Resonances Fill Stochastic Phase‐Space a

This is a presentation of recent results of collaborative research with R. S. MacKay and I. C. Percival.’.’ This research initiates a program to determine the transport properties of stochastic orbits in nonintegrable Hamiltonian systems. While most of the motion for an integrable Hamiltonian system is quasi-periodic and confined to invariant tori, a typical Hamiltonian is nonintegrable. Furthermore, although there may exist some invariant tori, many are destroyed and replaced by invariant Cantor sets, called “cantori”.’ Often, much of the remaining phase-space is filled with orbits that appear chaotic, or stochastic, and these orbits can leak through the cantori. In our first paper (reference I ) , we introduced the notion of flux for the case of area-preserving maps and showed how the flux through cantori could be calculated. Transport in chaotic regions is impeded by partial barriers that are formed on the framework of the cantori. Flux across these partial barriers takes place through “turnstiles”. Choosing a discrete set of partial barriers (corresponding to the “most important” cantori) partitions the phase-space into regions between which flux occurs, with the amount being calculable as the difference in action of certain orbits. This allows one to estimate transport rates. I will describe here the results of the second paper (reference 2) where we provide a less arbitrary partition of the phase-space. The corresponding partial barriers are formed from pieces of stable and unstable manifolds of hyperbolic periodic points, and the regions that they bound are called “resonances”. Consider an integrable system, for example, the simple pendulum. The unstable or hyperbolic orbit corresponds to the pendulum at rest with the bob on top (8 = r). This orbit has an unstable manifold, which corresponds to that orbit beginning at rest infinitesimally close to 8 = T . The orbit is identical to the stable manifold of the hyperbolic fixed point, which corresponds to the orbit beginning at 8 = 0 with just enough energy to approach the vertical position as t m. Together, the stable and unstable manifolds form the separatrix. The resonance is defined as the region of phase-space interior to the separatrix. It contains an orbit of the same period as the hyperbolic orbit, which in this case is the stable equilibrium point. This point is surrounded by “trapped” or “vibrational” invariant curves, which make an island in phase-space. In general, when the hyperbolic orbit is periodic with period n, the resonance is made up of a chain of n such “islands”.