Dynamics of perturbed elliptical billiard tables

Abstract

Dynamical billiards consist of a particle on a two-dimensional table, bouncing elastically off a boundary curve. The state of the system is given by two numbers: one describing the location along the curve where the bounce occurs, and another describing the incoming angle of the bounce. Successive bounces define a two-dimensional area preserving map, and iterating this map gives a dynamical system first studied by Birkhoff. One of the simplest smooth table shapes is that of an ellipse, in which case the dynamics of the billiard map is completely integrable. The longstanding Birkhoff conjecture is that elliptical tables are the only smooth convex table for which complete integrability occurs. In this spirit, we present an implicit real analytic method for iterating billiard maps on perturbed elliptical tables. This method allows us to compute local stable and unstable manifolds of periodic orbits using the parameterization method. Globalizing these local manifolds numerically provides insight into the dynamics of the table.