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.
Highlights
Billiard models, in which a point particle moves uniformly until it undergoes abrupt elastic collisions with a fixed boundary, are the playground of statistical physicists and mathematicians alike, whose work focuses on the interplay between their dynamical and statistical properties [1].There are two main categories of chaotic billiards
We introduce a class of higher-dimensional convex billiards, in which the particle motion occurs inside cylindrical domains bounded by oblique planes; we call these cylindrical stadium billiards
In the presence of two elliptic eigenvalues which generate linear oscillations in the plane transverse to the cylinder axis, nonlinear effects along the neutral directions may act as a restoring force, allowing for stable oscillations in all phase space directions
Summary
In which a point particle moves uniformly until it undergoes abrupt elastic collisions with a fixed boundary, are the playground of statistical physicists and mathematicians alike, whose work focuses on the interplay between their dynamical and statistical properties [1]. This has the consequence that its stability analysis yields, out of the four phase-space dimensions of the billiard map, two parabolic directions (with unit eigenvalue), corresponding to motion along the family Another example of a three-dimensional cavity whose periodic orbits belong to continuous parametric families is the three-dimensional billiard whose domain consists of the intersection of a sphere with a cuboid [28]. It is the goal of this paper to consider billiards whose phase space combines neutral directions with curved regions, associated with stable and unstable behaviors, and explore the conditions under which they can display hyperbolic regimes, i.e. such that all pairs of opposite Lyapunov exponents are non-zero To this end, we introduce a class of higher-dimensional convex billiards, in which the particle motion occurs inside cylindrical domains bounded by oblique planes; we call these cylindrical stadium billiards. Appendix A and Appendix B, provide exact forms obtained for the linear and non-linear analysis of some of the periodic orbits studied in sections 5 and 6
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