On Unbounded Motions in a Real Analytic Bouncing ball Problem

Abstract

We consider the model of a ball elastically bouncing on a racket moving in the vertical direction according to a given periodic function f(t). The gravity force is acting on the ball. We prove that if the function f(t) belongs to a class of trigonometric polynomials of degree 2 then there exists a one dimensional continuum of initial conditions for which the velocity of the ball tends to infinity. Our result improves a previous one by Pustyl’nikov and gives a new upper bound to the applicability of KAM theory to this model.

Dynamics of the Fermi–Ulam model in an external gravitational field

“…In the 1990s, Krüger et al [5] proposed a natural generalized model, the Fermi-Ulam model in an external gravitational field with periodic p(t), they proved that at all later moments of time, the energy of the ball is bounded along every orbit by applying the generalized version of Moser's invariant curve theorem given by Pustyl'nikov [13]. In addition, it is worth noting that there are a few recent results in a similar model (pingpong in gravity) which describes a ball bouncing elastically against an infinitely heavy moving wall in a gravity field; one can refer to [10][11][12]19] and the references therein.…”

Jumping Platonic Solids on a Vibrating Plate

“…The majority of them treated the simplest case of a jumping sphere, starting with early work by Zaslavski [ 18 ] and Holmes. [ 19 ] The mechanical properties of spheres considered in the literature range from fully inelastic [ 20 ] to completely elastic spheres, [ 21 ] the restitution coefficient being an important parameter. Different scenarios have been described, including synchronization of the particle jumps with the bottom plate oscillations or their subharmonics, so‐called sticking states, periodic motion, and chaotic or quasi‐chaotic trajectories.…”