On unbounded motions in a real analytic bouncing ball problem

TL;DR

This paper investigates a bouncing ball model with a periodic racket motion, proving the existence of unbounded velocities for certain initial conditions and refining the understanding of the model's dynamics beyond KAM theory.

Contribution

It demonstrates that for specific periodic functions, the ball's velocity can become unbounded, extending previous results and establishing new limits for KAM theory's applicability.

Findings

Existence of initial conditions leading to infinite velocity

Improved bounds on KAM theory applicability

Identification of unbounded motion in a specific bouncing ball model

Abstract

We consider the model of a ball elastically bouncing on a racket moving in the vertical direction according to a given periodic function $f(t)$. The gravity force is acting on the ball. We prove that if the function $f(t)$ belongs to a class of trigonometric polynomials of degree $2$ then there exists a one dimensional continuum of initial conditions for which the velocity of the ball tends to infinity. Our result improves a previous one by Pustyl'nikov and gives a new upper bound to the applicability of KAM theory to this model.