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

Abstract

We introduce a class of convex, higher-dimensional billiard models that generalize 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, associated with a pair of parabolic eigendirections: the periodic orbits are unstable in the presence of a defocusing mechanism, but are marginally stable otherwise. By performing 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 that admit nonlinearly 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. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.