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, disappearanceand 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 desireto 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 usefulto—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 dynamicalsystem described by an ordinary differential equation of the form x = f(x). Supposethere are equilibria at x = x0 and x = x1 then a solution φ : R→ R2 of the systemis called a heteroclinic trajectory from x0 to x1 if

limt→−∞

φ(t) = x0 and limt→+∞

φ(t) = x1.

Heteroclinic trajectories often act as separatrices—or boundaries—between dif-ferent 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 differentproperties 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’sdivide the plain into three pieces: those points on a trajectory, φ(t) such thatlimt→∞ φ(t) = x0, limt→−∞ φ(t) = x0 and all other points. The union of all pointsin the first group are called the stable manifold of x0. The second form the unstablemanifold. [2, 4].

At unstable equilibria, points where the eigenvalues of the Jacobian have positivereal parts, the stable manifold will be zero dimensional, while the unstable will betwo dimensional. For centers, points where the real part of the eigenvalues are zerohave zero dimensional stable and unstable manifolds.

Date: December 12, 2012.

1

2 EDDIE BECK AND ELEANOR DOWNS

For saddle nodes, however, the stable and unstable manifolds both have dimensionone pointing in the direction of the Jacobian’s eigenvectors associated with thepositive 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 ofthe other. Normally, the intersection of two one-dimensional manifolds gives us anisolated single point. However, this can’t happen. If they intersect, they do sonon-trivially.

Theorem 2.1. If the Existence and Uniqueness Theorem applies and if the unstablemanifold of one equilibrium point intersects the stable manifold of another, there isa 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 trajec-tory from that point to p and likewise a trajectory from p to the other point. Bythe Uniqueness and Existence Theorem, p can lie on at most one trajectory, theintersection of the two manifolds is at least one dimensional. �

Such intersections are heteroclinic trajectories. The appearance and disappear-ance 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 re-gards to the pendulum is straight-forward, the existence and location of heteroclinictrajectories 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 un-damped pendulum, whose motion is modeled by the usual system

θ +g

Lsin(θ) = 0.

We dedimensionlize this equation with the following change of variables:

τ = ωt, ω2 =g

L.

We do this to simplify the problem; the force due to gravity and the length ofthe pendulum are constants throughout our discussion and their presence is notterribly illuminating. Furthermore, by adding ν := θ, we turn this one-dimensional,second-order system into a first-order, two-dimensional system. So θ + sin(θ) = 0,becomes

θ = ν, ν = − sin(θ).

STABLE AND UNSTABLE MANIFOLDS, HETEROCLINIC TRAJECTORIES AND THE PENDULUM 3

3.1. Thinking Locally. While this paper is motivated by global bifurcation think-ing, we found that all of our observations about the global behavior of the system isrelated to the local behavior of equilibria. We see that the equilibria of the systemlie at (nπ, 0), for any integer n.

J =

[0 1

− cos(θ) 0

], ∆(J) = cos(θ), Tr(J) = 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 equilibriaat the even multiples of correspond to the pendulum hanging straight down, whilethe equilibria at the saddle nodes represent the pendulum balancing directly upside-down. Since the system is conservative and time-reversible we may conclude thateach linear center is, in fact, a non-linear center for the system. The eigenvalues ofthe matrix at each saddle node are 1 and −1, with eigenvectors (1, 1) and (1,−1),respectively.

-4.8 -4.4 -4 -3.6 -3.2 -2.8 -2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8

-3.2

-2.8

-2.4

-2

-1.6

-1.2

-0.8

-0.4

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

Figure 1. Undamped Pendulum: The horizontal axis is angle θ betweenthe pendulum straight down. The blue lines are the heteroclinic trajecto-ries.

We need only consider the system on the interval (−π, π). When we sketch thephase 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 besimple, closed curves surrounding the equilibrium at the origin. These represent theback-and-forth swinging motion of the pendulum. The solution curves above and

4 EDDIE BECK AND ELEANOR DOWNS

below these orbits are open waves which correspond to the pendulum making com-plete rotations about the pivot forever. The heteroclinic orbit is therefore importantin that, if we can find its path, we can determine which initial conditions will causethese different behaviors of the pendulum to occur.

