Bifurcations in the family of billiards associated with the curvature flow

TL;DR

This paper investigates how billiard dynamics on convex curves change under curvature flow, identifying bifurcations of periodic orbits and the destruction of non-convex caustics, revealing new dynamical behaviors during deformation.

Contribution

It provides a detailed analysis of bifurcations and orbit destruction in billiards deformed by curvature flow, a novel exploration of dynamical properties during geometric deformation.

Findings

Bifurcations of period two orbits identified.

Destruction of non-convex caustics in elliptical billiards.

Deformation causes significant changes in billiard dynamics.

Abstract

We describe some dynamical properties of one parameter families of billiards on convex curves (ovals) which are deformed by the curvature (curve-shortening) flow. We obtain the bifurcations of the period two orbits and some special non-Birkhoff orbits, the normal periodic orbits. We prove the destruction of non-convex caustics of the ellipse by deforming it through the curvature flow.