4.7  Summary

Surfaces of section of a typical Hamiltonian system exhibit a menagerie of features including fixed points, invariant curves, resonance islands, and chaotic zones. Integrable systems have much simpler surfaces of section. By adding small effects to integrable systems we get insight into how this complicated behavior emerges.

Surfaces of section for integrable systems display only certain characteristic orbit types. There are fixed points, which correspond to equilibria or periodic orbits. A fixed point may be stable or unstable, depending on the stability of the corresponding equilibrium or orbit. There are sets of points on the section that are asymptotic forward and backward in time to the unstable fixed point. And there are sets of trajectories that fall on invariant curves. If the rotation number of the invariant curve is irrational, each of these trajectories densely covers the invariant curve; if the rotation number is rational, then each trajectory visits only a finite number of points on the invariant curve.

Linear stability analysis addresses the nature of the motion near the fixed points on the section. These points correspond to either equilibrium points or periodic orbits. There are characteristic frequencies of the motion, each with an associated characteristic direction. For Hamiltonian systems only certain patterns of characteristic frequencies are possible. On two-dimensional area-preserving surfaces of section, as generated by Hamiltonian systems, fixed points are linearly stable (elliptic fixed points) or linearly unstable (hyperbolic fixed points).

With the addition of small effects, the surface of section changes in certain typical ways. One characteristic change occurs near the unstable fixed points. The stable and unstable manifolds, those curves consisting of the sets of points that are asymptotic to the unstable fixed points forward and backward in time, no longer join smoothly, but instead cross. A first crossing implies that there are an infinite number of other crossings, and the stable and unstable manifolds develop an extremely complicated tangle.

The Poincaré-Birkhoff construction shows how the infinite number of periodic orbits on an invariant curve with rational rotation number that is characteristic of an integrable system degenerates into a finite number of alternating stable and unstable fixed points when the system becomes nonintegrable. This phenomenon is recursive, so we find that it develops an infinite hierarchy of structure: The region around every stable fixed point is itself filled with commensurabilities with alternating stable and unstable fixed points.

Some invariant curves survive the addition of small effects to an integrable system. The Kolmogorov-Arnold-Moser theorem proves that some invariant curves persist upon perturbation. We can find invariant curves of particular rotation numbers by comparing the pattern of points generated for a candidate initial point on the invariant curve to the expected pattern of points for the invariant curve being sought. As the additional effect is made stronger, the invariant curves that survive longest are those with the most irrational rotation number. At the point of breakup, the probability of visitation of various points on the invariant curve develops a self-similar appearance. For larger perturbations, the invariant curve disappears, leaving an invariant set with an infinite number of holes.