An investigation of escape and scaling properties of a billiard system

TL;DR

This paper explores how particles escape from a billiard system with variable boundary holes, analyzing survival probabilities, scaling behaviors, and basin complexities to understand the system's statistical and dynamical properties.

Contribution

It introduces a detailed analysis of escape dynamics, survival probability scaling, and basin entropy relations in a billiard system with adjustable boundary holes.

Findings

Survival probability decays exponentially with power-law tails near stability islands.

Survival probability shows scaling invariance relative to hole size.

Basin entropy correlates with system parameters and exhibits scaling invariance.

Abstract

We investigate some statistical properties of escaping particles in a billiard system whose boundary is described by two control parameters with a hole on its boundary. Initially, we analyze the survival probability for different hole positions and sizes. We notice the survival probability follows an exponential decay with a characteristic power law tail when the hole is positioned partially or entirely over large stability islands in phase space. We find the survival probability exhibits scaling invariance with respect to the hole size. In contrast, the survival probability for holes placed in predominantly chaotic regions deviates from the exponential decay. We introduce two holes simultaneously and investigate the complexity of the escape basins for different hole sizes and control parameters by means of the basin entropy and the basin boundary entropy. We find a non-trivial relation between these entropies and the system's parameters and show that the basin entropy exhibits scaling invariance for a specific control parameter interval.