefore, - cos(w, + arg(C - co)) > sin

(j E G2)-

1

(-') for E D(Nh) and j E G1 U G2 .

Moreover, (XIII.2.34) implies

arg( - o) -

2(r + 1)(e+ e') +

2ry1

and (XIII.2.28) and (XIII.2.29) imply (r + 1)(t + t') < n - ry'. Hence,

I arg( - {o)

(XIII.2.44)

-

ir

- 27

Observe that idol = Nn+' implies 1812 = M2 + t2 + 2cMt, where M = Nh+1 and (cf. Figure 4).

c= -cos(rr-0) with 0=

Let b = sin ('). Then,

0
(XIII.2.45)

(cf. (XIII.2.44)). Set Y(r) = (M2 + 7-2 + 2aMr)h/2 exp(M,r)

or

(M2 + t2 + 26Mt)h/2

where M, = iµj I cos(Wj +arg( - co)) . Then,

r

(XIII.2.46)

dr = { Mj+ h M2 + T2

+ 2f Mr

IY + 1

dt,

XIII. SINGULARITIES OF THE SECOND KIND

416

and Af, < - rµ+l

'r + am

< 0. From (XIII.2.45), it follows that

i

2hµJIb21),

Mb (r > 0). If M satisfies an inequality M >

1. Since Y = 0 for r = 0, we obtain Y(r) < exp 1 112r+ x

JG,c

2(r + 1) Ipj lb

M2 + r + 2c"Mr then Y+ T- < r + 1) 2(1

(r > 0). This implies that

i 1)

ISI-h exp I

exp{i arg(e - o)}J I Idsl < 2(lu Ib l - r +tl

Therefore, (XIII.2.43) follows. In this case, H =

2(rub 1),

where y = min{p,).

In other cases (i.e., G3, G4, G1, G2, and G3), the lines L, are given by (XIII.2.39),

where 9(£) satisfies - 2 it + 27' < wj + 9(e) <

in D(Nh). Therefore,

cos(w, + 9(e)) > sin try' I =b

(XIII.2.47) for

27r -

E D(N),).

(i) Consider the case when 10(x) -

Isle =112 + t2 -

cos(sr - IB(S)

Therefore, I31-h

J,<

exp

Idyl <

r+1 p1 S

L

a given point { E D(Nh). Then,

f

exp [ -

r±1 ( +

Iexp

+tl

0

_

=141-h exp

1

- argil) ? IC12. I dt

(wj +

dt

exp l - Nj, 1I(r1}

< 1

`

r+1J

lµ)Ib

This implies (XIII.2.43).

(ii) Consider the case when 27r < 19(x) -

a given point t E TD(Nh).

Since the lines Lg are in D(Nh), the distance D from the origin to L,t is not less Then, IS12 can be written in the form 1x12 = D2+o2 (cf. Figure than M = Nh+i.

5).

FIGURE 4.

FIGURE 5.

3. PROOF OF THEOREM XIII-1-2

Consider a function Y(r) = (D2 + r2)h/2exp(Mjr) 100

417

xp + M, ) da, where

M2 = 1'U2I `os(+ + OW). Note that b > 0. Hence, for r > 0, (XIII.2.47) implies

that Y(r) < M . On the other hand, Y(r) satisfies the differential equation dY = {

M+h

if M >

D2 + r2 } h(r + 1) .

1i2 1b

< (r + 1)

1

M2

Y -1. Since

r D2+72

<

1

2D

,

it follows that

dY

dr

>

lit, Ib

2(r + 1)

Here, a use was made of the inequality M -< D. Since Y(0) <

we obtain Y(r) <

2(r i 1) (r < 0). Therefore, (XIII.2.43) follows. 1µ, b

1A2 ib

This completes the proof of Lemma XIII-2-6. In this case, again, H = where µ =min{µ2 }. 2

Y-1

2(r + 1) µb

,

0

We fix the paths of integration on the right-hand sides of (XIII.2.15) as explained above. The constant H in (XIII.2.13) is chosen to be equal to the constant H given in Lemma XIII-2-6. Note that H is independent of h.

XIII-3. Proof of Theorem XIII-1-2 Let us construct a solution u2 = 02 (x) of (XIII.2.15) so that (XIII.3.1)

as x

O2 (x)

0

(j = 1,2,... n)

oo in (XIII.1.15).

Observation XIII-3-1. As in the previous section, denote by Do(Nh) the domain in the x-plane which corresponds to the domain D(Nh) in the -pllane. The domain Do(Nh) depends on h. Set o i = Nh+1 exp

{i(r + 1)(e +

and denote by

(h = 1,2,3.... ), xoh) the corresponding point in Do(Nh ). Also, setting h' = r+1 consider the domain Do(Nh,). As mentioned in the previous section, the constant H in (XIII.2.13) is chosen to be equal to the constant H in Lemma XIII-2-6. Choose a positive constant 8 so that r

H 1

{ cb +

Nh2

<_ S.

Furthermore, by

111

virtue of (XIII.2.13), Nh, can be chosen so large that Nh, > No and b < d (cf. (XIII.2.12)). Fix Nh, in this way. Then, by a method similar to that in §XII-3 for

each j (j = 1, 2, ... , n), successive approximations can be defined to construct a solution u2 = O,(x) of (XIII.2.15) which is holomorphic in Do(Nh-) and 101(x)l < (j = 1,2,... ,n), where = xT+'. In this way, the existence of a bounded solution u2 = 02 (x) (j = 1, 2, ... , n) of system (XIII.2.15) is proved.

XIII. SINGULARITIES OF THE SECOND KIND

418

0 (j = 1, 2,... , n) as Observation XIII-3-2. In order to prove that Oj (x) x -+ oo in Do(Nh'), use the fact that, for each positive integer k, the functions 4j (x) (j = 1, 2, ... , n) also satisfy the integral equations Oj(x(k))exp

.(xr+1 _ (xok))r+1)I

1r+1 -JAI

+ f tTRj(t, i (t)) exp [r (XIII.3.2)

(tr+l

- xr+1) I dt

Ljc

uj(x) =

J

(j E G1 U G2),

I

[r+i(tr+1

trR1(t,, (t))exp

_xr+1)ldt JJ

UVG1UG2)

in the domain Do(Nk). Here, u(x) denotes the Cn-valued function whose entries are

(u1(x),u2(x),... ,un(x)). Note that we can assume Do(Nk) C Do(Nh') without loss of generality.

Upon applying successive approximations similar to those of §XII-3 together with Lemma XIII-2-6 to (XIII.3.2), it can be proved that (XIII.3.2) has a solution

u,(x) = 0., (x) (j = 1,2,... ,n) such that IiP, (x) I < CkIxI-k U = 1,2,... ,n) in Do(Nk), where Ck is a suitable positive number. Also, using Lemma XIII-2-6 and (XIII.2.13), it can be shown that ¢j (x) = ikj (x) (j = 1,2,... , n) in VO(Nk), since

Oj (x) - j (x) = j tr [RR (t, fi(t)) - Rj(t,,(t))I exp Ir +

1(tr+1 _

xr+1)l dt, J

where j = 1, 2,... , n, and ¢(x) and tG(x) denote en-valued functions whose entries are (01(x), 02(x), ... , On(x)) and (ti1(x), ,11 (x)), respectively. Thus, we conclude that 0, (x) 0 (j = 1, 2,... , n) as x -' oo in Do(Nh,). Now, let us return to Observation XIII-2-2. If we set pj(x) = Oj(x) + zj(x), the solution v , = pj(x) (j = 1, 2, ... , n) of (XIII.1.1) satisfies all of the requirements of

Theorem XIII-1-2. 0 Remark XIII-3-3. Let Ax, y', µ, e) be a Cn-valued function of a variable (x,

e)

E C x Cn x Cn x C such that (a) the entries of f are bounded and holomorphic in a domain Do = { (x, b, µ, e) : IxI > Ro, Iu-1 < do, lµl < p o, IeI < Eo, I arg EI < po}, where R0, 60, µo, co, and po are some positive constants, (b) f admits a uniform asymptotic expansion in Do s/ r/x, a/,

e)

ti

+0 Ehfh(x, . p)

as

e - 0,

h=0

where coefficients fh (x, y, i) are single-valued, bounded, and holomorphic in the domain

Ao = {(x, 9, #) : IxI > Ro,

I vi < 6o,

Iµl < po},

3. PROOF OF THEOREM XIII-1-2

419

(c) if f (x, y, µ', e) = fo (x, #,c) + A(x, µ, e)y + O(I y12), then

fo(oo,µ',e) = 0

and

detA(oo,0,0) 0 0.

Let )q, A2, ... , )1n be n eigenvalues of A(oo, 0', 0) and set wf = arg A, (j _ 1, 2, ... , n). Note that the w, are not unique. Set also S(pl, p2) _ {x : Pi < argx < p2}. Let a and r be two non-negative integers. Assume that the sector S(pl, p2) is contained in the sector 32

- min{wj} + apo < (r+ 1)argx <

32

- max{wj} - apo

f o r a suitable choice of {wj : j = 1, 2, ... , n}. Under these assumptions, we can prove the following theorem in a way similar to the proof of Theorem XIII-1-2.

Theorem XIII-3-4. The system e° fy = x"f (x,

c) has two solutions

and i (x, µ, e) such that (i) these two solutions are bounded and holomorphic in the domain

¢(x, 91, e)

So = {(x, 9, e)

if

:

Ix1 > N, x E S(p1, p2), 1141 < ul, 0 < IEI < e1,1 arg el < p0},

p1 and E1 are sufficiently small,

(ii) the solution ¢(x, µ, e) admits an uniform asymptotic expansion +oo

F, x-k&(µ, e)

as

x -. oo

k=1

in So, where coefficients &(µ,e) are bounded, holomorphic, and admit uniform asymptotic expansions in powers of a as c -+ 0 in the domain {(µ, e) Iµ1 < {31, 0 < lei < el, I argel < po}, (iii) the solution 1'(x, N, e) admits an uniform asymptotic expansion: +oo

(x, , e) '

ehWh(x, µ)

as

f -4 0

h=0

in So, where coefficients >Gh(x,#) are bounded, holomorphic, and admit uni-

form asymptotic expansions in powers of x'1 as x - oo in the domain {(x,µ') : Ixf > N, x E S(p1,p2), Jill < p1}. FLrther more, there exist functions tc(x, µ, e) for t = 0,1, ... such that (a) these functions ¢'c satisfy conditions (i) and (is) given above, (b)

h=0

A complete proof of Theorem XIII-3-4 is found in [Si7J and (Si101.

420

XIII. SINGULARITIES OF THE SECOND KIND

Remark XIII-3-5. In the proof of Theorem XHI-1-2, we used the assumption that the matrix Ao on the right-hand side of (XIII.1.10) is invertible (cf. Assumption III of §XIII-1). Without such an assumption, we can prove the following theorem. Theorem XIII-3-6. Let F(x, yo, yr, ... , yn) be a nonzero polynomial in yo, yl, 00

,

an,x-°`

yn whose coefficients are convergent power series in x'1, and let p(x) = M--0

E C[(x-1)) be a formal solution/ of the differential equation (XIII.3.3)

F(x,y,Ly,..., fn)=0. \arg

Then, for any given direction x = 0, there exist two positive numbers 6 and a and a function 0(x) such that (i) ¢(x) is holomorphic in the sector S = {x E C:0< [xj <6, (arg x - 0[ < a}, (ii) 0(x) admits the formal solution p(x) as its asymptotic expansion as x o0 in S, (iii) 0(x) satisfies differential equation (XIII.3.3) in S. The main idea of the proof is similar to the proof of Theorem XIII-1-2. However,

we need the thorough knowledge of the structure of solutions of a linear system

xd = A(x)y that we shall explain in §XIII-6. Also, it is more difficult to define the paths of integration (cf. [I] and (RS11). A complete proof of Theorem XIII-3-6 is found in [RS1). See, also, §XIII-8.

