Chaos in cylindrical stadium billiards via a generic nonlinear mechanism

TL;DR

This paper generalizes the chaotic behavior of stadium billiards to higher dimensions, identifying nonlinear mechanisms that lead to chaos or stability depending on geometry.

Contribution

It introduces conditions for chaos in higher-dimensional convex billiards, extending the classical stadium model to three or more dimensions.

Findings

Higher-dimensional billiards can be fully chaotic under certain geometric conditions.

A nonlinear instability mechanism can induce chaos or stable oscillations.

Specific geometries lead to stable nonlinear oscillations like whispering-gallery modes.

Abstract

We describe conditions under which higher-dimensional billiard models in bounded, convex regions are fully chaotic, generalizing the Bunimovich stadium to dimensions above two. An example is a three-dimensional stadium bounded by a cylinder and several planes; the combination of these elements may give rise to defocusing, allowing large chaotic regions in phase space. By studying families of marginally-stable periodic orbits that populate the residual part of phase space, we identify conditions under which a nonlinear instability mechanism arises in their vicinity. For particular geometries, this mechanism rather induces stable nonlinear oscillations, including in the form of whispering-gallery modes.