Simple model of bouncing ball dynamics: displacement of the table assumed as quadratic function of time

TL;DR

This paper analyzes the nonlinear dynamics of a bouncing ball with a quadratic moving limiter, revealing fixed points, cycles, and chaos, and explaining chaos emergence via homoclinic structures in bifurcations.

Contribution

It introduces an analytical and numerical study of bouncing ball dynamics with a quadratic displacement limiter, highlighting the formation of chaotic bands from unstable cycles.

Findings

Chaotic bands arise from homoclinic structures in bifurcations.

Multiple dynamical modes including fixed points, cycles, and chaos are identified.

Analytical and numerical methods are combined to study the system.

Abstract

Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Displacement of the limiter is a quadratic function of time. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands appear due to homoclinic structures created from unstable 2 - cycles in a corner-type bifurcation.