XIII-4. A block-diagonalization theorem Consider a system of linear differential equations (XIIi.4.1)

dg dx

= x' A(x}y,

where r is a positive integer, y" E C', and A(x) is an n x n matrix. The entries of A(x) are holomorphic with respect to a complex variable x in a sector (XIII.4.2)

jxl > No,

f argx[ < ao,

where No and ao are positive numbers. Assume that the matrix A(x) admits an asymptotic expansion in the sense of Poincare 00

(X1II.4.3)

A(x) E x'"A &'=o

as x - oo in sector (XIII.4.2), where coefficients A are constant n x n matrices. Suppose also that Ao = A(oo) has a distinct eigenvalues A1, A2, ... , J1t with mul-

tiplicities n1, n2, ... , nt, respectively (nl + n2 +

+ nt = n). Without loss of

generality, assume that Ao is in a block-diagonal form: ()UII.4.4)

AO = diag [Al, A2, ... , At) ,

4. A BLOCK-DIAGONALIZATION THEOREM

421

where Aj are of x n, matrices in the form A,

(XIII.4.5)

(j=1,2,...,1).

=A,In, +Nj

Here, A , is a lower-triangular nilpotent n, x n, matrix. The main result of this section is the following theorem (cf. (Si7J).

Theorem XIII-4-1. Under the assumption (XIIL4.3) and (XIII.4.4), there exists an n x n matrix P(x) whose entries are holomorphic in a sector (XIII.4.6)

I argxj < al,

IxI > N1,

where Ni 1 and a1 are sufficiently small positive numbers, such that (i) P(x) admits an asymptotic expansion 00

(XIII.4.7)

P(x)

Ex-"p-

(Pp = In),

"=0

as x -' oo in sector (X111.4.6), where coefficients P" are constant n x n matrices,

(ii) the transformation y = P(x)zi

(XIII.4.8) reduces system (X111.4.1) to

di

(XIII.4.9)

,

= x B(x)i,

where B(x) is in a block-diagonal form (XIII.4.10)

B(x) = diag [BI (x), B2(x),... , Bt(x)J

.

Hen, B,(x) an n, x n. matrices and admit asymptotic expanswns 00

(XIII.4.11)

B, (x) ^-- E xBj" "=o

as x

oo in (XIII.4.6), where coefficients B,, are constant n) x n) matrices.

Proof.

From (XIII.4.1), (XIII.4.8), and (XIII.4.9), we derive the equation (XIII.4.12)

dP

= xr(A(x)P - PB(x)]

that determines the matrices P(x) and B(x). Set (XIII.4.13)

A(x) = Ao + E(x),

B(x) = A0 + F(x),

P(x) = In + Q(x).

XIII. SINGULARITIES OF THE SECOND KIND

422

Then, E(x) = O(x-1), F(x) = O(x'1), and Q(x) = O(x-1). Furthermore, (XIII.4.12) becomes dQ = x''[AoQ - QAo + E - F + EQ - QF]. dx

(XIII.4.14)

Write each of three matrices E(x), F(x), and Q(x) in a block-matrix form according to that of Ao in (XIII.4.4), i.e., (XIII.4.15)

F(x) = diag]Fl,F2,... ,Ft], Ell E11 E12 ... ... E2t E21 E22

Q11

E(x) _

Q12

...

Qlt

Q(x) = Ell

E2

Eu

.

Qtt where EJk and QJk are nJ X nk matrices and F, are nJ x nJ matrices. Set

QJJ = 0

(XIII.4.16)

Q11

Q2t

(j = 1,2,... ,t).

From (XIII.4.4), (XIII.4.14), (XIII.4.15), and (XIII.4.16), it follows that

(j=1,2,...,t)

F,=EJJ+FEJhQh,

(XIII.4.17)

h#J and

(XIII.4.18)

dQjk

= x' [43Qik - QJkAk + EJk +

EJhQhk - QJkAk]

(j j4 k).

h*k

Substituting (XIII.4.17) into (XIrII.4.18), a system of nonlinear differential equations dQJk dx

(XIII.4.19)

= x' LAJQJk - QJkAk +

EJhQhk h*k

{

- QJk (Ekk +

EkhQhk) + EJkJ

(j # k)

h*k

is obtained. Since it is assumed that .11i ... , Al are distinct eigenvalues of Ao and that A0 is in the block-diagonal form (XIII.4.5), upon applying Theorem XIII-1-2 to (XIII.4.19) we can construct a desired holomorphic solution Qjk(x) of (XIII.4.19) 00

which admit an asymptotic expansion Qjk(x) > x "Qjk" (j, k = 1, 2,... , 3; j "=1

k), where Qjk" are constant nj by nk matrices. Defining Fj by (XIII.4.17) and then B(x) by (XIII.4.13), the proof of Theorem XIII-4-1 is completed. 0

Theorem XIII-4-1 concerns the behavior of solutions of system (XIII.4.1) near x = oo. Since it is useful to give a similar result concerning behavior of solutions near x = 0, we consider, hereafter in this section, a system of differential equations (XIII.4.20)

xa+1 dY = A(x)y,

423

4. A BLOCK-DIAGONALIZATION THEOREM

where d is a positive integer and the entries of n x n matrix A(x) are holomorphic in a neighborhood of x = 0. Also, assume that A(0) is in a block-diagonal form

()III.4.21)

+N2,...

A(0) = diag[A11,,,

+JVt],

where a1, ... , .1t are distinct eigenvalues of A(0) with multiplicities n1, n2, ... , nt, + nt = n), and, for each j, .W is a lower-triangular and respectively (n1 + n2 +

nilpotent nj x nj matrix. Comparing the present situation with that of Theorem XIII-4-1, we notice the following two differences:

(a) singularity is at x = 0 in the present situation, while singularity is at x = 00 in Theorem XIII-4-1,

(b) The power series expansion of A(x) is convergent in the present situation, while A(x) in Theorem XIII-4-1 admits only an asymptotic expansion in a sector containing the direction arg x = 0. We can change any singularity at x = 0 to a singularity at x = oo by changing 1. Also, any direction argx = 8 can be changed the independent variable x by x to the direction arg x = 0 by rotating the independent variable x. Furthermore, the asymptotic expansion P of Po(x) and the expansion b of Be are formal power series satisfying the equation xd+i dP = AP - PB. This implies that two matrices P and b are independent of P. Hence, using Corollary XIII-1-3, the following result is obtained.

Theorem XIII-4-2. Let A(x) be an n x n matrix whose entries are holomorphic in a neighborhood of x = 0. Also, let d be a positive integer. Assume that the matrix A(0) is in block-diagonal form (XIII.4.21), where al, ... , at are distinct eigenvalues of A(0) with multiplicities n1, n2, ... , nt, respectively (n1 + n2 +, + nt = n), and for each j, JVj is a lower-triangular and nilpotent n, x nj matrix. Fix a real number 0 so that (a, - Xk)e-utO V EP for j 96 k. Then, there exists two positive numbers be

and ee and an n x n matrix Pe(x) such that (a) the entries of Po(x) are holomorphic and admit asymptotic expansions in powers of x as x - 0 in the sector Se = Ix: 0 < IxI < be, I arg x - 01 < + to }, 00

(b) if >2 xmP,,, is the asymptotic expansion of Pe(x), then this expansion is m=0

independent of 8 and Po = I,,,

(c) the transformation (XIII.4.22)

PB(x)u" changes system (XIIL4.20) to a system xd+1

= Be(x)u,

where the matrix Be(x) is in a block-diagonal form

Be(x) = diag (B1e(x), B2e(x), ... , Bte(x)] For each,, Bje is an n, xn2 matrix which admits also an asymptotic expansion (XIII.4.23)

in x as x--i0 in So. The main claims of this theorem are

XIII. SINGULARITIES OF THE SECOND KIND

424

(i) the asymptotic expansion of Pe(x) is independent of 0, (ii) the opening of the sector So is larger than k.

Proof of Theorem XIII-4-2 is left to the reader as an exercises.

Remark XIII-4-3. Using Theorem XIII-3-4, we can generalize Theorem XIII4-1 to the system e°dx = xr'A(x, µ, e)y", where r and o are non-negative integers, y" E C^, and A(x, µ', e) is an n x n matrix with the entries that are bounded and holomorphic with respect to the variable (x, µ, e) E C x C'° x C in a domain Do = {(x, µ, e) : 1x1 > N, Iµ1 < µo, 0 < 1e1 < eo, 0 < 1 arg e1 < po}. Also, assume 00

that A admits a uniform asymptotic expansion A(x, µ, e) - > ehAh(x, µ) in Do h=0

as a -+ 0, where coefficients Ah (X, µ) are bounded and holomorphic in the domain {(x,µ) : 1x1 > N, 1µ1 < µo}. A complete proof of this result is found in [Si7] and [Hs2].

XIII-5. Cyclic vectors (A lemma of P. Deligne) In the study of singularities, a single n-th-order differential equation is, in many

cases, easier to treat than a system of differential equations. In this section, we explain equivalence between a system of linear differential equations and a single n-th-order linear differential equation. Let us denote by 1C the field of fractions of the ring C[(x]] of formal power series in x, i.e., K=

I

p E C[Ix]J, 9 E C[Ix]j, 9 34 0} 9:

Also, denote by V the set of all row vectors (cl(x), c2(x), ... , c,,(x)), where the entries are in the field K. The set V is an n-dimensional vector space over the field 1C.

Define a linear differential operator C : V - V by G[vl = by + 7l(x) (v' E V ), where b = x and S2(x) is an n x n matrix whose entries are in the field 1C. We

d

first prove the following lemma.

Lemma XIII-5-1 (P. Deligne [Del]). There exists an element iio E V such that {v"o, Gv"o, G2v"o,

... , G"-lvo} is a basis for V as a vector space over IC.

Proof.

For each nonzero element v of V, denote by µ(v'' the largest integer t such that {ii, Cii, C2v, ... ,.CeV) is linearly independent over K. In two steps, we shall derive

a contradiction from the assumption that max{µ(v') : v' E V} < n - 1.

5. CYCLIC VECTORS

425

Step 1. First, we introduce a criterion for linear dependence of a set of elements of V. Consider a set {i11, ... vm } C V, where m is a positive integer not greater than

n = dim, V. Let 6j = (c31,c12,... ,Cin) (j = 1,2,... ,m). Set .7 = {(jl,... ,jm)

1 < it < j2 < . < jm < n}, and introduce a linear order .7 -' { 1, 2,... , (M-)) in the set J. Let us now define a map

(.) Vm={(v1,V2,...,Vm): 61 EV (j=l,...,m)} -+ Id.) (VI, ... , Vm) --+ V1 A v2 A ... A vm, where

1,AV2A ... A vm

f

cI31

det C-i 1

Cj32

...

clim

:

:

Cmj2

C-3-

(jl,... ,jm) E .7

:

A v,,,:

It is easy to verify the following properties of V1 A v"2 A

1m_1 A Um

Vk_1 A

=

(1)

Vk_1 A

+

Vm_1 A v"m

v'k_1 A

Vm_1 A Vm,

V1 A ... A Vk_1 A (a Vk) A vk+I A ... A Vm_1 A Vm (2)

=

6k_1 A irk

A vm),

for all aEIC,

VIA.-.A irk A ...A...AVjA ... A Vm = -(11A...A iY A

(3)

A Vm),

(4) a sety{ {6j, 62,... , Vm } E V is linearly dependent if and only if v'1 AV2 A 6 in K('^).

A 17m =

Step 2. Fix an element Vo of V such that µ(6o) = max{p(V) : v" E V} < n-1. Since p(VO) < n-1, another element w of V can be chosen so that {VO, Cv"o, CZi3o, ... , CnOv',

w"} is linearly independent, where no = max{µ(v") : v E V}. Set v' = vo+Ax"'O E V,

where A E C and m is an integer. Then, Cwt = Cw"o + C'(Ax-ti) = Crvo + Axm(C + m)tw. Since {V, Cv", ... , Cn0+1V} is linearly dependent, it follows that v A DU A A CnOv' A C"O+I V = 6 for all A E C and all integers in. Note that v A,06 A A C"OiYA C"°+16 is a polynomial in A. Since this polynomial is identically zero, each coefficients must be zero. For example, the constant term of this polynomial is i6o A CVO A ... A 00+I V"o. This is zero since {iio, Ciio..... G"0+Iuo} is linearly dependent. Compute the coefficient of the linear term in A of the polynomial. Then, w A C60 A ... A Cna+1v'o + v'Y A ... A 00 Vo A (C +

m)n0,+1ti

no

+ 1:60A...ACi-IVoA (C+m)lw A CJ+'A A...ACnb+Ii = 0 J=1

XIII. SINGULARITIES OF THE SECOND KIND

426

identically for all integers m. The left-hand side of this identity is a polynomial in in of degree no + I. Hence, each coefficient of this polynomial must be zero. In particular, computing the coefficient of mn0+1, we obtain vOAGvOA

. A GnbtloAw' =

0. This is a contradiction, since {vo, Gvo, ... , L"Ovo, w} is linearly independent. This completes the proof of Lemma XIII-5-1. Definition XIII-5-2. An element vo E V is called a cyclic vector of G if {6o, Lu0, L2v'o, ... , Ln-1%) is a basis for V as a vector space over X.

