Probing the statistics of transport in the H\'enon Map

TL;DR

This paper investigates the statistical properties of boundary circles and flux ratios in the Hénon map, revealing non-random distributions that influence transport in mixed phase space.

Contribution

It provides new statistical analyses of boundary circle rotation numbers and flux ratios, identifying universality classes affecting transport in the Hénon map.

Findings

Higher probability of small continued fraction elements in boundary circles.

Two universality classes in flux ratio distributions.

Implications for transport models in mixed phase space.

Abstract

The phase space of an area-preserving map typically contains infinitely many elliptic islands embedded in a chaotic sea. Orbits near the boundary of a chaotic region have been observed to stick for long times, strongly influencing their transport properties. The boundary is composed of invariant "boundary circles". We briefly report recent results of the distribution of rotation numbers of boundary circles for the H\'enon quadratic map and show that the probability of occurrence of small elements of their continued fraction expansions is larger than would be expected for a number chosen at random. However, large elements occur with probabilities distributed proportionally to the random case. The probability distributions of ratios of fluxes through island chains is reported as well. These island chains are neighbours in the sense of the Meiss-Ott Markov-tree model. Two distinct universality families are found. The distributions of the ratio between the flux and orbital period are also presented. All of these results have implications for models of transport in mixed phase space.