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

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 there are preferred regions of escaping in the polar angle, hence given a partial answer to an open problem: Where to place a hole in order to maximize or minimize a suitable defined measure of escaping.