Observation XI II-5-3. Let uo be a cyclic vector of Land let P(x) be the n x n maUo

trix whose row vectors are {vo,G'o,... ,L' 'io}, i.e., P(x) =

Gvo

. Then,

Gn-lvo

nv"o o

L21 yo

=

and, hence, setting A(x) = L[P(x)]P(x)-1 = bP(x)P(x)-l +

P(x)f2(x)P(x)-1, we obtain 0 0

1

0

0

1

0 0

..

...

0 0

A 0

ap

0 0 0 ... 1 al a2 a3 ... an-1

with the entries a, E C. Thus, we proved the following theorem, which is the main result of this section.

Theorem XIII-5-4. The system of differential equations y1

(XIII.5.1)

by' = fl(x)y", where g _ Y.

becomes

bu" = A(x)u,

(XIII.5.2)

if y is changed by u = P(x)y". System (XIll.5.2) is equivalent to the n-th-order differential equation n-1

(XHI.5.3)

bnq -

arbrq = 0, where q = yl. 1=o

5. CYCLIC VECTORS

427

Example XIII-5-5. (1) Let us consider the system y1

by" = 0,

(a)

y=

where

yn

The transformation

u = diag[l, X'... , xn-11y

(T)

changes system (a) to

bu = diag[0,1,... ,n - 1]u.

(E)

Further, the transformation 1

2 22

--

(r)

U

2n-1

changes (E) to the form 0

1

0

0

0

1

0 0

0

0 w,

0

0

ao

a1

0 a2

0 a3

1

...

an-1

where ao, a,,.. . , an-1 are integers. Hence, the transformation

w" =

1

1

1

...

1

0

1

2

..

n- 1

0

1

22

0

1

2n-1

...

(n - 1)2

... (n -

,x°-1

y

1)n-1

changes (a) to (E'). This implies that vo = (1, x, this case. (II) Next, consider the system (b)

diag [1, x,...

x2'. ..

xn-1) is a cyclic vector in

by = Ay,

where A is a constant diagonal n x n matrix. Choose a transformation similar to (T) of (a) to change system (b) to (E")

oiZ = A'u

XIII. SINGULARITIES OF THE SECOND KIND

428

so that A' is a diagonal matrix with n distinct diagonal entries. Then, a transformation similar to (T') can be found so that (E") is changed to (E') with suitable constants ao, al, ... , an_ I .

XIII-6. The Hukuhara-Turrittin theorem In this section, we explain a theorem due to M. Hukuhara and H. L. TLrrittin that clearly shows the structure of solutions of a system of linear differential equations of the form (XIII.6.1)

where the entries of the n x n matrix A are in K (cf. §XIII-5). In order to state this theorem, we must introduce a field extension f- of K. To define L, we first set

+a E a.nxm1" : a.., E C and M E Z M=M

I

,

+a where 7L is the set of all integers. For any element a = F a,nxn'/° of K,,, we M=M +00

define x

ji by x da = F M=M

\ v) amx'nl '. Then, K, is a differential field. The field

+oo

L is given by L = U K which is also a differential field containing K as a subfield. L=1

Furthermore, L is algebraically closed. The Hukuhara-T rrittin theorem is given as follows.

Theorem XIII-6-1 ({Huk4j and [Tu1j). (XIII.6.2)

There exists a transformation

y' = UI

such that

(i) the entries of the matrix U are in L and det U ;j-1 0, (ii) transformation (XIII.6.2) changes system (X111.6.1) to (XIII.6.3)

xd

= Bz',

where B is an n x n matrix in the Jordan canonical form (XIII.6.4)

B=diag[Bl,B2,...,Bpj, Bj=diag[BJi,Bj2i...,B,,n,], Bjk =AI In,,. + Jn,,,.

429

6. THE HUKUHARA-TURRITTIN THEOREM

Here, I,,,, is the njk x njk identity matrix, J.,,, is an n 1k x n3k nilpotent matrix of the form

(XIII.6.5)

0 0

1

0

0

1

0

0

0

0

0

0

Jn",

and the Aj are polynomials in xfor some positive integer s, i.e., d,

(XIII.6.6)

Aj = > A

x-'18

where

all E C

(j = 1, 2, ... , p)

r=O

and (XII1.6.7) A,d,

0

if dJ > 0

A., - A, are not integers if i 0 j.

and

Proof.

Without loss of generality, assume that the matrix A of system (XIII.6.1) has the form 0 0

0 0

1

0

0

1

0 0

...

0

0

...

1

a1

0 a2

0

00

a3

"'

an-1

A

where ak E 1C (k = 0, 1, ... , n - 1) (cf. Theorem XIII-5-4). Set E= A Then,

an- 1

n In.

trace [E] = 0.

(XIIi.6.8)

Consider the system

X E = E.

(XIII.6.9)

Case 1. If there exists an n x n matrix S with the entries in X such that det S 0 0 dii = A(x)u", where the and the transformation w = Sii changes (XIII.6.9) to x entries of A are in C[[x]], then there exists another n x n matrix S with the entries in X such that det S 96 0 and the transformation w = Suu (XIII.6.10) dig

= Aoii, where the entries of the matrix A0 are in C. changes (XIII.6.9) to x F irthermore, any two distinct eigenvalues of A0 do not differ by an integer (cf. Theorem V-5-4). Hence, in this case, system (XIII.6.1) is changed by transformation (XIII.6.10) to dil

xaj = [an 1 In+Ao]u.

This proved Theorem XIII-6-1 in this case.

XIII. SINGULARITIES OF THE SECOND KIND

430

Case 2. Assume that there is no n x n matrix S with entries in X such that det.S 0 and the transformation w = Su' changes (XM.6.9) to xdu = A(x)u, where the entries of A are in C([x]]. Since 1

0

0

0

1

0

0

- ln an_1 E = 0

0

0

ap

al

a2

- ln an_1

1

an-2

a3

an-1 --

-an-1,

a cyclic vector can be found for system (XIII.6.9) by using the matrix W defined by

win

w11

with writ

Wnn

[wll ... Win] = (10 ... 01, [w,,1

... wJn] = V1-1[1 0

01,

cn]) = x[c1 - - cn]E. The matrix W is lower[c1 triangular and the diagonal entries are {1, ... ,1}, i.e., where V([c1

1

(XIII.6.11)

0 1

W=

0 0

0 0

1

If (XIII.6.9) is changed by the transformation v" = Wtv", then (XIU.6.12)

X!LV

_

x

+ WEJ W-Y.

It follows from (XIII.6.8) and (XIII.6.11) that ( XIII . 6 . 13 )

t racel x

+ WE]W- l =

0.

Also,

x 11

0 0

1

0

0

1

0

0

0

0 0

0 0

+ WE] W-1 = V[ W]W-1

W i31 /32

0

l

22

An-1

A

6. THE HUKUHARA-TURRITTIN THEOREM

431

where 1k E )C (k = 0, 1,... , n - 1). In particular, from (XIII.6.13), it follows that On-1 = 0. Under our assumption, not all Qt are in C[[x]]. Set 9 = (t: at V C[[x]]} +00

and set Also, /3t = x-u" E

Qtmxm

(t E 3), where, for each t, the quantity µt is a

m=0 +oo

positive integer, 1: /3tmxm E C[[xJ] and #to 54 0. Set m=0

k=max(n µt t

:IE91. J

Then, ut < k(n - t) for every t E J and µt = k(n - t) for some t E J. This implies that

Of =

(I)

L. m>-k(n-t)

Qt,mxm

(t=0,1,... n- 1)

and

Qt = x-k(n-t) (Ct + xqt)

(11)

for some t such that k(n - t) is a positive integer, ct is a nonzero number in C, qt E C[[x]], and Qt,m E C. We may assume without any loss of generality that k = h for some positive integers h and q. 9

Let us change system (XIII.6.12) by the transformation

v" = diag [1, x-k, ... , x-(n-1)kI u". Then, (XIII.6.14) where

(XIII.6.15)

0

1

0

0

0

0

1

0

F=

+ kxkdiag [0,1, ... , n 0

0

0

0

70

71

72

73

1]

and

7t = xk(n-t)o,

=:

m>-k(n-t)

In particular, 7n-1 = 0.

$,,mxm+k(n-t)

E C[ [x'1911

(0 < t < n - 1).

XIII. SINGULARITIES OF THE SECOND KIND

432

+oo

Setting F = E xm/9Fm, where the entries of F,,, are in C, we obtain m=o 0

1

0

0

C1

C2

C3

01

0

Fo= CO

...

0

Cn_2

where the constants co, cl ... , cn_2 are not all zero. This implies that the matrix F0 must have at least two distinct eigenvalues. Hence, there exists an n x n matrix T such that

(1) T =

XM19Tm, where the entries of the matrices Tm are in C and To is m=o

invertible,

(2) the transformation

y = Ti

(XIII.6.16)

xd

changes system (XIII.6.14) to a system

a block-diagonal form G = [

0

0

,

= x-kGxi with a matrix G in

where Gl and G2 are respectively

G2 J

n1 x nl and n2 x n2 matrices with entries in C[[x1/91and that nl + n2 = n and n., > 0 (j = 1, 2) (cf. §XIII-5). Therefore, the proof of Theorem XIII-6-1 can be completed recursively on n. 0 Observation XIII-6-2. In order to find a fundamental matrix solution of (XIII.6.1), let us construct a fundamental matrix solution of (XIII.6.3) in the following way:

Step 1. For each (j, k), set 4'1k = xA,0 eXp[AJ (x)] exp[(log x)Jf, ], where

-t/s

if

dj = 0,

if

d1>0.

Step 2. For each j, set

Dj = diag ['j1, where J. = diag [J,,,,, J,,72 , ... J,mJ ] .

xAJ0 exp[A,(x)] exp[(logx)Jj],

6. THE HUKUHARA-TURRITTIN THEOREM

433

Step 3. Set A

= ding [Al(x)In A2(x)I,,,, ... ,

C = diag [,\101., + J1, 1\20I., + J2, ...

,

Then, (XIII.6.17)

4) = diag [451, 4i2, ... , 4ip] = xC exp[A]

is a fundamental matrix solution of (XIII.6.3), where xC = diag [xA1OxJ, xl\2OxJ2'

... 'X aPOxJP]

,

xJ, = exp[(logx)JjJ.

The matrix (XIII.6.18)

U4i = Uxc exp[A]

is a formal fundamental matrix solution of system (XIII.6.1), where U is the matrix of transformation (XIII.6.2) of Theorem XIII-6-1. The two matrices exp[A] and xC commute.

Observation XIII-6-3. Theorem XIII-6-1 is given totally in terms of formal power series. However, even if the matrix A(x) of system (XIII.6.1) is given analytically, the entries of U of transformation (XIII.6.2) are, in general, formal power series in xl1", since the entries of the matrix T(x) of transformation (XIII.6.16) are formal power series in general. Transformation (XIII.6.16) changes system (XIII.6.14) to a block-diagonal form. Therefore, in a situation to which Theorem XIII-4-1 applies, transformation (XIII.6.2) can be justified analytically. The following theorem gives such a result.

Theorem XIII-6-4. Assume that the entries of an n x it matrix A(x) are holomorphic in a sector So = {x E C 0 < lxJ < ro, I argxi < ao} and admit asymptotic expansions in powers of x as x 0 in So, where ro and ao are positive numbers. Assume also that d is a positive integer and y" E C'. Let S be a subsector of So whose opening is sufficiently small. Then, Theorem XIII-6-1 applies to the system (XIII.6.19)

xd+1dy = A(x)f dx

with transformation (XIII.6.2) such that the entries of the matrix U of (XIII.6.2) are holomorphic in S and each of them is in a form x,00(x) where p is a rational number and O(x) admits an asymptotic expansion in powers of xl"° as x 0 in S, where s is a positive integer.

