Regular and Chaotic Dynamics of a Piecewise Smooth Bouncer

TL;DR

This paper analyzes the dynamics of a particle bouncing on a piecewise constant velocity wall, revealing regular, chaotic, and novel bifurcation behaviors, including inelastic collapse and border-collision bifurcations, which differ from sinusoidal models.

Contribution

It introduces a piecewise linear bouncer model, providing analytical insights and uncovering unique bifurcation phenomena not observed in sinusoidal systems.

Findings

Existence of unbounded orbits (Fermi acceleration) in elastic case

Observation of inelastic collapse and sticking solutions

Discovery of border-collision bifurcations due to non-smooth dynamics

Abstract

The dynamical properties of a particle in a gravitational field colliding with a rigid wall moving with piecewise constant velocity are studied. The linear nature of the wall's motion permits further analytical investigation than is possible for the system's sinusoidal counterpart. We consider three distinct approaches to modeling collisions: (i) elastic, (ii) inelastic with constant restitution coefficient and (iii) inelastic with a velocity-dependent restitution function. We confirm the existence of distinct unbounded orbits (Fermi acceleration) in the elastic model, and investigate regular and chaotic behavior in the inelastic cases. We also examine in the constant restitution model trajectories wherein the particle experiences an infinite number of collisions in a finite time i.e., the phenomenon of inelastic collapse. We address these so-called "sticking solutions" and their relation to both the overall dynamics and the phenomenon of self-reanimating chaos. Additionally, we investigate the long-term behavior of the system as a function of both initial conditions and parameter values. We find the non-smooth nature of the system produces novel bifurcation phenomena not seen in the sinusoidal model, including border-collision bifurcations. The analytical and numerical investigations reveal that although our piecewise linear bouncer is a simplified version of the sinusoidal model, the former not only captures essential features of the latter but also exhibits behavior unique to the discontinuous dynamics.