Chaos and stability in a two-parameter family of convex billiard tables

TL;DR

This paper investigates a two-parameter family of convex billiard tables, revealing complex dynamical behaviors and proposing the existence of new ergodic billiard classes near skewed stadia, supported by numerical and heuristic analysis.

Contribution

It introduces a two-parameter generalization of oval billiards, explores their dynamical properties, and conjectures new ergodic classes near skewed stadia, backed by numerical simulations.

Findings

Identification of complex dynamical phenomena during deformation from circular to stadia billiards.

Conjecture of new ergodic billiard classes in certain parameter regions.

Numerical confirmation using Lyapunov-weighted dynamics.

Abstract

We study, by numerical simulations and semi-rigorous arguments, a two-parameter family of convex, two-dimensional billiard tables, generalizing the one-parameter class of oval billiards of Benettin--Strelcyn [Phys. Rev. A 17, 773 (1978)]. We observe interesting dynamical phenomena when the billiard tables are continuously deformed from the integrable circular billiard to different versions of completely-chaotic stadia. In particular, we conjecture that a new class of ergodic billiard tables is obtained in certain regions of the two-dimensional parameter space, when the billiards are close to skewed stadia. We provide heuristic arguments supporting this conjecture, and give numerical confirmation using the powerful method of Lyapunov-weighted dynamics.