Observation XIII-6-5. In the case when the entries of the matrix A(x) on the right-hand side of (XIII.6.19) are in C[[x]l and A(0) has n distinct eigenvalues, the matrix A(x) also has n distinct eigenvalues A1(x), A2(x), ... , which are in C((xJJ. Furthermore, the corresponding eigenvectors p"1(x), p"2(x), ... ,15n(x) can be constructed in such a way that their entries are in Chill and that p""1(0),7"2(0), ... ,

XIII. SINGULARITIES OF THE SECOND KIND

434

p",,(0) are n eigenvectors of A(O). Denote by P(x) the n x n matrix whose column

vectors are p1(x), p2(x), ... , p,+(x). Then, detP(O) 36 0 and P(x)-lA(x)P(x) = diag[A1(x),A2(x),... ,An(x)]. This implies that the transformation y = P(x)ii changes system (XIII.6.19) to (XIII.6.20)

xd+1 dx

{diagiAi(x)A2(x).... , An(x)]

- xd+1P(x)-1 d ( )

It is easy to construct another n x n matrix Q(x) so that (a) the entries of Q(x) are in C[(x]], (b) Q(0) = 4,, and (c) the transformation i = Q(x)v changes system (XIII.6.20) to (XIII.6.21)

xd+1 dv

dx

= diag [ft1(x), µ2(x), ... , µn(x)] v,

where µ1(x), µ2(x), ... , i (x) are polynomials in x of degree at most d such that A, (x) = it) (x) + O(xd+1) ( j = 1, 2, ... , n). Therefore, in this case, the entries of the matrix U of transformation (XIII.6.2) are in 1C.

Observation XIII-6-6. Assume that the entries of A(x) of (XIII.6.19) are in C([x]]. Assume also that A(O) is invertible. Then, upon applying Theorem XIII-6-1

to system (XIII.6.19), we obtain following theorem.

d1 s

= d for all j. Using this fact, we can prove the

Theorem XIII-6-7. Let Q;i1(x) and Qi2(x) be two solutions of a system (XIII.6.22)

xd+1 dy = A(x)yf + x f (x),

dx

where d is a positive integer, the entries of the n x n matrix A(x) and the C"-valued function 1 *(x) are holomorphic in a neighborhood of x = 0, and A(O) is invertible. Assume that for each j = 1, 2, the solution ¢,(x) admits an asymptotic expansion

in powers of x as x - 0 in a sector S3 = {x E C : lxl < ro, aj < arg x < b3 }, where ro is a positive number, while aJ and bi are neat numbers. Suppose also that S1 n S2 0. Then, there exist positive numbers K and A and a closed subsector S = {x : lxl < R,a < argx < b} of Sl nS2 such that K exp[-Alxl -d] in S. Proof.

Since the matrix A(O) is invertible, the asymptotic expansions of1(x) and ¢2(x) are identical. Set 1 (x) = ¢1(x) -$2(x). Then, the C"-valued function >G(x) satisfies system (XIII.6.19) in S, n$2 and ii(x) ^_- 6 as x 0 in S1nS2. By virtue of Theorem XIII-6-4, a constant vector 66 E C" can be found so that r%i(x) = U4D(x)6, where lb(x) is given by (XIII.6.17). Now, using Observation XIII-6-6, we can complete the proof of Theorem XIII-6-7.

435

6. THE HUKUHARA-TURRITTIN THEOREM

Observation XIII-6-8. The matrix A = diag [AI In,, A2In3, ... , API,y] on the right-hand side of (XIII.6.18) is unique in the following sense. Assume that another formal fundamental matrix solution Uxcexp[A] of system (XIII.6.1) is constructed with three matrices U, C, and A similar to U, C, and A. Since the matrices Uxc exp[A] and UxO exp[A] are two formal fundamental matrices of sys-

tem (XHI.6.1), there exists a constant n x n matrix r E GL(n, C) such that Uxc exp[A] = U5C exp[A]I' (cf. Remark IV-2-7(1)). Hence, exp[A]T exp[-A] = x-CU-IUxC. Using the fact that r is invertible, it can be easily shown that A = A if the diagonal entries of A are arranged suitably. For more information concerning the uniqueness of the Jordan form (XIII.6.3) and transformation (XIII.6.2), see, for example, [BJL], [Ju], and [Leve].

Observation XIII-6-9. The quantities A.,(x) are polynomials in x1l'. Set w = 2a[!] and x 1/a = wx 1/a Then, i = x. Therefore, if z 1/e in Ur Cexp[A] is exp replaced by zI/', then another formal fundamental matrix of (XIII.6.1) is obtained. This implies that the two sets {Aj (i) : j = 1, 2,... , p} and (A.,(x) : j = 1, 2,... , p) are identical by virtue of Observation XIII-6-8.

Observation XIII-6-10. A power series p(x) in x1/' can be written in a form a-1

Ax) = Exh1'gh(x), where ql(x) E C[[x]] (j = 0, 1, ... , s - 1). Using this fact and h=0

Observation XIII-6-9, we can derive the following result from Theorem XIII-6-1.

Theorem XIII-6-11. There exist an integer q and an n x n matnx T(x) whose entries are in C[[x]] such that (a) det T (x) 96 0 as a formal power series in x, dil

(b) the transformation y" = T(x)t changes system (XII1.6.1) to x = E(x)iZ with an it x it matrix E(x) such that entries of x9E(x) are polynomials in x. The main issue here is to construct, starting from Theorem XIII-6-1, a formal transformation whose matrix does not involve any fractional powers of x in such a way that the given system is reduced to another system with a matrix as simple as possible. A proof of Theorem XIII-6-11 is found in [BJL]. Changing the independent variable x by x-1, we can apply Theorem XIII-6-1 to singularities at x = oo. The following example illustrates such a case.

Example XIII-6-12. A system of the form P(x) where P(x) = xm+

y = lyd,

0]b,

ahxin is a positive odd integer, and the ah are complex h=1

numbers, has a formal fundamental matrix solution of the form x

F(x)

1

0

0 f1 x-1/2 ] l I

1

-111

e

0

0 eE(t,a) I

XIII. SINGULARITIES OF THE SECOND KIND

436

where 1/2

m

+

E

k=1

+00

+ E bk(a) xk

ak xk

k=1

2

E(x, a) = (m 2) x(m+2)/2 + +

i

1
(a)x(m+2-2h)/2,

(.m + 2 - 2h)

bh

+00

x-h Fr, with an integer q and 2x2 constant matrices Fh such that

and F(x) = xq> h--O

+00

det h=O x-h Fh ,-6 0 as a formal power series in x-l. For details of construction, see [HsSj and [Sil3j.

XIII-7. An n-th-order linear differential equation at a singular point of the second kind Let us look at the formal fundamental matrix solution (XIII.6.18) of system (XIII.6.1). First, notice that if we set k = max {

as :j = 1,2,... pthen k is the JJJ

order of singularity of system (XIII.6.1) at x = 0 (cf. Definition V-7-8). Let (XIII.7.1) 0
.

set th =

n, (h = 1, 2,... , q). It is easy to see that lh > 0 (h = 1, 2,... , q) d /s=kh

q

P

and Elh = En, = n. h=1

7=1

Observation XIII-7-1. System (XIII.6.1) has th linearly independent formal solutions of the form (XIII.7.2)

'ih,v(x) = x'1h - exp[Qh,v(x)Wh,,,(x)

(v = 1, ... ,;h = 1, ...

, q),

where -yh,,, E C, Qh,,,(x) is either equal to 0 or a polynomial in x-1/' of the form I2h,vx-kh(I +O(x1/')) (11h,v E C and I2h,, # 0), Qh.v(x) = and the entries of lh,,,(x) are polynomials in logx with coefficients in Define q + 1 points (Xh, Yh) (h = 0,1, ... , q) recursively by (Xo,Y0)

(010),

(h = 1, 2,... , q). (Xh, Yh)== (Xh-1 + eh, Yh-t + kheh) Let us denote by N the polygon whose vertices are q + 1 points (Xh, Yh) (h =

0,1, ... , q). The polygon N has q distinct slopes kh given in (XIII.7.1).

7. AN N-TH-ORDER LINEAR DIFFERENTIAL EQUATION

437

Definition XIII-7-2. The polygon N is called the Newton polygon of system (XIII.6.1) at x = 0. Observation XIII-7-3. In §XIII-5, it was shown that system (XIII.6.1) is equivalent to an n-th-order linear differential equation n-1

(XIII.7.3)

anb" r) + E atbtrl = 0, t=o

where b = x operator

, at E C((xj), and an # 0 (cf. Theorem XIII 5 4). For the differential n-1

(XIII.7.4)

C(>,J = anb"r) + j:atdtrt. t=o

the Newton polygon N(C) is defined in the following way.

If a power series a = E c,,x'n E C((xj] is not 0, we set v(a) = min{m : cn m=0

0}.

If a = 0, set v(0) = +oo. For operator (XIII.7.4), consider n + 1 points

(Q, v(at)) (t = 0, 1, ... , n) on an (X, Y)-plane. Set

Pt = ((X, Y) : 0 < X < e, Y > v(at)},

and

P = UP,. t=o

Definition XIII-7-4. The boundary curve C of the smallest convex set containing P is called the Newton polygon of the operator C at x = 0. Denote by N(C) the Newton polygon of C at x = 0.

Definition XIII-7-5. Two Newton polygons are said to be identical if the two polygons become the same by moving one or the other upward in the direction of the Y-axis. Now, we prove the following theorem.

Theorem XIII-7-6. If system (XIII.6.1) and differential equation (X111.7.3) are equivalent in the sense of Theorem X111-5-4. then the two Newton polygons N and N(C) are identical. Proof.

The proof of this theorem will be completed if the following three statements are verified:

(a) If./V(C) has only one nonvertical side with slope k, then differential equation

= A(x)g with a matrix A(x) (XIII.7.3) is equivalent to a system xk+1 whose entries are power series in x11", and A(0) is invertible if k > 0, where s is a positive integer such that sk is an integer.

XIII. SINGULARITIES OF THE SECOND KIND

438

(b) If (XIII.7.1) gives slopes of all nonvertical sides of N(L), then the operator L is factored in the following way: (XIII.7.5)

where, for each h, N(Lh) has only one nonvertical side with slope kh. (c) If L is factored as in (b) and each differential equation Lh[r)h] = 0 is equiv-

alent to a system xdih = Ah(x)uh, then the differential equation (XIII.7.3) is

equivalent to x !L = diag [A, (x), A2(x), ... , Aq(x)] y'. Statement (a) can be proved by an idea similar to the argument which is used to reduce system (XIII.6.12) to (XIII.6.14) in Case 2 of the proof of Theorem XIII-6-1. A proof of Statement (b) is found in [Mall], [Si16] and [Si17, Appendix 1]. Look

at the system Lq]u] = V,

L1... L9_1[v] = 0.

Then, Statement (c) can be verified recursively on q without any complication. The proof of Theorem XIII-7-6 in detail is left to the reader as an exercise. Combining Observation XIII-7-1 and Theorem XIII-7-6, we obtain the following theorem.

Theorem XIII-7-7. If the distinct slopes of the nonvertical sides of N(L) are given by (XIII.7.1), then the n-th-order differential equation (XIIL7.S) has, at x = 0, n linearly independent formal solutions of the form (XIII.7.6)

rlh,v(x) = xtin.,. exp[Qh.v(x)]Oh,v(x)

where

(i) if

{(X, Yh-I + kh(X - Xh_1)) : Xh-1 C X < Xh}

is the nonvertical side of N(L) of slope kh, then

It, = Xh - Xh-1

(h = 1, 2,... , 9),

(ii) ryh,v E C, Qh,v(x) is either equal to 0 or a polynomial in x- 1/11 of the form

Qh,v(x) = lAh,vx-k"(1 + O(xi'°))

(I
and the quantities Oh,v(x) are polynomials in log x with coefficients in C[[x'/']] Here, s is a positive integer such that skh (h = 1,... , q) are integers, and PI + P2 + + Pq = n.

