A piecewise smooth Fermi–Ulam pingpong with potential

Abstract

In this paper we study a Fermi–Ulam model where a pingpong ball bounces elastically against a periodically oscillating platform in a gravity field. We assume that the platform motion $f(t)$ is 1-periodic and piecewise $C^3$ with a singularity, $\dot {f}(0+)\ne \dot {f}(1-)$ . If the second derivative $\ddot {f}(t)$ of the platform motion is either always positive or always less than $-g$ , where g is the gravitational constant, then the escaping orbits constitute a null set and the system is recurrent. However, under these assumptions, escaping orbits co-exist with bounded orbits at arbitrarily high energy levels.

On Unbounded Motions in a Real Analytic Bouncing ball Problem

“…Now we want to pass to the limit in (18). Notice that in order to pass the limit into the last sum we have to use the dominated convergent theorem noticing that for every s ≥ n…”

“…This model has inspired many authors as it represents a simple mechanical model exhibiting complex dynamics; see for example [5,6,10,[12][13][14][15][16] where results on periodic or quasiperiodic motions are proved together with, in some case, topological chaos. Moreover, for some f presenting some singularities it is possible to study statistical and ergodic properties [2,18].…”

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.…”

Some remarks on the periodic motions of a bouncing ball

“…Bounded motions can be regular (periodic and quasiperiodic, see [13]) and chaotic (see [12,15,22]). Moreover, the non periodic case is studied in [8,9], the case of different potentials is considered in [4] and recent results on ergodic properties are present in [23].…”