3.2. Thinking Globally. The fact that it is conservative makes drawing trajecto-ries, both heteroclinic and otherwise, terribly easy as each trajectory will be a levelcurve of the conserved quantity E = 1

2ν2 − cos(θ). Since (nπ, 0) are isolated points

and (nπ, 0) is a minimum of E, we know that these points cannot be the endpointsof heteroclinic orbits. (That, of course, also follows from them being centers.) Let’sconsider now (nπ, 0) for odd n. Then E = 1 and we can explicitly express therelationship between ν and θ.

1

2ν2 − cos(θ) = 1

ν2 − 2 cos(θ) = 2

ν2 = 2 + 2 cos(θ)

ν = ±√

2 + 2 cos(θ)

So, we can explicitly express these trajectories.1 Notice that the heteroclinictrajectories bound the region of the plain consisting entirely of closed trajectoriesand every trajectory not so bounded never changes direction; ν will oscillate, but ifν > 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; tobe 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 inthat direction and any less wont spin at all, but swing back and forth. This verytiny change in initial conditions about that trajectory results in radical differenttrajectories.

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 predictscenters at the even multiples. In fact, the type of equilibria we find at these pointsis now dependent on the value of b. When b2 − 4 < 0, we will have stable spirals.

1The ease of this situation is terribly unusual.

STABLE AND UNSTABLE MANIFOLDS, HETEROCLINIC TRAJECTORIES AND THE PENDULUM 5

J =

[0 1

− cos(θ) −b

], ∆(J) = cos(θ), Tr(J) = −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 thissystem approach that equilibrium corresponding to the pendulum hanging straightdown. Other trajectories will represent the pendulum initially making completecircles around the pivot, but will eventually be sucked into a spiral and come to restat the stable equilibrium.

-15 -10 -5 0 5 10 15

-15

-10

-5

5

10

15

Figure 2. The canonical damped pendulum when b = .5. Blue are theunstable 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 energyfunction given our new system.

E(θ, ν) =1

2ν2 − cos(θ)

E(θ, ν) = νν + sin(θ)θ

= ν(−bν − sin(θ)) + sin(θ)(ν)

= −bν2

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 indefinitelyas before.

6 EDDIE BECK AND ELEANOR DOWNS

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 canhave any attracting fixed points. [4]

While the stable and unstable manifolds of our saddle points still play a rolebreaking up the plane into trajectories of different behavior, though the differenceis nothing as severe as before when we had heteroclinic orbits. The stable manifoldspartition the plane by which equilibrium point the trajectory is heading. The un-stable manifolds are much less informative; they further separate the plain by whatdirection 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 manytimes it spun in a direction, there really is no difference between trajectories: allwill slow down, oscillate and stop.

4.1. Saddle-to-∞ trajectories when b < 0. While it doesn’t really make senseto have b < 0 from a physics perspective, we’re mathematicians and we believe innegative 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 thedamped pendulum from something else, we examine here what that something elseis. In this case, it’s the stable trajectories dividing the plane, separating trajectoriesthat tend to (∞,∞) from those limiting to (−∞,−∞).

-4.8 -4.4 -4 -3.6 -3.2 -2.8 -2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8

-3.2

-2.8

-2.4

-2

-1.6

-1.2

-0.8

-0.4

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

Figure 3. When b = −.5. Blue are the unstable manifolds and red arethe stable manifolds.

STABLE AND UNSTABLE MANIFOLDS, HETEROCLINIC TRAJECTORIES AND THE PENDULUM 7

4.2. The Swayless Pendulum When b > 2. If b > 2, the equilibria turn from sta-ble spirals into stable nodes and the pendulum will no longer oscillate before comingto rest. We note this local bifurcation with subtle albeit real global implications—notrajectory will ever change directions—to provide a standard for comparison in thenext 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 to2.5. Something interesting to note, as b went from −.5 to 0, we can see differenttrajectories, not crossing, but being shuffled around.

And despite being only a couple frames of the almost 200 frame video, the cameoappearance of the heteroclinic trajectories visually sharp whereas the change of thestable 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 theUniversity of Georgia.

The authors thank Edward Azoff, Malcolm Adams, Robert Varley and Ted Shifrinfor their patient treatment curing the authors of many confusions related to thismaterial. That which is correct is much to their credit. As to the errors, whichsurely 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 anIntroduction 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 Dynamicalsystems