Homoclinic and heteroclinic intersections for lemon billiards

TL;DR

This paper investigates the complex dynamics of billiards on symmetric lemon-shaped tables, demonstrating the existence of crossing homoclinic and heteroclinic intersections for a range of parameters, which implies chaotic behavior.

Contribution

It proves the existence of homoclinic and heteroclinic intersections in lemon billiards near a specific parameter range, revealing chaotic dynamics.

Findings

Existence of crossing homoclinic and heteroclinic intersections

Positive topological entropy for certain lemon billiards

Chaotic dynamics in lemon billiard systems

Abstract

We study the dynamical billiards on a symmetric lemon table $\mathcal{Q}(b)$, where $\mathcal{Q}(b)$ is the intersection of two unit disks with center distance $b$. We show that there exists $\delta_0>0$ such that for all $b\in(1.5, 1.5+\delta_0)$ (except possibly a discrete subset), the billiard map $F_b$ on the lemon table $\mathcal{Q}(b)$ admits crossing homoclinic and heteroclinic intersections. In particular, such lemon billiards have positive topological entropy.