Stability and instability results in a model of Fermi acceleration

TL;DR

This paper analyzes a mathematical model of a particle bouncing on an oscillating plate under a potential, revealing conditions for chaotic escape and stability islands, with implications for understanding Fermi acceleration phenomena.

Contribution

It provides new results on the measure and structure of escaping orbits and elliptic islands in a Fermi acceleration model with power-law potentials and sinusoidal plate motion.

Findings

Escaping orbits have full Hausdorff dimension for powers less than 1.

Existence of elliptic islands of period 2 at high energies for many motions.

Conditions under which the measure of elliptic islands is finite or infinite.

Abstract

We consider the static wall approximation to the dynamics of a particle bouncing on a periodically oscillating infinitely heavy plate while subject to a potential force. We assume the case of a potential given by a power of the particle's height and sinusoidal motions of the plate. We find that for powers smaller than 1 the set of escaping orbits has full Hausdorff dimension for all motions and obtain existence of elliptic island of period 2 for arbitrarily high energies for a full-measure set of motions. Moreover we obtain conditions on the potential to ensure that the total (Lebesgue) measure of elliptic islands of period 2 is either finite or infinite.