从双曲到非双曲开放台球：一种熵与标度律方法
From Hyperbolic to Non-Hyperbolic Open Billiards: An Entropy and Scaling Law Approach
AI总结：
本文用逃逸盆地、盆地熵、平均逃逸时间和生存概率研究重力场中开放圆形台球，揭示其随能量升高由双曲混沌经 KAM 岛转向非双曲混合相空间并趋近可积的动力学规律。
摘要：
我们研究了在均匀引力场影响下开放圆形台球中的逃逸动力学。该系统的性质被作为粒子总能量以及边界上两个对称放置孔洞尺寸的函数来考察。利用包括逃逸盆地、盆地熵（$S_b$）、平均逃逸时间（$\arτ$）和生存概率（$P(n)$）在内的一套定量工具，我们刻画了一个从低能量下完全混沌的双曲体制向较高能量下非双曲、混合相空间转变的系统。我们的结果表明，这一转变以 Kolmogorov-Arnold-Moser（KAM）岛的出现为标志。我们显示盆地熵和平均逃逸时间都对该转变敏感：随着黏性 KAM 岛出现，前者达到峰值，而后者急剧增加。生存概率分析证实了这一动力学图像，其从双曲体制中的纯指数衰减转变为混合体制中带有饱和平台的类幂律衰减，这直接量化了被俘获轨道的测度。在高能极限下，系统动力学趋近于可积情形，导致由 $S_b$ 和 $\arτ$ 共同衡量的复杂性相应降低。
英文摘要：
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 $\barτ$.
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