A complete proof of Theorem XIII-7-7 is found in [St]. We can construct formal solutions (XIII.7.6), using an effective method with the Newton polygon N(L). The following example illustrates such a method.

439

7. AN N-TH-ORDER LINEAR DIFFERENTIAL EQUATION

Example XIII-7-8. Consider the differential operator C = x63 - x62 - 5 - 1 or the third-order differential equation C[q] = 0. The Newton polygon N(C) is given

by Figure 6. In this case, q = 2, kl = 0, and k2 = 1.

3

1

FIGURE 6.

+00

0). Then,

(i) For k1 = 0, set i _ E c,,,xX+m (co M=0 +00

C[q] _

{ (A + m)3 - (,\ + m)2} c,,,xa+m+1 M=0 t o0

(A + m + 1)C,,,xa+m = 0. m=o

Hence, ((A + 1)co = 0, 1

(A+m+1)Cm = {(A+m-1)3-(A+m-1)2}C,,,_1

for m > 1.

Therefore,

A = -1

and

2)2(m -3) c,,, _ (m c,,,_1 m

for m > 1.

-

Thus,

A = -1,

cl = -2co,

c, ,, = 0

for m>2.

This implies that q = CO (x-1 - 2) is a solution of C[r7] = 0.

(ii) For k2 = 2, set n =

exp[Ax-112](.

b"['1] = exp[Ax-112]

\a -

Then, x_1 2) [S]

(n =0,1,2.... ).

Therefore, C[n] = 0 is equivalent to 3

ft Cb -

Zx-1/2) - x I tS -

-\X-112

)2_

(b - 2x-1/2) - 1

0.

XIII. SINGULARITIES OF THE SECOND KIND

440

Since 2

- \x-112d

2x-1/21

J 3

2x-1/2

/

=d3

-

+

Ax-1/2

(x_1/2

tax-1/252 +

-

18,\x-1/2

+ 42x-1, 4,\2x-I

/

+

3,\2X-1 + 8A3x-3/21 +

d

,

it follows that

/

2_

XId-

2x-1/2/3

= x63 +1

2

- x I d - 2x-1/2

(xhI2 + x 2

62

+

(d

(.x2 - 1 + 4

- g31 x-1/2 - 1 + g-) +

-

x-1/2) - 1

4Ax1/2)

d

x1/2.

The Newton polygon of this operator has only one nonvertical side of slope 2 for arbitrary A (cf. Figure 7-1). However, if A is determined by the equation A

\3

2

8

0,

then the Newton polygon has a horizontal side (cf. Figure 7-2).

FIGURE 7-1.

FIGURE 7-2.

Hence, we can find two formal solutions of equation £[77] = 0: exp [2x-1 12] xv, 11 + x1 /2f, J,

exp [ - 2x-1/2]xl2[1 + x1/2f2J,

where p1 and p2 are constants and f1 and f2 are formal power series in

C[[x1/2JJ.

8. GEVREY PROPERTY OF ASYMPTOTIC SOLUTIONS

441

XIII-8. Gevrey property of asymptotic solutions at an irregular singular point In this section we prove a result which is more precise than Theorem V-1-5 ([Mai]). In §XIII-3, we stated an existence theorem of asymptotic solutions for a given formal solution of an algebraic differential equation (cf. Theorem XIII-3-6). If differential equation (XIII.3.3) has a formal solution, we can transform (XIII.3.3) to the form

f_[y] = x"'G(x,y,by,... b"-1y)

(XIII.8.1)

(b =

x;),

where n

G = F_ ah(x)bh,

(XIII.8.2)

h=o

and G(x, yo, yl, ... , y.-I) is a convergent power series in (x, yo, yl ... y,,-,) (cf. (SS31 and [Mal2j). Here, it can be assumed without any loss of generality that (i) ah (h = 0,1, ... , n) are convergent power series in x and a # 0, 00

(ii) differential equation (XIII.8.1) has a formal solution p(x) = E a,,,x"' E m=0

CI[x]I,

(iii) M is an integer such that for any differential operator At of order not greater than n, the two Newton polygons N(C - xMIC) and N(C) are identical (cf. Definitions XIII-7-4 and XILI-7-5). Using Theorem XIII-3-6, we can find (a) a good covering {S1, $2, ... , SN } at x = 0, (b) N solutions 01(x), 02(x), ... , ON (x) of (XIII.8.1) in S1, S2,... , SN, respec-

tively such that of are holomorphic and admit the formal solution p(x) as their asymptotic expansions as x -+ 0 in St, respectively. Set ui = 01 - 0t+1 on Se n Sjt+1. Then, ue are flat in the sense of Poincare in sectors Se n S(+1, respectively, where Sv+1 = S1. Furthermore, if we define differential operators IQ by

E

-(x,... h

(C - x"'K t)[ut] = 0

on

Kt

O
bh (tot + (1 - t)-Ol+l I.... )dt bh,

then

se n St+l

(f = 1, 2, .... N).

If the Newton polygon N(,C) has only one nonvertical side of slope 0, then x = 0 is a regular singular point of C[iI] = 0. Therefore, in this case the formal solution p(x) is convergent (cf. Theorem V-2-7). Let us assume that./V(C) has at least one side of positive slope. In such a case, let (XIII.8.3)

0 < kl < k2 <

.

.

. < k9 < +00

442

XIII. SINGULARITIES OF THE SECOND KIND

be all of the positive slopes of the Newton polygon N(,C). Then, since N(1 C -xMJCe)

and N(G) are identical, we must have lut(x)l : -yeXp(-A

Ixl-k]

on

Sr n Se+1

for a non-negative number -y and a positive number A, where k E {k1, k2 ... , ky} (cf. Theorem XIII-7-7). Now, by virtue of Theorem XI-2-3, we obtain the following theorem.

Theorem XIII-8-1. Under the assumptions given above, the formal solution p(x) is a formal power series of Gevrey order 1 and, for each t, the solution 0e admits

p(x) as its asymptotic expansion of Gevrey order k as x -. 0 in S1, where k E {k1ik2... ,kq}. This theorem was originally prove in (Raml] for a linear system. For nonlinear cases, see, for example, ISi17, §A.2.4, pp. 207-211J.

Remark XIII-8-2. In Exercise XI-14, we gave the definition of a k-summable power series. As stated in Exercise XI-14, if a formal power series f E C((x]] is k-summable in a direction argx = 0, there exists one and only one function F E Al /k (Po, 0

-

2k

-e,6+ 2k + e) such that J(F] = f , where Po and a are

positive unmbers. This function F is called the sum of f in the direction arg x = 0. If we use the idea of Corollary XIII-1-3 and Theorem XIII-6-7, we can prove the following theorem concerning the k-summability of a formal solution of a nonlinear system (XIII.8.4)

xk+1 dy

dx

= A(x)y" + xb(x, yj.

Theorem XIII-8-3. Under the assumptions (i) k is a positive integer, (ii) A(x) is an n x n matrix whose entries are holomorphic in a neighborhood of x = 0 and b(x, y-) is a C"-valued function whose entries are holomorphic in a neighborhood of (x, yl = (0, (iii) A(0) is invertible, 00

system (XIII.8.4) has one and only one formal solution y = p(x)

xmpm and m=1

p(x) is k-summable in any direction arg x = 0 except a finite number of values of 6. Furthermore, the sum of p(x) in the direction arg x = 0 is a solution of (XIII.8.4).

To prove this theorem, it suffices to choose the good covering {S1, S2, ... , SN }

at x = 0 in the proof of Theorem XIII-8-1 so that opening of each of sectors {S1, S2, ... , SN } is larger than k . We can prove a more general theorem.

443

EXERCISES XIII

Theorem XIII.8.4. Let a linear differential operator L = 6 - A(x) be given, where 6 =

,

and A(x) is an n x n matrix whose entries are meromorphic in a

neighborhood of x = 0. Also, let (X111.8.3) be all the positive slopes of the Newton polygon N(C) of the operator C. Assume that

(1) k1 ?

51

(2) C[ f] is meromorphic at x = 0 for a f (x) E C[[xiln. Then, there exist a finite number of directions arg x = 0 (1= 1,2,... , p) such that i f 0 34 Ot f o r I = 1 , 2, ... , p, there exist q formal power series j, (v = 1, 2, ... , q) satisfying the following conditions: in the direction argx = 0, (a) for each v, the power series f is (b)f=fi+fz+...+fq.

A complete proof of this theorem is found in [BBRS]. In this case, f is said to be {k1, k2 ... , kq }-multisummable in the direction arg x = 0. We can also prove multisummability of formal solutions of a nonlinear system. For those informations, see, for example, [Ram3], [Br), [RS21, and [Bal3j.

EXERCISES XIII

XIII-1. Using Observation XIII-6-5, diagonalize the following system:

dg dx

_

x+ 5 3

x+ 8

`yil

where

-x+ 1 y

l yz J

for large )x[.

XIII-2. Show that there is no rational function f (x) in x such that (a"f)(r) + rzf(x) = x

Hint. One method is to show that the given equation has a unique power series solution in X-1 which is divergent at x = oo. Another method is to observe that any solution of this equation has no singularity in ]x] < +oo except possibly at x = 0. Furthermore, if p(x) is a rational solution, then some inspection shows that p(x) does not have any pole at x = 0. This implies that p(x) must be a polynomial. But, we can easily see that this equation does not have any polynomial solution (cf. {Si20J).

XIII-3. Find a cyclic vector for the differential operator £[ 3 = x;ii + 'A, with a

0

constant n x n matrix A of the form A = I At

1,

where for each j = 1, 2, the

XIII. SINGULARITIES OF THE SECOND KIND

444

quantity Aj is an of x n, matrix of the form 0 0

... ...

0

0

a2,j

C13,j

... ...

0

1

0

0

0

1

0

0 a 1,)

0 0

A?= CYO,

1

ant -1j

with complex constants ah,j.

XIII-4. Find the Newton polygon and a complete set of linearly independent formal solutions for each of the following three differential equations: (a) x(52y + 45y - y = 0,

where b = x

(b) x2(52y + xby - y = 0,

(c) xb2y + x6y - y = 0,

d .

XIII-5. Find the Newton polygon of the following system:

dx

0

1

1

0

x

x

10

0

x2

x3 d =

Hint. For the given system, a cyclic vector is (1, 0, 0). This implies that the transformation 0 x-2

1

X-2

(T)

x-4 + x-3

X-4-2x-2

0

0

-

2x-2

x-

changes the given system to (E)

where

(5 - 1)(x5 - 1)(x2(5- 1)u1 = 0,

b = x

.

Note that transformation (T) is equivalent to u1 = yl, u2 = dyl, u3 = 52y1, and the given system can be written in the form 1/2 = (x25 -

(5 - 1)1/3 = 0.

1/3 = (xb - 1)1/2,

The Newton polygon of (E) has three sides with slopes 0, 1, and 2, respectively.

On the other hand, The standard form of the given system in the sense of Theorem XIII-6-1 is 3 du

a(x)

0

0

b(x)

0

0

0

x2

X - = dx

0 u,

where

a(x) = 1 + 0(x),

b(x) = x + 0(x2).

Hence, this also shows that the Newton polygon of the given system has three sides with slopes 0, 1 and 2, respectively.

EXERCISES XIII

445

XIII-6. Let A(x) be an n x n matrix whose entries are holomorphic and bounded in a domain Ao = {x : 3x! < ro} and let j (x) be a Cn-valued function whose entries are holomorphic and bounded in the domain Ao. Also, let Al, A2, ... , An be eigenvalues of A(O). Assume that det A(0) # 0. Assume further that two real numbers 01 and 82 satisfy the following conditions: (1) 01 < 02,

(2) none of the quantities Aye-'ke (j = 1, 2, ... , n) are real and positive for a positive integer k if 01 < 8 < 02, Aqe-'k93 (3) Ape-ike' and are real and positive for some p and q.

