When we try to represent the figure formed by these two curves and their intersections in a finite number, each of which corresponds to a doubly asymptotic solution, these intersections form a type of trellis, tissue, or grid with infinitely serrated mesh. Neither of these two curves must ever cut across itself again, but it must bend back upon itself in a very complex manner in order to cut across all of the meshes in the grid an infinite number of times. The complexity of this figure will be striking, and I shall not even try to draw it.
Henri Poincaré, New Methods of Celestial Mechanics, volume III , chapter XXXIII, section 397
We have seen rather complicated features appear as part of the Poincaré sections of a variety of systems. We have seen fixed points, invariant curves, resonance islands, and chaotic zones in systems as diverse as the driven pendulum, the non-axisymmetric top, the Hénon-Heiles system, and the spin-orbit coupling of a satellite. Indeed, even in the standard map, where there is no continuous process sampled by the surface of section, the phase space shows similar features.
The motion of other systems is simpler. For some systems conserved quantities can be used to reduce the solution to the evaluation of definite integrals. An example is the axisymmetric top. Two symmetries imply the existence of two conserved momenta, and time independence of the Hamiltonian implies energy conservation. With these conserved quantities, determining the motion is reduced to the evaluation of definite integrals of the periodic motion of the tilt angle as a function of time. Such systems do not exhibit chaotic behavior; on a surface of section the conserved quantities constrain the points to fall on curves. We may conjecture that if points on a surface of section do not apparently fall on curves then not enough conserved quantities exist to reduce the solution to quadratures.
We have seen a number of instances in which the behavior of a system changes qualitatively with the inclusion of additional effects. The free rigid body can be reduced to quadratures, but the addition of gravity-gradient torques in the spin-orbit system yields the familiar mixture of regular and chaotic motions. The motion of an axisymmetric top is also reducible to quadratures, but if the top is made non-axisymmetric then the divided phase space appears. The system studied by Hénon and Heiles, with the classic divided phase space, can be thought of as a solvable pair of harmonic oscillators with nonlinear coupling terms. The pendulum is solvable, but the driven pendulum has the divided phase space.
We observe that, as additional effects are turned on, qualitative changes occur in the phase space. Resonance islands appear, chaotic zones appear, some invariant curves disappear, but others persist. Why do resonance islands appear? How does chaotic behavior arise? When do invariant curves persist? Can we draw any general conclusions?