Chaotic properties of the truncated elliptical billiard

TL;DR

This paper explores the chaotic dynamics of a two-parameter truncated elliptical billiard, analyzing stability, chaos extent, and the influence of boundary shape, revealing differences from elliptical stadium billiards and proposing a new generalized shape.

Contribution

It provides a detailed numerical and analytical study of the chaotic properties of truncated elliptical billiards, including stability analysis and the introduction of a new three-parameter boundary shape.

Findings

Full chaos occurs in elongated shapes with unstable or neutral orbits.

Circular arcs mark the transition to fully chaotic behavior.

A new three-parameter shape is proposed for further study.

Abstract

Chaotic properties of symmetrical two-dimensional stadium-like billiards with elliptical arcs are studied numerically and analytically. For the two-parameter truncated elliptical billiard the existence and linear stability of several lowest-order periodic orbits are investigated in the full parameter space. Poincar\' e plots are computed and used for evaluation of the degree of chaoticity with the box-counting method. The limit of the fully chaotic behavior is identified with circular arcs. Above this limit, for flattened elliptical arcs, mixed dynamics with numerous stable elliptic islands is present, similarly as in the elliptical stadium billiards. Below this limit the full chaos extends over the whole region of elongated shapes and the existing orbits are either unstable or neutral. This is conspicuously different from the behavior in the elliptical stadium billiards, where the chaotic region is strictly bounded from both sides. To examine the mechanism of this difference, a generalization to a novel three-parameter family of boundary shapes is proposed and suggested for further evaluation.