Show that there exist one and only one solution f = fi(x) of the system x'`+1 ds = A(x)f+ x f (x) such that the entries of fi(x) are holomorphic and admit asymptotic expansions in powers of x as x - 0 in the sector S = {x : 0 < !xj < r0, 81 < 2k

argx<82+2k Hint. If we use Corollary XIII-1-3 at x = 0, it can be shown that for each 8 in the interval B1 < B < B2 + 7, a solution ¢'(x; 8) is found so that the entries of 2k

O(x; 8) are holomorphic and admit asymptotic expansions in powers of x as x - 0 'r in the sectorial domain Sa = x : 0 < jxj < ro, I arg x - 81 < 2k + ea }, where Ee is a sufficiently small positive constant depending on 8. If 10 - 8'l is sufficiently small,

then ¢(x; 0) = d(x; 8).

XIII-7. (a) Show that j (x) _

(-1)m(m!)xr+L is a formal solution of m=o

2 dy

(E)

dx+y-x=0

and that f is not convergent except at x = 0. (b) For a given direction 8, find a solution fe(x) of (E) such that fe(x) ^- j (x) as x 0 in the direction arg x = 8. (c) Calculate fe, (x) - fe, (x) for two given directions 81 and 02. Hint. For Part (b), use the following steps: Step 1. Apply Theorem XIII-1-2 to the given differential equation. To do this, we

must change x = 0 to t = oo by x = 1. Then, the differential equation becomes du

1

= y - t . In this case, n = 1, r = 0, and the eigenvalue is ,u = 1. Set arg u = 0.

Then, the domain D(N, ry) (cf. (XI 11. 1. 15)) is

D(N,7) = {t:iti > N, I argt - 2gir! < 32

-

f}

XIII. SINGULARITIES OF THE SECOND KIND

446

where q is an integer, N is a sufficiently large positive number, and -y is a sufficiently small positive number. In terms of x, the domain D(N, y) becomes

S(N, -t) = {x:O < IxI < N, I arg x - 2pirI <

2 - 'Y}

where p = -q. Since there is no singular point of the given differential equation in the domain 0 = {x: 0 < Ixi < oo}, we can conclude that, for each fixed integer p, there exists a solution ¢p(x) of the given differential equation such that (i) op(x) is analytic (but not single-valued) in !2, (ii) -ip(x) j (x) as x 0 in the domain Dp =

{x: 0 < xl < oc, argx - 2pl < _

37r

2

Step 2. It is easy to see that f (x) = el/=J 2t-1e-1l`dt (x > 0) satisfies the given 0

differential equation and has the asymptotic property f (x) = j (x) as x

0. Since

¢p(x) - f (x) is a solution of the homogeneous differential equation x2dy + y = 0, it follows s that ¢p(x) = f (x) + cell=, where c is a constant. From this, op(x) _ elI1

t-le-ll`dt follows for argx = 2pir.

Step 3. Using an argument similar to that of Step 2,

fe(x) =

rt

4p(x)

if

10 - 2p7ri <

Op(x) + cge,/,

if

2 < 0 - 2pir < 2

21 ,

is obtained, where co is an arbitrary constant.

Step 4. Note that IC

t-le-1'`dt = 27ri,

where C is a counterclockwise oriented circle with the center at x = 0. Hence, using analytic continuation of f (x), we obtain

fi(x) = ¢p(x) + 2pirie'

for

argx = 2pir.

XIII-8. Show that the following differential equation has a nontrivial convergent power series solution: 3

d 2x

+ dX+ y = Q. 2

EXERCISES XIII

447

Hint. Step 1: The given differential equation has three linearly independent formal solutions

+-

+oo

1(x) = ell=

1+

amxm

+oo

2(x) = 1+ L. bmxm, and o3(x) = x+ m=2

M=1

amxm. m=3

In fact, Ol can be found through some calculation with the Newton polygon. The other two can be found by solving the equations

al + 603 = 0, f ao + 2a2 = 0, (l a,, + (m + 1)m(m - 1)0,,,+1 + (m + 2)(m + 1)am+2 = 0

for m > 2

+00

for a formal solution E a,,,x"`. m=o

Step 2: The given differential equation has three linearly independent actual solu3ir tions such that e-'/=O1(x) ^ as x 0 in the sector j argx + 7r1 <

T,

and 02(x)

fi(x) and 03(x) ^_- 3(x) as x -. 0 in the sector I argxI < 32

Step 3: In the direction arg x = -7r, 02(2e2x,)

02(x) -

= C201(x)

03(xe2ai)

and

03(x) -

= C301 (-T)

for some constants c2 and C3. Then, c3 (x) - c2.3(x) is a convergent power series solution of the given differential equation.

Remark: (1) See [HIJ. (2) This result was originally proved for a more general case in [Per1]. (3) There is another proof based on Exercises V-18 and V-19 (cf. [HSWI).

XIII-9. Consider a system of differential equations (E)

xp+1

du dx = F(x, y, u),

where p is a positive integer, x is a complex independent variable, y is a complex parameter, it and F are n-dimensional vectors (i.e., E C'), and entries of F(x, y, u") are holomorphic with respect to (x, y, u-) in a neighborhood of (x, y, u) = (0, 0, 0). Assume that there exists a formal solution of system (E) 00

u = V, (X, y) _

yh+Gh(x),

h=0

where coefficients zlih(x) are R'-valued functions whose entries are holomorphic in

a neighborhood of x = 0. Assume also that O0(0) = 0 and det

[(oo)] o9p

Show that i (x, y) is convergent in a neighborhood of (x, y) = (0, 0).

54 0.

XIII. SINGULARITIES OF THE SECOND KIND

448

Hint. See ]Si14] and [Si21].

XIII-10. Consider a Pfaffian system

J xl

(S)

aaxu

= F(x, y, u,

y4+t ala = G(x, y, u-),

where p and q are positive integers, x and y are two complex independent variables,

iZ, F, and d are n-dimensional vectors (i.e., E C"), the entries of F and G are holomorphic with respect to (x, y, 11) in a neighborhood of (x, y, u) _ (0, 0, system (S) is completely integrable, i.e., F and d satisfy the condition yq+I

5F (x, y, v') +

041

(x, y, u)G(x, y, u = xp+l

Assume that F(0, 0, 0) = det

(0, 0, 0)J

j4 0.

(x, y, u) + 5 (x, y, U-)F(x, y, u).

d(0, 0, 0) = 0, det

0,

and

[(oo)J

36

0,

and

Show that system (S) has one and only one solution

L

11 = (x, y) such that i (0,O) = 0 and that entries of

y) are holomorphic

with respect to (x, y) in a neighborhood of (x, y) = (0, 0).

Hint. This is an application of Exercise XIII-9. Step 1. Construct a formal power series solution +00

(FS)

11 = '1G(x, y) = E yhrGh(x) h=o

of the system yq+1

ay

= G (x y ,

,

in such a way that coefficients t/ih(x) are holomorphic in a neighborhood of x = 0. Step 2. For t _ l%i(x, y), note that yl+9ay (i+ P

ax -

F(x,y,11))

= yl+qxl+p 0211

y1+qaF

= yi+qxl+p 8211 0;-ax

= xl+pyl+q 8211

1+q Y

aF ay TX

ac 011 ad ax + au ax

-

(xl+P

ax 811

OF'yl+ga11 su

-

ay

OF &a

G(x,y,ur)

x1+paG

5y-ax

= x1+p

-

ay

8i-ax

F(x,y,i)

-

x

auF(x,y,u)

+paG

ax

-

ad

su F'(x, y, u)

449

EXERCISES XIII

Using this result, it can be shown that (FS) also satisfies the system

xP+i a = Ax, y, u)

(E)

Step 3. Upon applying Exercise XIII-9 to (E), the convergence of t%i(x, y) is proved.

XIII-11. Complete the proof of Theorem XIII-7-6 by verifying rigorously statements (a), (b), and (c) in the proof.

XIII-12. Show that the series

E

(FS)

+00

(3)h

y = P(x) = x-1/4 1 + >( -1)h

54hh!r(h +)

[

is a formal solution of the differential equaton function.

d2-xZ

XIII-13. Show that the differential equation

xexp [_x3/2} 3

-xy = 0, where r is the Gamma-

d2-xZ

- xy= 0 has a unique solution

O(x) such that (1) b(x) is entire in x and (2) ¢(x)exp [3x3/21 admits the formal series p(x) exp [x3I2J as its asymptotic exansion as x -oc in the sector I arg xj <

r, where p(x) is givenby (FS).

Remark. Ai(x) =

2(--X)

v/Fr

is called the Airy function (cf. [AS, p. 446), [Wasl, pp.

124-1261, and [01, pp. 392-394]).

XIII-14. Using the same notations in Exercises XIII-12 and XIII-13, show that if 3 J then ¢(w'ix) and Q(wx) are two solutions of equation (S). Also, w =xpel(2ril (i) derive asymptotic expansions of ¢(w-lx) and m(wx), (ii) show that {Q(x),y5(w-lx)}, {0(w-1x),O(wx)}, and {d(wx),Q(x)} are three fundamental sets of solutions of (S), (iii) show that if we set m(x) = c1O(w-1x)+c2y5(wx), then c2 = -w and

[ct w

is equal to the 2 x 2 identity matrix, (iv) using (iii), show that cl = -w-1. XIII-15. Show that if O(x, A) is an eigenfunction of the eigenvalue problem

(EP)

d2 y

da l

+00

La

then

( i)

Q(x) is entire in x and Q(x) exp

ITJ

is a polynomial,

1

O

11J

3

XIII. SINGULARITIES OF THE SECOND KIND

450

(ii) all negative odd integers are eigenvalues of (EP) and there is no other eigenvalue,

(iii) for every non-negative integer n, Hn(x) = (-1)ne=2 Un (e-y2) is a polynor

2

mial, and -0n(x) = H.(x)exp I - 2 is an eigenfunction of (EP) for the eigenvalue -(2n + 1),

J L

+00

r+00

On(x)/m(x)dx = 0 if n 9k m, and / On(x)2dx = 2nn!y'. J o00 J 0o Remark. The polynomials Hn(x) are called the Hermit polynomials (cf. [AS, p. (iv)

775] and [01, p. 49]).

XIII-16. Construct Green's function of the boundary-value problem

- x2y = f (x),

j

+00 r+00

(i)

00

J

j

y(x)2dx < +00. Show also that o0

G(x,)2dxd< +oo,

00

+00 +

(ii) if f (x) is real-valued, f (x), f'(x), and f"(x) are continuous,

f (x)2dx < o0 00

+00

+00, and

f{f"(x) - x2 f (x)}2dx < +oo, then the series Y 2nn!

(f, On)

00

xOn(x) converges to f(x) uniformly on the interval -oo < x < +oo, where On (x) are defined in Exercise XIII-15, and (f, g) _

Hint. See §VI-4.

j

+00

f (x)g(x)dx. 00

n-1

XIII-17. Consider a differential operator C[y] _ dxn + 1: an-h(x)

h

h , where

h=0

+00

aj(x) =

ajmx-'n

E C[[x1]. Also, assume that aj,_m1 # 0 if a,(x) is not

m=-mj equal to zero identically, while mj = -oo if aj(x) is zero identically. In the (X, Y)-

plane, consider the points Pj = (j, mj) (j = 1,... , n). Construct a convex polygon II whose vertices are (0, 0), pi, p2, ... , p, such that each pk is one of those points Pj, and that all other points Pj are situated below the polygon. Set po = (0, 0) and Pk = (ak, Pk) (k = 1) ... , s), where an = n. Denote by pk the slope of the segment

k k+1 (k = 0, ... , s - 1). Then, Po > Pl > P2 > ... > p,-,. Assume

that pk>-1 (kvo). Show that

(i) the differential equation £[y] = 0 has n - at,,, linearly independent formal soMe

lutions of the form ye(x) = xe" E Req(x)(logx)q (e = a,,. + 1,... , n), where q=o

Rrq(x) E C[[x-1]], b E C, and the Mt are non-negative integers,

(ii) if k < vo, then the differential equation £[y] = 0 has ak+1 - ak linearly me

independent formal solutions of the form yr = e^<(x)x6, E Rrq(x)(log x)q (e = q-0

EXERCISES XIII

451

1 + ok,... , nk+1), where At(x) = Arxl+1 b + terms of lower degree, At E C, A, 96 0, bt E C, M, are non-negative integers, and Req E C[[x-lIP]], p being a positive integer. Hint. Compare the polygon in this problem with the Newton polygon at x=0 defined in §XIII-7.

