Dynamics of some piecewise smooth Fermi-Ulam Models

TL;DR

This paper analyzes the high energy behavior of piecewise smooth Fermi-Ulam models, revealing conditions for hyperbolic or elliptic dynamics, and characterizing the measure and dimension of Fermi accelerating orbits.

Contribution

It introduces a normal form for the models and characterizes the measure, dimension, and stability of orbits depending on the parameter, advancing understanding of Fermi acceleration phenomena.

Findings

Fermi acceleration orbits have zero measure but full Hausdorff dimension in hyperbolic case.

Almost all orbits eventually fall below a fixed energy threshold.

Stable high-energy periodic orbits exist in the elliptic case.

Abstract

We find a normal form which describes the high energy dynamics of a class of piecewise smooth Fermi-Ulam ping pong models; depending on the value of a single real parameter, the dynamics can be either hyperbolic or elliptic. In the first case we prove that the set of orbits undergoing Fermi acceleration has zero measure but full Hausdorff dimension. We also show that for almost every orbit the energy eventually falls below a fixed threshold. In the second case we prove that, generically, we have stable periodic orbits for arbitrarily high energies, and that the set of Fermi accelerating orbits may have infinite measure.