Boundary Circles of Mixed Phase Space, Hamiltonian Systems

TL;DR

This paper studies the distribution of boundary circles in mixed phase space of Hamiltonian systems, revealing non-random patterns in their continued fraction expansions that influence transport properties.

Contribution

It provides the first detailed analysis of boundary circle rotation number distributions in the Henon map, highlighting non-random occurrence of small continued fraction elements.

Findings

Small continued fraction elements are more common than random expectation.

Large elements occur with probabilities similar to random distributions.

Results impact understanding of transport 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 circles, called "Boundary circles." We investigate the distribution of rotation numbers of boundary circles for the Henon 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. These results have implications for models of transport in mixed phase space.