STABLE AND UNSTABLE MANIFOLDS, HETEROCLINIC TRAJECTORIES AND THE PENDULUM EDDIE BECK AND ELEANOR DOWNS
Abstract. Heteroclinic trajectories are an object of study in many different areas of theoretical and applied mathematics providing useful tools in understanding dynamical systems. A cornerstone of global bifurcation theory, their appearance, disappearance and behavior help us understand the global behavior of dynamical systems, not unlike the study of fixed points in local bifurcation theory. This work is motivated by a desire to more deeply understand these creatures, but tragically, we can add no more to the subject than what has already been said. What follows is a flat treatment of the subject confined to two-dimensional systems produced by—and hopefully accessible and useful to—other undergraduate students of mathematics and the other sciences.
1. Global behavior, separatrices and heteroclinic trajectories Heteroclinic trajectories are solutions of systems of differential equations which “connect” two equilibria of the system. More formally, consider the dynamical system described by an ordinary differential equation of the form x˙ = f (x). Suppose there are equilibria at x = x0 and x = x1 then a solution φ : R → R2 of the system is called a heteroclinic trajectory from x0 to x1 if lim φ(t) = x0 and lim φ(t) = x1 .
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Heteroclinic trajectories often act as separatrices—or boundaries—between different types of behavior in the phase plane. A separatrix is itself a phase curve, unique in that it acts as a demarcation between other phase curves with different properties determined by their initial condition. [1, 5, 3] 2. From local to global: stable and unstable manifolds Given a saddle point, x0 , assuming the Uniqueness and Existence Theorem, let’s divide the plain into three pieces: those points on a trajectory, φ(t) such that limt→∞ φ(t) = x0 , limt→−∞ φ(t) = x0 and all other points. The union of all points in the first group are called the stable manifold of x0 . The second form the unstable manifold. [2, 4]. At unstable equilibria, points where the eigenvalues of the Jacobian have positive real parts, the stable manifold will be zero dimensional, while the unstable will be two dimensional. For centers, points where the real part of the eigenvalues are zero have zero dimensional stable and unstable manifolds. Date: December 12, 2012. 1
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For saddle nodes, however, the stable and unstable manifolds both have dimension one pointing in the direction of the Jacobian’s eigenvectors associated with the positive and negative eigenvalues, respectively. For there to be a heteroclinic trajectory from an equilibrium point to another, the stable manifold of one would have to intersect with the unstable manifold of the other. Normally, the intersection of two one-dimensional manifolds gives us an isolated single point. However, this can’t happen. If they intersect, they do so non-trivially. Theorem 2.1. If the Existence and Uniqueness Theorem applies and if the unstable manifold of one equilibrium point intersects the stable manifold of another, there is a trajectory between the two. Proof. To show this, let’s consider one point, p, where the two manifolds intersect. As p is on the unstable manifold of one of the fixed points, there exists a trajectory from that point to p and likewise a trajectory from p to the other point. By the Uniqueness and Existence Theorem, p can lie on at most one trajectory, the intersection of the two manifolds is at least one dimensional. Such intersections are heteroclinic trajectories. The appearance and disappearance of heteroclinic orbits often accompany global changes in the phase portrait. Just as changing parameters can change the number or types of equilibria we have, it may also determine whether or not a heteroclinic orbit exists, and if they do, which equilibria are connected by them. While our application of these ideas in regards to the pendulum is straight-forward, the existence and location of heteroclinic trajectories is an active area of research with very few universal results.
3. The Frictionless Pendulum As an illustration, consider the basic example of the phase portrait of the undamped pendulum, whose motion is modeled by the usual system g θ¨ + sin(θ) = 0. L We dedimensionlize this equation with the following change of variables: τ = ωt,
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We do this to simplify the problem; the force due to gravity and the length of the pendulum are constants throughout our discussion and their presence is not ˙ we turn this one-dimensional, terribly illuminating. Furthermore, by adding ν := θ, second-order system into a first-order, two-dimensional system. So θ¨ + sin(θ) = 0, becomes θ˙ = ν, ν˙ = − sin(θ).
