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

TL;DR

We study escape dynamics in an open circular billiard under gravity with two symmetric boundary holes, examining dependence on total energy $E$ and hole size $h$. By analyzing basin entropy $S_b$, mean escape time $\bar{\tau}$, and survival probability $P(n)$, we detect a hyperbolic-to-non-hyperbolic transition marked by the emergence of KAM islands. The results show $S_b$ peaking near $E \approx 1.0$, $\bar{\tau}$ increasing due to stickiness, and $P(n)$ transitioning from exponential decay to a stretched-exponential form with a saturation plateau, with high-$E$ dynamics tending toward integrability and reduced complexity. Additionally, the authors uncover scaling laws for the escape rate with hole size $h$, finding $\kappa \propto h^z$ and universal curves under $n' = n h^z$, indicating robust scaling across dynamical regimes.

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 ($\barτ$), 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 $\barτ$.

Figures (8)

• Figure 1: Schematic of the open circular billiard under a uniform gravitational field. The diagram illustrates a particle parabolic trajectory between two collisions, defining the angular position $\theta_n$ and the reflection angle $\alpha_n$. The angle of the tangent at the collision point is $\phi_n$. The inset shows a magnified view of one of the two holes, each with an angular width of $2h$.
• Figure 2: Escape basins in the $(\theta, \alpha)$ phase space for a fixed hole size of $h=0.04$ at four characteristic total energies: (a) $E=0.5$ and (b) $E=0.7$ in the hyperbolic-like regime, and (c) $E=1.3$ and (d) $E=2.0$ in the mixed phase space regime. Initial conditions that escape through the hole at $\theta=0$ are colored orange, while those escaping at $\theta=\pi$ are black. White regions correspond to initial conditions that do not escape within $10^6$ collisions, indicating the presence of KAM islands.
• Figure 3: Escape basins for the open billiard at a fixed total energy of $E=1.0$ for four different hole sizes: (a) $h=0.01$, (b) $h=0.04$, (c) $h=0.07$, and (d) $h=0.1$. The color scheme is the same as in Fig. \ref{['fig:big_Basins']}. As the hole size increases, the basin structures become larger and more defined, reflecting a decrease in the system unpredictability.
• Figure 4: (a) Basin entropy $S_b$ as a function of the total energy $E$ for four different hole sizes, as indicated in the legend. A peak in $S_b$ is observed near the transition energy $E \approx 1.0$. (b) A comparison between the phase space area of non-escaping orbits ($A_w$, black curve, left axis) and the basin entropy ($S_b$, red curve, right axis) for a fixed hole size of $h=0.1$. The local minima of $S_b$ correlate with sharp increases in the area of the KAM islands.
• Figure 5: Mean escape time $\overline{\tau}$ as a function of total energy $E$ for four different hole sizes $h$. The sharp increase in $\overline{\tau}$ for $E \ge 1.0$ is a direct consequence of the 'stickiness' effect introduced by the emergence of KAM islands in the phase space.