Numerical study of billiard motion in an annulus bounded by non-concentric circles

Abstract

This paper exposes a simple, focusing-dispersing billiard system whose phase space simultaneously exhibits the full range of possible Hamiltonian syste, orbit behavior over a continuous range of system parameter values. Specifically, we find that billiard motion between two non-concentric circles is characterized by three distinct phase space regions: (1) a rigorously integrable region in which billiard orbits undergo collisions only with the outer circle boundary; (2) a KAM near-integrable region in which billiard collisions strictly alternate between inner and outer circles; and (3) a chaotic region produced by orbital sequences of billiard collisions which randomly alternate between the integrable and near-integrable patterns. The properties of the integrable, near-integrable, and chaotic regions, the presence of island chains, and the homoclinic points associated with certain hyperbolic fixed points are discussed. Perhaps the most interesting feature of this billiard system, however, is the fresh view of the source of chaos it provides; specifically, the chaotic orbits of region (3) exhibit “random” jumping between the extended invariant curves of regions (1) and (2).