Chaotic dynamics in a simple bouncing ball model

TL;DR

This paper analyzes the chaotic behavior of a bouncing ball system with a periodically moving table, deriving the Poincare map and exploring how the system transitions to chaos analytically.

Contribution

It introduces an analytical derivation of the Poincare map for a bouncing ball with a piecewise constant velocity table and investigates chaos transition scenarios.

Findings

Derived the Poincare map for the system

Identified conditions for transition to chaos

Provided analytical insights into chaotic dynamics

Abstract

We study dynamics of a ball moving in gravitational field and colliding with a moving table. The motion of the limiter is assumed as periodic with piecewise constant velocity - it is assumed that the table moves up with a constant velocity and then moves down with another constant velocity. The Poincare map, describing evolution from an impact to the next impact, is derived and scenarios of transition to chaotic dynamics are investigated analytically.