Some remarks on the periodic motions of a bouncing ball

TL;DR

This paper investigates the existence, stability, and multiplicity of periodic bouncing motions of a ball on a periodically moving racket, providing new insights into their stability properties and the structure of such motions.

Contribution

It offers a detailed analysis of periodic motions in a bouncing ball system with a periodic forcing function, including stability results and multiplicity conditions.

Findings

One periodic motion is proven to be unstable for analytic forcing functions.

Conditions for the existence and multiplicity of periodic motions are established.

The set of periodic motions has specific structural properties under analytic conditions.

Abstract

We consider the vertical motion of a free falling ball bouncing elastically on a racket moving in the vertical direction according to a regular $1$-periodic function $f$. For fixed coprime $p,q$ we study existence, stability in the sense of Lyapunov and multiplicity of $p$ periodic motions making $q$ bounces in a period. If $f$ is real analytic we prove that one periodic motion is unstable and give some information on the set of these motions.