Transport rates of a class of two-dimensional maps and flows

Abstract

A method is developed for estimating the transport rates of phase space areas for a class of two-dimensional diffeomorphisms and flows. The class of diffeomorphisms we considered are defined by the topological structure of their stable and unstable manifolds, and hence are universal. We show how to estimate the transport rates for a class of diffeomorphisms found by Easton and for an extension of this class of diffeomorphisms which is found via a “perturbation” of the topology of the stable and unstable manifolds. This is done by introducing symbolic dynamics and transfer matrices which in turn relate transport phenomena in phase space to Markov processes in a precise manner. In addition to the transport rates, we use the transfer matrices to obtain estimates for the topological entropy, averaged stretching rates, and the elongation rate of the unstable manifold. The flows we consider are two-dimensional, time-periodic flows which can be reduced via Poincaré section to the extended family of maps. We develop an analytical method, based on Chirikov's Whisker map, to classify a given flow according to the structure of its manifolds in its Poincaré section. This allows the techniques developed here for maps to be directly applied to time-periodic flows.