Regular and chaotic dynamics of a piecewise smooth bouncer

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.

A Three Dimensional Gravitational Billiard in a Cone

“…One-dimensional driven gravitational systems have been thoroughly investigated [20][21][22][23][24][25]. The seminal example is the so-called gravitational bouncer, consisting of a particle impacting a periodically driven wall in the presence of a constant gravitational field.…”

“…The seminal example is the so-called gravitational bouncer, consisting of a particle impacting a periodically driven wall in the presence of a constant gravitational field. The gravitational bouncer was introduced as a variant of the well-known Fermi-Ulam model [26,27], and has been studied for several types of driving motions including sinusoidal [20][21][22] and piecewise linear [23][24][25]. To model the Feldt experiments, two-dimensional driven gravitational billiards were studied numerically in [14,28]; in the former theoretical study rotational effects were ignored, while in the latter rotational effects were included in the model, making analytical computations e.g., periodic orbits difficult.…”