Dynamics of a Piecewise Linear Bouncer

The dynamical properties of a particle in a gravitational field colliding with a rigid wall moving with piecewise constant velocity are studied. We consider three distinct approaches to modeling the collision; elastic, inelastic with constant restitution coefficient and inelastic with a velocity-dependent restitution function. We confirm the existence of Fermi acceleration in the elastic model, and find periodic, quasi-periodic, and chaotic behavior in both inelastic models. We also examine the phenomenon of inelastic collapse. We address the related ``sticking solutions'' and their connection 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. The analytical and numerical investigations reveal that our model captures the essential features of the well-studied sinusoidally driven version and also exhibits behavior unique to the discontinuous dynamics.