A Piecewise Smooth Fermi-Ulam Pingpong with Potential

TL;DR

This paper investigates a Fermi-Ulam pingpong model with a piecewise smooth oscillating platform, revealing conditions for recurrence and coexistence of escaping and bounded orbits at high energy levels.

Contribution

It introduces a piecewise smooth model of the Fermi-Ulam system and analyzes the coexistence of escaping and bounded orbits under specific curvature conditions.

Findings

Escaping orbits form a null set under certain conditions.

System exhibits recurrence despite high energy levels.

Escaping and bounded orbits coexist at high energies.

Abstract

In this paper we study a Fermi-Ulam model where a pingpong bounces elastically against a periodically oscillating platform in a gravity field. We assume that the platform motion $f(t)$ is piecewise $C^3$ with a singularity $\dot{f}(0+)\ne\dot{f}(1-)$. If the second derivative of the platform motion is either always positive $\ddot{f}(t)>0$ or always $\ddot{f}(t)<-g$ 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 coexist with bounded orbits at arbitrarily high energy level.