Linear and nonlinear stability of periodic orbits in annular billiards

TL;DR

This paper analyzes the stability of periodic orbits in annular billiards, revealing conditions for stability, bifurcations, and non-ergodic behavior depending on the scatterer's position and size.

Contribution

It provides analytical results on the stability and bifurcation of periodic orbits in annular billiards with varying scatterer positions and sizes, including the existence of KAM islands.

Findings

Existence of linearly stable periodic orbits near the boundary.

Stability changes from elliptic to hyperbolic with scatterer displacement.

Presence of non-ergodic intervals related to scatterer radii.

Abstract

An annular billiard is a dynamical system in which a particle moves freely in a disk except for elastic collisions with the boundary, and also a circular scatterer in the interior of the disk. We investigate stability properties of some periodic orbits in annular billiards in which the scatterer is touching or close to the boundary. We analytically show that there exist linearly stable periodic orbits of arbitrary period for scatterers with decreasing radii that are located near the boundary of the disk. As the position of the scatterer moves away from a symmetry line of a periodic orbit, the stability of periodic orbits changes from elliptic to hyperbolic, corresponding to a saddle-center bifurcation. When the scatterer is tangent to the boundary, the periodic orbit is parabolic. We prove that slightly changing the reflection angle of the orbit in the tangential situation leads to the existence of KAM islands. Thus we show that there exists a decreasing to zero sequence of open intervals of scatterer radii, along which the billiard table is not ergodic.