Partitioning two-dimensional mixed phase spaces

Abstract

Hamiltonian systems typically exhibit a mixture of chaos and regularity, complicating any scheme to partition phase space and extract a symbolic description of the dynamics. In particular, the dynamics in the vicinity of stable islands can exhibit complicated topology that is qualitatively distinct from that away from the islands. We develop an approach to partition the chaotic phase space of a general dynamical system represented by a two-dimensional map (homeomorphism). This approach can accommodate mixed phase space structure with an arbitrarily high degree of accuracy. The partitioning scheme is built around networks of nested heteroclinic tangles—fundamental geometric objects that organize phase–space transport. These tangles can be used to progressively approximate the dynamics in the vicinity of stable island chains. The net result is a symbolic approximation to the dynamics in the chaotic sea, and an associated phase–space partition, which includes the influence of stable islands and which is approximately Markov.