Geometrical origin of chaoticity in the bouncing ball billiard

TL;DR

This paper investigates the geometric origins of chaos in the bouncing ball billiard, identifying how specific configurations of the floor influence the system's chaotic behavior and estimating bounds on Lyapunov exponents.

Contribution

It provides a geometric explanation for chaos in the bouncing ball billiard and estimates Lyapunov bounds using semi-analytical methods for curved floors.

Findings

Chaoticity is linked to the geometrical structure of the billiard system.

A lower bound for the maximal Lyapunov exponent is estimated.

Chaos occurs within specific frequency intervals for circular arc floors.

Abstract

We present a study of the chaotic behavior of the bouncing ball billiard. The work is realised on the purpose of finding at least certain causes of separation of the neighbouring trajectories. Having in view the geometrical construction of the system, we report a clear origin of chaoticity of the bouncing ball billiard. By this we claim that in case when the floor is made of arc of circles - in a certain interval of frequencies - a lower bound for the maximal Ljapunov can be evaluated by sem-ianalitical techniques.