Nonpersistence of resonant caustics in perturbed elliptic billiards

TL;DR

This paper proves that resonant elliptical caustics in billiards are destroyed under generic perturbations, using Melnikov theory and complex analysis of elliptic functions.

Contribution

It demonstrates the nonpersistence of resonant caustics in perturbed elliptic billiards through a rigorous mathematical approach.

Findings

Resonant caustics are generically destroyed by perturbations.

Melnikov analysis confirms the nonpersistence of these caustics.

Complex elliptic function analysis underpins the proof.

Abstract

Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics ---the ones whose tangent trajectories are closed polygons--- are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This result follows from a standard Melnikov argument and the analysis of the complex singularities of certain elliptic functions.