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3.1. Thinking Locally. While this paper is motivated by global bifurcation thinking, we found that all of our observations about the global behavior of the system is related to the local behavior of equilibria. We see that the equilibria of the system lie at (nπ, 0), for any integer n. 0 1 J= , ∆(J) = cos(θ), Tr(J) = 0 − cos(θ) 0 The Jacobian matrix, J, of the system predicts a linear center at (0, 0) and each (nπ, 0) for even n and a saddle node at each (nπ, 0) where n is odd. The equilibria at the even multiples of correspond to the pendulum hanging straight down, while the equilibria at the saddle nodes represent the pendulum balancing directly upsidedown. Since the system is conservative and time-reversible we may conclude that each linear center is, in fact, a non-linear center for the system. The eigenvalues of the matrix at each saddle node are 1 and −1, with eigenvectors (1, 1) and (1, −1), respectively. 3.2
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Figure 1. Undamped Pendulum: The horizontal axis is angle θ between the pendulum straight down. The blue lines are the heteroclinic trajectories.
We need only consider the system on the interval (−π, π). When we sketch the phase portrait, we see that there are two heteroclinic orbits connecting the equilibria (−π, 0) and (π, 0). Within the cycle formed by these orbits, the trajectories will be simple, closed curves surrounding the equilibrium at the origin. These represent the back-and-forth swinging motion of the pendulum. The solution curves above and
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below these orbits are open waves which correspond to the pendulum making complete rotations about the pivot forever. The heteroclinic orbit is therefore important in that, if we can find its path, we can determine which initial conditions will cause these different behaviors of the pendulum to occur. 3.2. Thinking Globally. The fact that it is conservative makes drawing trajectories, both heteroclinic and otherwise, terribly easy as each trajectory will be a level curve of the conserved quantity E = 12 ν 2 − cos(θ). Since (nπ, 0) are isolated points and (nπ, 0) is a minimum of E, we know that these points cannot be the endpoints of heteroclinic orbits. (That, of course, also follows from them being centers.) Let’s consider now (nπ, 0) for odd n. Then E = 1 and we can explicitly express the relationship between ν and θ. 1 2 ν − cos(θ) = 1 2 ν 2 − 2 cos(θ) = 2 ν 2 = 2 + 2 cos(θ) p ν = ± 2 + 2 cos(θ) So, we can explicitly express these trajectories.1 Notice that the heteroclinic trajectories bound the region of the plain consisting entirely of closed trajectories and every trajectory not so bounded never changes direction; ν will oscillate, but if ν > 0 on a trajectory then it always will, likewise if it’s negative. It is worth noting that this demarcation line of a heteroclinic is terribly fine; to be on one requires the precision on par with throwing a bottle and having it land, balanced, upside-down. With any more speed, our pendulum will keep spinning in that direction and any less wont spin at all, but swing back and forth. This very tiny change in initial conditions about that trajectory results in radical different trajectories. 4. The Damped Pendulum We consider our system for the undamped pendulum and add a linear damping, thereby adding a parameter to our system. The equation then becomes θ¨ + bθ˙ + sin(θ) = 0, where b > 0 represents the damping strength. Again, this can be rewritten θ˙ = ν,
ν˙ = −bν − sin(θ).
Our Jacobian still predicts saddles at odd multiples of π, but no longer predicts centers at the even multiples. In fact, the type of equilibria we find at these points is now dependent on the value of b. When b2 − 4 < 0, we will have stable spirals. 1The ease of this situation is terribly unusual.
