Statistics of the Island-Around-Island Hierarchy in Hamiltonian Phase Space

TL;DR

This paper investigates the statistical properties of boundary circle winding numbers and flux distributions in Hamiltonian phase space, revealing universal patterns and differences in their level and class distributions.

Contribution

It introduces a statistical approach to analyze the complex boundary structures in Hamiltonian systems, focusing on winding number distributions and fluxes, with evidence for universality.

Findings

Distinct level and class distributions observed.

Universal patterns identified in flux distributions.

Analytical fits support the universality of the distributions.

Abstract

The phase space of a typical Hamiltonian system contains both chaotic and regular orbits, mixed in a complex, fractal pattern. One oft-studied phenomenon is the algebraic decay of correlations and recurrence time distributions. For area-preserving maps, this has been attributed to the stickiness of boundary circles, which separate chaotic and regular components. Though such dynamics has been extensively studied, a full understanding depends on many fine details that typically are beyond experimental and numerical resolution. This calls for a statistical approach, the subject of the present work. We calculate the statistics of the boundary circle winding numbers, contrasting the distribution of the elements of their continued fractions to that for uniformly selected irrationals. Since phase space transport is of great interest for dynamics, we compute the distributions of fluxes through island chains. Analytical fits show that the "level" and "class" distributions are distinct, and evidence for their universality is given.