XIII-18. Consider a differential operator C[yl = x9+1 dx + 1(x)y', where q is a 00

positive integer and f2(x) _ > xlf2, with Hi being n x n constant matrices. As in t=o

§V-4, the operator C can be represented by the matrix H1 f12

01

J,, +A

and J_ _ [Oq]

where H =

I...

Here, 0, is the nr x oo zero matrix and I,,,, is the oc x oo identity matrix. Let

Qo=So+No and A=S+N be the S-N decomposition of 0o and that of A in the sense of §V-3. Also, let Po, P1, P2, ... , Pm, ... be n x n constants matrices and let AO be an n x n diagonal

matrix such that det Po 0 0 and that

(1)

SoPo = PoAo,

Ao =

A 1

0

0

0

0

0

A2

0

0

0

0

0

A3

0

0

0

0

0

0

An

and (2)

SP=PAo,

where P1

P=

P2 P0

P.

Note that At are eigenvalues of So. Hence, A, are eigenvalues of 00. Show that

(a) the matrix S represents a multiplication operation: y -+ o(x)y for an n x n matrix o(x) = So + O(x) whose entries are formal power series in x,

XIII. SINGULARITIES OF THE SECOND KIND

452

(b) the matrix N represents a differential operator Lo[yM = x4+1 fy + v(x)y for an n x n matrix v(x) = No +O(x) whose entries are formal power series in x, (c) C[f, = £o[yl + o(x)y and Go[o(x)yj = a(x)Go[yl, +00

(d) if we set P(x) _ F xmP,,,, then P(x)-1o(x)P(x) = A0, m=0

(e) if we set /C[uZ = P(x)-'C[P(x)uZ and Ko[u] = P(x)-1Go[P(x)ul, then K[u"j = Ko[ul + Aou,

(f) if we set Ko[i ] = xq+1

du

Vii

+ vo(x)u,

vo(x) =

.

vn1

then

vjk(x) = 0 if .\.7 # Ak (cf. [HKS]).

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\

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(Sc] L. Schwartz, Theorie des Distributions, 3rd ed., Hermann, Paris, 1966. [Sil] Y. Sibuya, Sur un systcme d'6quations differentielles ordinaires lnnaires h coefficients periodiques et contenants des parami tres, J. Fac. Sci. Univ. Tokyo, I, 7(1954), 229-241. [Si2] Y. Sibuya, Nonlinear ordinary differential equations with periodic coefficients, Punk. Ekv., 1(1958), 77-132. [Si3] Y. Sibuya, Sur reduction analytique d'un systeme d'i quations differentielles ordinaires lnnaires contenant un parametre, J. Fac. Sci. Univ. Tokyo, 7(1958), 527-540.

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INDEX

Asymptotic solution, 374 Asymptotic stability, 236, 300 a sufficient condition for, 241 Atiyah, M. F., 109 Autonomous system, 279

,A.(r,a,b), 353 Abel's formula, 80 Abramowitz, M., 141, 449, 450 Adjoint equation, 79 Airy function, 449 Algebraic differential equation, 109, 353 Analytic differential equation, 20 Analytic simplification of a system, at a singular point, 421 with a parameter, 391 Approximate solutions, c-, 10 Approximations, successive, 3, 20, 21, 378, 418 Arzelh, C., 9

Balser, W., 371, 391, 401, 435, 443 Baire's Theorem, 303 Banach space, 25, 26, 103, 366, 394 differential equation in, 25, 394 Bartle, R. G., 303 Basis of the image space, 73 Bellman, R., 71, 144, 148, 197 Bendixson, 1., 27, 279, 291, 304 Bendixson center, 261 Bessel function, 125 Bessel inequality, 162 Birkhoff, G. D., 372, 390 Block-diagonalization theorem, 190, 204, 380, 422, 423

Arzelb-Ascoli Lemma, 9, 13, 30 Ascoli, G., 9

Associated homogeneous equation, 97, 99 Asymptotics, Gevrey, 353 Poincar6, 342

Asymptotic behavior of solutions, 197 of a linear second-order equation, 226 of linear systems, 209, 213, 219 Asymptotic (series) expansion, 343, 353 definition, 343 differentiation, 347 integration, 347 inverse of, 346 of a holomorphic function, 345 of Ei(z), 352 of Gevrey order s, 353 of Log(o(z)), 352 of uniformly convergent sequences, 348 product, 344 sum, 344 Taylor's series as, 345 uniform with respect to a parameter,

Block solution, 107

Blowup of solutions, local, 17 Borel, E., 349, 356 Borel-Ritt Theorem, 349, 356 Boundary-value problem, 144 of the second-order equation, 305 Sturm-Liouville, 148 Bounded set of functions, 9 Bourbaki, N., 69 Braaksma, B. L. J., 359, 355, 443 Branch point, 44

C, field of complex numbers, 20 C", set of all n-column vectors in C, 20 C[xJ, set of all polynomials, 109 C{x}, set of all convergent power series, 109 C[[xJJ, set of all formal power series, 109 C{x}", set of all convergent power series with coefficients in C", 110 C[[xfJ", set of all formal power series with coefficients in C", 110 C[[xJJ set of all power series of Gevrey

343, 357, 361

of Gevrey order s, 357 uniqueness, 343 Asymptotic reduction of a linear system with singularity, 405 of a singularly perturbed linear system, 381

order s, 353 462

463

INDEX C-algebra, 70

Cayley-Hamilton Theorem, 71, 220 Canonical transformation of a Hamiltonian system, 92, 94 Cauchy, A. L., 1 Cauchy-Euler differential equation, 140 Caratheodory, C., 15 Center, 255, 257, 261, 272 perturbation of, 271 Cesari, L., 197 Characteristic exponents of an equation, 89, 203 Characteristic polynomial of a matrix,

in S-N decomposition, 74, 119, 122 Diagonalization theorem, Levinson's, 213 Differentiability with respect to initial values, 35 Differential operator C, 120, 151, 169, 424, 437, 443 S-N decomposition, 122 normal form, 121 Dirac delta function, 8 Distribution, of L. Schwartz, 8 of eigenvalues, 159 Dulac, H., 235, 251 Dwork, B., 8

71, 72

of a function of a matrix f(A), 84, 85 partial fraction decomposition of the inverse of, 72, 76 Chevalley, Jordan-, decomposition, 69 Chiba, K., 197 Classification of singularity of a homogeneous linear system, 132 Coddington, E. A.,14, 28, 36, 58, 65, 69, 108, 137, 138, 144, 148, 153, 162, 191, 197, 225, 233, 246, 266 274, 281, 290, 293, 298, 313

Commutative algebra, 109 Commutative differential algebra, 109, 362

Commutative differential module, 110 Comparison theorem, 58, 144 Complexification, 252 Contact point, exterior (interior), 295 Conti, R., 15, 235, 279

Continuity of a solution, 17, 29 with respect to the initial point and initial condition, 29, 31 with respect to a parameter, 31 Convergence of formal solution, 113, 118 Coppel, W. A., 144, 148, 197 Covering, good, 354 Cullen, C. G., 71 Cyclic vector, 424, 426

Date, E., 144 Deligne, P., 403, 424 Denseness of diagonalizable matrices, 70 Dependence on data, 28, Devinatz, A., 197 Diagonalizable matrix, 70

E(s, A), 366 E(s, A)", 367 Eastham, M. S. P., 197, 217 Eigenfunction, 162 Eigenvalues of a matrix, 70, 71 Eigenvalue-problem, 153, 154, 156 of a boundary-value problem, 153, 186 Eigenvectors of a matrix, 70 Elliptic sectorial domain, 296, 302 Equicontinuity, 9 equicontinuous set, 47, 55, 63

Euler, Cauchy-, differential equation, 140 Existence, 1, 12 of solution of an initial-value problem, 1, 3, 12, 21

with Lipschitz condition, 3 without Lipschitz condition, 12 of solution of a boundary-value problem, 148 of S-N decomposition of a matrix, 74 Existence theorem for a nonlinear system at a singular point, 405 Existence theorem for a singularly perturbed nonlinear system, 420 cxp[Al, exponential of a matrix A, 80 exp[tAl with S-N decomposition, 82 Extension of a solution, 16 of a nonhomogeneous equation, 17 Finite-zone, potentials, 189 First variation of an equation, 284 Flat of Gevrey order s, 342, 367, 385 Flatto, L., 341 Floquet, G., Theorem, 69, 80, 225 Formal power series, 109, 343, 349

INDEX

464

Formal power series (cont.) as an asymptotic series, 350 of Gevrey order s, 353, 355

with a parameter as an asymptotic series, 351 Formal solution, 109, 342 convergence, 115, 118 Taylor's series as, 345 uniqueness, 115 Fourier series, 165, 166, 167 Fuchs, L., 137

Fundamental matrix solution, 78, 105, 125, 122 of a system with constant coefficient, 81

with periodic coefficient, 87, 88

of an equation at the singularity of the first kind, 125 of an equation at the singularity of the second kind, 136, 432, 433 Fundamental theorem of existence and uniqueness, 3, 12 without Lipschitz condition, 8 Fundamental set of linearly independent solutions, 78 normal, 202 Gel'fand, 1. M., 181

Gel'fand-Levitan integral equation, 181 General solution, 7, 105 Gerard, R., 108, 112, 125 Gevrey asymptotics, 353 Gevrey order 8, 356, 367 asymptotic expansion of, 353 flat of, 342, 367, 385 formal power series, 354 uniformly on a domain, 357, 361 Gevrey property of asymptotic solutions at an irregular singularity, 441

solutions in a parameter, 385 Gingold, H., 104, 197, 217, 231 Global properties of solutions, 15 GL(n, C), general linear group, 69 Good covering, 354 Green's function, 150, 154 Gronwall, T. If., Lemma, 3, 398

Haber, S., 304

Hamiltonian system with periodic coefficients, 90

Harr inequality, 27 Harr's uniqueness theorem, 27 Harris, W. A., Jr., 142, 143, 197, 433, 447 Hartman, P., 14, 27, 28, 39, 58, 69, 144, 148, 153, 162, 197, 246, 281, 293, 298

Heaviside function, 8 Hermit polynomials, 450 Higher order scalar equations, 98 Hille, E., 103 Hirsch, M. W., 69, 251

Holomorphic function asymptotic to a formal series, 349, 356 Homogeneous systems, 78 Homomorphism of differential algebras, 359, 362, 365 Howes, F. W., 304, 339

Hsieh, P. F., 78, 104, 108, 130, 197, 217, 231, 372, 424, 436, 452 Hukuhara, M, 25, 26, 50, 108, 131, 197, 212, 218, 261, 403, 428, 447 Hukuhara-Nagumo condition, 218 for bounded solution of a second order equation, 212 for bounded solution of a system, 218 Hukuhara-Turrittin theorem, 428 Humphreys, J. E., 69 Hyperbolic sectorial domain, 296, 302 Hypergeometric series, 138 Improper node, 252, 254 perturbation of, 261 stable, 252, 253, 256 unstable, 252, 253 Independent solutions, 78 Index of isolated stationary point, 293 Index of a Jordan curve, 293 Indicial polynomial, 112 Infinite-dimensional matrix, 118 Infinite-dimensional vector, 120, 393 Infinite product of analytic function, 196 Initial-value problem, complex variable, 21 partial derivative of a solution as a solution of, 33 Initial-value problem, real variable, 1, 28, 32, 39

INDEX

nonhomogeneous equation, 96 nonlinear, 1, 16, 32, 39 second order equation, 324, 333 Inner product, 151 Instability, 243 Instability region, 189 Integral inequality, 377, 418 Invariant set, 280, 299 Irregular singular point, 134, 441 Iwano, M., 447 Iwasaki, 1., 138

J, map of A,(r,a,b) to C[[x]] 353 one-to-one, 359 onto, 356 Jacobson, N., 69 Jordan-Chevalley decomposition, 69 Jordan canonical form, 421, 426 Jordan curve, index of, 293 Jost solution, 168, 180, 194 Jurkat, W., 136, 391, 401, 402, 435