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0 1 J= , ∆(J) = cos(θ), Tr(J) = −b − cos(θ) −b With this damping of our system, any trajectory which in our original system (without damping) represented the pendulums back-and-forth motion will in this system approach that equilibrium corresponding to the pendulum hanging straight down. Other trajectories will represent the pendulum initially making complete circles around the pivot, but will eventually be sucked into a spiral and come to rest at the stable equilibrium. 15
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Figure 2. The canonical damped pendulum when b = .5. Blue are the unstable manifolds and red are the stable manifolds.
However, there’s a serious change in the global behavior of the system. Now, we have no closed trajectories. To prove this, we take the derivative of our energy function given our new system. 1 E(θ, ν) = ν 2 − cos(θ) 2 ˙ E(θ, ν) = ν ν˙ + sin(θ)θ˙ = ν(−bν − sin(θ)) + sin(θ)(ν) = −bν 2 That is that E˙ is always negative. So no trajectory can hit the same point twice, completing the proof. It follows that the pendulum can’t keep spinning indefinitely as before.
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At this point, we know it’s impossible for our system to remain conservative. Even if we were to hope to find another energy function, no conservative system can have any attracting fixed points. [4] While the stable and unstable manifolds of our saddle points still play a role breaking up the plane into trajectories of different behavior, though the difference is nothing as severe as before when we had heteroclinic orbits. The stable manifolds partition the plane by which equilibrium point the trajectory is heading. The unstable manifolds are much less informative; they further separate the plain by what direction will the trajectory be heading when it makes its last spin over the top. Unless there was a thread attached to the pendulum keeping track of how many times it spun in a direction, there really is no difference between trajectories: all will slow down, oscillate and stop. 4.1. Saddle-to-∞ trajectories when b < 0. While it doesn’t really make sense to have b < 0 from a physics perspective, we’re mathematicians and we believe in negative numbers and we add this section to show what’s on the other side of b = 0. As b is changing, as the brief appearance of the heteroclinic trajectories separates the damped pendulum from something else, we examine here what that something else is. In this case, it’s the stable trajectories dividing the plane, separating trajectories that tend to (∞, ∞) from those limiting to (−∞, −∞). 3.2
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Figure 3. When b = −.5. Blue are the unstable manifolds and red are the stable manifolds.
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4.2. The Swayless Pendulum When b > 2. If b > 2, the equilibria turn from stable spirals into stable nodes and the pendulum will no longer oscillate before coming to rest. We note this local bifurcation with subtle albeit real global implications—no trajectory will ever change directions—to provide a standard for comparison in the next section that includes discusses an animation of the phase portrait. 5. Animation and Computations To help us understand this process, we created an animation at http://www. youtube.com/watch?v=mc8n8Uo4RLE of the phase portrait as b went from −.5 to 2.5. Something interesting to note, as b went from −.5 to 0, we can see different trajectories, not crossing, but being shuffled around. And despite being only a couple frames of the almost 200 frame video, the cameo appearance of the heteroclinic trajectories visually sharp whereas the change of the stable spirals to stable nodes is remarkably understated. Acknowledgments All figures and animations were created using Grapher on Quad-Core Mac Pro. The animation was finalized on the same using iMovie and uploaded to Youtube. All hardware and software was provided by the Mathematics Department at the University of Georgia. The authors thank Edward Azoff, Malcolm Adams, Robert Varley and Ted Shifrin for their patient treatment curing the authors of many confusions related to this material. That which is correct is much to their credit. As to the errors, which surely remain, belong entirely to the authors. References [1] W.-J. Beyn. The Numerical Computation of Connecting Orbits in Dynamical Systems IMA Jounral of Numerical Analysis. 1990. [2] Morris Hirsch, Stephen Smale and Robert Devaney. Differential Equations, Dynamical Systems and an Introduction to Chaos. [3] H.L. Smith. Stable and Unstable Manifolds of Planar Dynamical Systems 2011. [4] Steven Strogatz. Nonlinear Dyamics and Chaos With Applications to Physics, Biology, Chemistry and Engineering. [5] George Tigan. On a Method fo Finding Homoclinic and Heteroclinic Orbits in Multidemsional Dynamical systems
