Dynamics in chaotic zones of area preserving maps: Close to separatrix and global instability zones

Abstract

The purpose of this paper is to study phenomena in chaotic zones of area preserving maps using simpler models which are easier to analyse theoretically and numerically. First of all the study of the dynamics in a neighbourhood of the separatrices of a resonant zone is carried out. The well-known separatrix map, defined on a figure eight when needed, is used to determine the location of rotational invariant curves (r.i.c.) inside and outside the resonance. The interest in this part is on a quantitative description of the dynamics in a neighbourhood of the separatrices: to produce theoretical estimates of the width of the stochastic zone, distance to the r.i.c., existence of tiny islands close to the separatrices, …In every one of the studied items one has tried to complement the limit analytic study with realistic numerical simulations, describing the analogy when possible. After this study, we focus on the formation of larger domains without r.i.c. (e.g. Birkhoff domains). To this end we introduce the biseparatrix map model. Although this is a qualitative model, the mechanism of destruction of the “last” r.i.c., and hence the process of creation of zones without r.i.c., is clarified by means of this simple model. Several numerical examples illustrate the results obtained and are used as a test of the theoretical quantitative predictions.