K, the field of fractions of C[[x[], 424 K[u'J, differential operator, 123 Kaplan, J., 197 Kato, J., 44, 66 Kato, T., 103 Kimura, II., 138 Kimura, T., 197 Kneser, H., theorem, 41, 47, 305 Kohno, M., 78, 108, 130, 452

Komatsu, H., 8

C, differential operator, 120, 151, 169, 424, 437, 443 normal form, 121 calculation of, 130 C+, limit-invariant set, 280 Laplace transform, inverse, 103 LaSalle, J., 279, 281 Lebesgue-integrable function, 15 Lee, E. B., 68, 106 Lefschetz, S., 279, 281 Legendre, equation, 141 polynomials, 141, 192 Leroy transform, incomplete, 354 Lettenmeyer, F., 142 Levelt, A. H. M., 108, 125, 435 Levin, J. J., 340

465

Levinson, N, 14, 28, 36. 65, 69, 108, 137, 138, 144, 153, 162, 191, 197, 225, 233, 246, 266, 274, 281, 290, 293, 298, 304, 340, 341

Levinson's diagonalization theorem, 21:3 Levitan, B. M., 181 Levitan, Gel'fand-, integral equation, 181 Liapounoff, A., 197 Liapounoff function, 239, 309, 311 Liapounoff's direct method, 281 Liapounoff's type number, 198 calculation of, 203 multiplicities, 202 of a function, 198

of a system at t = oc, 201, 204 of a solution, 199 properties, 198 Lie algebra, 90 Limit cycle, 292 Limit invariant set, 280 Lin, C.-H., 370 Lindelbf, E., 1, 28, 359 Lipschitz condition, 3, 43, 64

constant, 3 sufficient condition for, 5 existence without, 8 Local blowup of solutions, 17 Logarithm of a matrix, 86 log[ 1 + ltf ], 86, 88 log[8(w)[, 88 log[8(w)2], 88, 94 log[s], 86, 88 log[S2], 88 Lutz, D. A., 136, 197, 391, 401, 402, 435

M,,(C), set of all n x n matrices in C, 69 MacDonald, 1. C., 109 Magnus, W., 196 Mahler, K., 112 Maillet, E., 112, 353, 441 Malgrange, B., 438 Manifold, 45 differential equation on, 45 stable, 243 unstable, 246 Markus, L., 68, 69, 96, 106 Matrix, 69 functions of a matrix f (A), 84

466

INDEX

Matrix (cont.) diagonalizable, 70

in S-N decomposition, 74 infinite-dimensional, 118 nilpotent, 71 in S-N decomposition, 74 norm of a matrix A, 11A11, 69 semisimple, 70

symplectic, 93, 95 upper-triangular, 70 Maximal interval, 16 Maximal solution, 52, 54, 56, 66 Minimal solution, 52, 54, 66 Milnor, J. W., 298 Moser, J., 135 Mullen, F. E., 232 Multiplicity of an eigenvalue of a matrix, 72, 74 Multiplicity of Liapounoff's type number, 201 of a system of equations with constant coefficients, 202 of a system of equations with periodic coefficients, 202 Multipliers, of an autonomous system, 284 of a system with periodic coefficients, 89, 183, 203 of periodic orbits, 318, 319

N(1), Newton polygon of G, 437, 439 Nagumo, M., 27, 50, 197, 212, 218, 304, 306, 327, 330, 333, 334, 342 Nagumo condition for uniqueness of solution, 64, 65 Nevanlinna, F., 342 Nevanlinna, R., 359 Newton polygon, 437, 439 Nikiforov, A. F., 141 Nilpotent matrix, 71 in S-N decomposition, 74 Node, 253, 254 improper, 253, 255 proper, 253, 256 Nonhomogeneous equation, 17 Nonhomogeneous initial-value problem, 96

Nonlinear equation, 18, 372, 384

Nonlinear initial-value problem, 17, 21 28, 32, 41

Nonstationary point, 291, 292 Nonuniqueness of solution of an initial-value problem, 41 Norm, (in C[[x]], 366

in C[[x]l", 367

in CI[x]j 366 of a continuous function, 162 in E(s, A)", 367 of a matrix A, 1PAIl, 69

of a vector, 1, 345 of an infinite-dimensional vector, 394 Normal form of a differential operator, 121

Normal fundamental set, 202 Null space of a homomorphism, 366

Olver, F. W. J., 138, 141, 449, 450 O'Malley, R. E., 304 Orbit, 251, 279 periodic, 304, 318, 319

of van der Pol equation, 313, 318 Orbitally asymptotical stability, 283 Orbital stability, 283 Orthogonal sequence, 166 Osgood, W. F., 63 Osgood condition for uniqueness of solution, 63, 65

Palka, B. P., 196 Parabolic sectorial domain, 296 Parseval inequality, 166 Partial differential equation, 39 Peano, G., 1, 28 Periodic coefficients, system with, 87 Periodic orbit, 291, 297, 318, 319 of van der Pol equation, 313, 318 Periodic potentials, 183 Periodic solution, 184, 288

of van der Pol equation, 313, 318 Perron, 0., 57, 212, 443 Perturbation, of initial-value problem, 43 of a center, 271 of a proper node, 266 of a saddle point, 263 of a spiral point, 270 of an improper node, 263 Peyerimhoff, A., 391, 401, 402

INDEX

Pfaffian system, 448 Phase plane, 251 Phase portrait of orbits, 251, 302 Phillips, R. S., 103 Phragmen-Lindelof theorem, 359, 370 Picard, E., 1 Poincarc, H., 279, 291, 304 Poincarc asymptotics, 342 Poincarc-Bendixson Theorem, 291, 293 Poincarc's criterion, 300 Popken, J., 112 Potentials, 175, 178 finite-zone, 189, reflectionless, 175, 177, 178, 181 periodic, 183

Projection, P,(A), 72 properties, 73 Proper node, 253, 254 perturbation of, 266 stable, 253, 255, 267 unstable, 253 R, real line, I R", set of all n-column vectors in R, 1 Rabenstein, A. L., 36, 70, 101 Ramis, J.-P., 342, 360, 363, 371, 386, 420, 443

Reflection coefficient, 175 Reflectionless potentials, 177, 178, 181 construction, 178

Regular singular point, 131, 133, 134, 394 Ritt, J. F., 349, 356

S,,, set of all diagonalizable matrices, 70 S-N decomposition, 74, 75 existence, 74

of a differential operator, 121

of a function of a matrix f(A), 84 of a matrix for a periodic equation, 88 of infinite order, 118, 120 of a real matrix, 75 uniqueness, 75 Saddle point, 252, 255, 256 perturbation of, 261 Sansone, G., 15, 235, 279

Saito, T., 274 Sato, Y., 336

Scalar equations, higher order, 98 Scattering data, 172, 175

467

Schwartz, L., 8 Second-order equation, boundary-value problem, 304, 308 Sectorial region, 296 elliptic, 296, 302 hyperbolic, 296, 302 parabolic, 296 Self- adjoi nt ness, 151, 155

Semisimple matrix, 70 Shearing transformation, 209 Shimomura, S., 138 Sibuya, Y., 44, 66, 69, 78, 96, 108, 112, 130, 131, 136, 142, 143, 197, 217, 227, 232, 304, 342, 360, 363, 365, 369, 371, 372, 381, 382, 390, 391, 392, 399, 419, 420, 424, 436, 438, 447, 448, 452

Singular solution, 42 Singular perturbation of van der Pol equations, 330 Singularity of a linear homogeneous system, 132, 134 of the first kind, 113, 132, 136 of the second kind, 132, 134, 136, 403 n-th-order linear equation, 436 regular, 131, 133, 134, 394 irregular, of order k, 134, 441 Smale, S., 69, 251 Solution, asymptotic, 374 asymptotically stable, 236, 241 independent, 78 fundamental matrix, 78, 105, 125, 222 periodic, 184, 288 of van der Pol equation, 313, 318 singular, 42 trivial, 236, 243 uniqueness, 1, 3, 21 Solution curves, 42, 49 Spiral point, 254, 255

perturbation of, 270 stable, 254, 256, 270, 272 unstable, 254, 272 Sperber, S., 112 Stability, 235 Stability region, 189 Stationary point, 280, 291, 312 isolated, 294 Stable manifolds, 243 analytic structure, 246

INDEX

468

of solution of an initial-value problem, 1, 3, 21, 79 Osgood condition, 63, 65 sufficient conditions for, 61, 63 Unstable manifold, 246 Upper-triangular matrix, 70 Urabe, M., 288

Stegun, 1. A., 141, 449, 450 Sternberg, W., 403, 438 Stirling formula, 358, 364 Sturm-Liouville problem, 148 Subalgebra, 109 Successive approximations, 3, 20, 21, 378, 418

Sufficient condition, for asymptotic stability, 241 for uniqueness, 61, 63 for unstability, 243 Summability of a formal power series, 371, 442, 443 Symplectic group Sp(2n, R), 92, 93 matrix of order 2n, 92, 93 107

Uvarov, V. Q., 141

Tahara, H., 112 Tanaka, S., 144 Taylor's series, 345 Transform, incomplete Leroy, 354 Transformation, shearing, 209 Transversal, to orbit, 291, 292 with respect to a vector, 291 Trivial solution, 236, 243 Turrittin, H. L., 391, 403, 428 Turrittin, Hukuhara-, theorem, 428 Two dimensional system with constant

Vector, infinite-dimensional, 120, 393 norm of, 393 Vector space of solutions, 78

V image of P, (A), 73 direct sum for C", 73 van der Pol equation, 312 for a large parameter, 322 for a small parameter, 319 singular perturbation of, 330 Variation of parameters, formula of, 101

Wasow, W., 108, 304, 342, 372, 382, 449 Watson, G. N., 141, 342, 359

Weinberg, L., 142, 447 Weyl, H., 69 Whittaker, E. T., 141 Winkler, S., 196 Wintner, A., 197 Wronskian, 145, 149, 168, 172

coefficients, 251

Unbounded operators, 103 Uniformly asymptotic series, 343, 357,

Xie, F., 197, 217

x"C{x}", 110

361

x"C((Fx)l", 110

Uniqueness, Lipschitz condition, 1, 3, 64 Nagumo condition, 64, 65 of S-N decomposition of a matrix, 75 of asymptotic expansion, 343 of formal solution, 115 of solution of a boundary-value problem, 150

Yoshida, M., 138 Zeros, of eigenfunctions, 185 of solutions, 144 Zygmund, A., 167

468

Universitext

(continued )

Moise: Introductory Problems Course in Analysis and Topology Morris: Introduction to Game Theory Polster: A Geometrical Picture Book Porter/Woods: Extensions and Absolutes of Hausdorff Spaces Ramsay/Rlchtmyer: Introduction to Hyperbolic Geometry Reisel: Elementary Theory of Metric Spaces Rickart: Natural Function Algebras Rotman: Galois Theory Rubel/Colliander: Entire and Meromorphic Functions Sagan: Space-Filling Curses Samelson: Notes on Lie Algebras

Schif: Normal Families Shapiro: Composition Operators and Classical Function Theory Simonnet: Measures and Prohabilit

Smith: Power Senes From a Computational Point of View Smorytiski: Self-Reference and Modal Logic Stillwell: Geometry of Surfaces

Stroock: An Introduction to the Theory of Large Deviations Sunder: An Invitation to von Neumann Algebras Tondeur: Foliations on Riemannian Manifolds Wong: Wcyl Transforms Zhang: Matrix Theory Basic Results and Techniques Zong: Strange Phenomena in Convex and Discrete Geometry Zong: Sphere Packing-,

Universitext The authors' aim is to provide the reader with the very basic knowledge necessary to begin research on differential equations with professional ability. The selection of topics should provide the reader with methods and results that are applicable in a variety of different fields. The text is suitable for a one-year graduate course, as well as a reference book for research mathematicians. The book is divided into four parts. The first covers fundamental existence, uniqueness, and smoothness with respect to data, as well as nonuniqueness. The second part describes the basic results concerning linear differential equations, and the third deals with nonlinear equations. In the last part, the authors write about the basic results concerning power series solutions.

Each chapter begins with a brief discussion of its contents and history. The book has 114 illustrations and 206 exercises. Hints and comments for many problems are given.

ISBN 0-387-98699-S

ISBN 0-387-98699-5 www.springer-ny.com

III

9 78038709869990>