Structure, size, and statistical properties of chaotic components in a mixed-type Hamiltonian system

TL;DR

This paper investigates the structure, size, and statistical properties of chaotic regions in a family of billiard systems, revealing insights into fractal boundaries, stickiness, and chaotic measures through numerical analysis.

Contribution

It provides a detailed quantitative analysis of chaotic components in mixed-type Hamiltonian systems, including fractal dimensions, recurrence times, and measures of chaos, with comparisons to a random motion model.

Findings

Fractal dimension of chaotic borders varies with billiard shape.

Cell recurrence times effectively quantify phase space stickiness.

Chaotic measure estimates are consistent across different methods.

Abstract

We perform a detailed study of the chaotic component in mixed-type Hamiltonian systems on the example of a family of billiards [introduced by Robnik in J. Phys. A: Math. Gen. 16, 3971 (1983)]. The phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times. The structure of the chaotic component is discerned from the cells visited by the chaotic orbit. The fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box-counting method. The cell-filling dynamics is compared to a model of uncorrelated motion, the so-called random model [Robnik et al. J. Phys. A: Math. Gen. 30, L803 (1997)], and deviations attributed to sticky objects in the phase space are found. The statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run. The stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times. The recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed. The standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale. Three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter. Lastly, the decay of correlations and the diffusion of momenta is analyzed.