A bouncing ball model with two nonlinearities: a prototype for Fermi acceleration

TL;DR

This paper investigates a nonlinear bouncing ball model with external forces, revealing how control parameters influence chaos and Fermi acceleration phenomena through Lyapunov exponents and wall velocity discontinuities.

Contribution

It introduces a two-dimensional nonlinear map for the bouncing ball with two nonlinearities and analyzes how control parameters affect chaos and acceleration.

Findings

Lyapunov exponent decreases with control parameter increase

Fermi acceleration occurs within specific parameter ranges

Discontinuity in wall velocity influences dynamical behaviors

Abstract

Some dynamical properties of a bouncing ball model under the presence of an external force modeled by two nonlinear terms are studied. The description of the model is made by use of a two dimensional nonlinear measure preserving map on the variables velocity of the particle and time. We show that raising the straight of a control parameter which controls one of the nonlinearities, the positive Lyapunov exponent decreases in the average and suffers abrupt changes. We also show that for a specific range of control parameters, the model exhibits the phenomenon of Fermi acceleration. The explanation of both behaviours is given in terms of the shape of the external force and due to a discontinuity of the moving wall's velocity.