From Hyperbolic to Non-Hyperbolic Open Billiards: An Entropy and Scaling Law Approach

TL;DR

This paper studies how escape dynamics in an open circular billiard change from hyperbolic to non-hyperbolic regimes as energy varies, revealing the emergence of KAM islands and their impact on system complexity and escape behavior.

Contribution

It introduces a comprehensive analysis of the transition from hyperbolic to mixed phase space in open billiards using entropy, escape times, and survival probabilities, highlighting the role of KAM islands.

Findings

Basin entropy peaks at the transition point.

Mean escape time increases sharply with the emergence of KAM islands.

Survival probability shifts from exponential to power-law decay in the mixed regime.

Abstract

We investigate the escape dynamics in an open circular billiard under the influence of a uniform gravitational field. The system properties are investigated as a function of the particle total energy and the size of two symmetrically placed holes in the boundary. Using a suite of quantitative tools including escape basins, basin entropy ($S_b$), mean escape time ($\bar{\tau}$), and survival probability ($P(n)$), we characterize a system that transitions from a fully chaotic, hyperbolic regime at low energies to a non-hyperbolic, mixed phase space at higher energies. Our results demonstrate that this transition is marked by the emergence of Kolmogorov-Arnold-Moser (KAM) islands. We show that both the basin entropy and the mean escape time are sensitive to this transition, with the former peaking and the latter increasing sharply as the sticky KAM islands appear. The survival probability analysis confirms this dynamical picture, shifting from a pure exponential decay in the hyperbolic regime to a power-law-like decay with a saturation plateau in the mixed regime, which directly quantifies the measure of trapped orbits. In the high-energy limit, the system dynamics approaches an integrable case, leading to a corresponding decrease in complexity as measured by both $S_b$ and $\bar{\tau}$.