Classical Dynamics of the Time-Dependent Elliptical Billiard

TL;DR

This paper investigates the nonlinear dynamics of static and driven elliptical billiards, revealing decay behaviors, saturation effects, and controllable particle emission rates influenced by boundary oscillations.

Contribution

It introduces a detailed analysis of how harmonic boundary oscillations affect decay saturation and emission rates in elliptical billiards, expanding understanding of their nonlinear dynamics.

Findings

Algebraic decay in static ellipse due to phase space structure

Saturation value in decay can be tuned by eccentricity

Harmonic boundary oscillations destroy saturation, enabling control of emission rates

Abstract

In this work we study the nonlinear dynamics of the static and the driven ellipse. In the static case, we find numerically an asymptotical algebraic decay for the escape of an ensemble of non-interacting particles through a small hole due to the integrable structure of the phase space of the system. Furthermore, for a certain hole position a saturation value in the decay that can be tuned arbitrarily by varying the eccentricity of the ellipse is observed and explained. When applying harmonic boundary oscillations this saturation value caused by librator type orbits is gradually destroyed via two fundamental processes which are discussed in detail. As a result, an amplitude dependent emission rate is obtained in the long time behavior of the decay, suggesting that the driven elliptical billiard can be used as a controllable source of particles.