Existence of stable periodic orbits in billiards close to lemon and moon billiards

TL;DR

This paper demonstrates that modifying lemon and moon billiards with small curvature arcs can create billiard tables with nonlinearly stable periodic orbits, expanding understanding of stability in billiard systems.

Contribution

It introduces the possibility of stable periodic orbits in billiards with more general boundary components of small curvature, beyond the classical lemon and moon billiards.

Findings

Replacing small curvature arcs can produce stable periodic orbits.

Classical lemon and moon billiards are hyperbolic with small curvature.

Generalized boundary components can alter stability properties.

Abstract

It is known that at lemon and moon billiards that have a sufficiently small curvature on one of their circular arcs are hyperbolic. In this paper we show that replacing this circular arc by a more general boundary component of small curvature could produce billiard tables that admit nonlinearly stable periodic orbits.