Influence of stability islands in the recurrence of particles in a static oval billiard with holes

TL;DR

This paper investigates how the placement and movement of a hole in an oval billiard affect particle recurrence and escape probabilities, revealing exponential decay behavior and preferred escape regions related to stability islands.

Contribution

It introduces a study of particle recurrence in an oval billiard with a moving boundary hole, analyzing how stability islands influence escape dynamics and identifying optimal hole placement.

Findings

Survival probability follows an exponential decay law.

Decay slope is proportional to the hole's relative size.

Preferred escape regions are associated with stability islands.

Abstract

Statistical properties for the recurrence of particles in an oval billiard with a hole in the boundary are discussed. The hole is allowed to move in the boundary under two different types of motion: (i) counterclockwise periodic circulation with a fixed step length and; (ii) random movement around the boundary. After injecting an ensemble of particles through the hole we show that the surviving probability of the particles without recurring - without escaping - from the billiard is described by an exponential law and that the slope of the decay is proportional to the relative size of the hole. Since the phase space of the system exhibits islands of stability we show that there are preferred regions of escaping in the polar angle, hence given a partial answer to an open problem: {\it Where to place a hole in order to maximize or minimize a suitable defined measure of escaping}.