Stable and unstable regimes in higher-dimensional convex billiards with cylindrical shape

TL;DR

This paper introduces higher-dimensional convex billiard models with cylindrical shapes, analyzing their stability and chaos, revealing conditions for stable oscillations and fully chaotic regimes.

Contribution

It generalizes stadium billiards to higher dimensions with cylindrical boundaries, providing a nonlinear stability analysis of periodic orbit families and identifying parameter regions with stable or chaotic behavior.

Findings

Existence of nonlinearly stable whispering gallery modes.

Regions with fully chaotic dynamics with positive Lyapunov exponents.

Conditions for stability related to the geometry and parameters of the models.

Abstract

We introduce a class of convex, higher-dimensional billiard models which generalise stadium billiards. These models correspond to the free motion of a point-particle in a region bounded by cylinders cut by planes. They are motivated by models of particles interacting via a string-type mechanism, and confined by hard walls. The combination of these elements may give rise to a defocusing mechanism, similar to that in two dimensions, which allows large chaotic regions in phase space. The remaining part of phase space is associated with marginally stable behaviour. In fact periodic orbits in these systems generically come in continuous parametric families, sociated with a pair of parabolic eigen-directions: the periodic orbits are unstable in the presence of a defocusing mechanism, but marginally stable otherwise. By performing the stability analysis of families of periodic orbits at a nonlinear level, we establish the conditions under which families are nonlinearly stable or unstable. As a result, we identify regions in the parameter space of the models which admit non-linearly stable oscillations in the form of whispering gallery modes. Where no families of periodic orbits are stable, the billiards are completely chaotic, i.e.\ the Lyapunov exponents of the billiard map are non-zero.