Periodic orbits in Hamiltonian chaos of the annular billiard

Abstract

We consider the motion of trajectories in the annular billiard, constituted of a circle with an internal, perfectly reflecting, eccentrically located secondary circle, displaying a generic Hamiltonian behavior (including periodic orbits, invariant curves, and chaotic areas). Periodic orbits embedded in the phase space are systematically investigated, with a focus on inclusion-touching periodic orbits, up to symmetrical orbits of period 6. Candidates for periodic orbits are detected by investigating grayscale distance charts and, afterward, each candidate is validated (or rejected) by using analytical and/or numerical methods. This Hamiltonian problem with Hamiltonian chaos (mechanical language) may equivalently be viewed as an optical problem with optical chaos (expressed with a geometrical optics language). It then may be extended to the study of interaction between a laser beam (or a plane wave as a limit) and a sphere with an eccentrically located spherical inclusion, this interaction being described by a generalized Lorenz-Mie theory recently established. Inclusion-touching periodic orbits in the annular billiard may generate a new class of morphology-dependent resonances in the associated extended generalized Lorenz-Mie theory problem.