论文标题
折射银河台球中的混沌动力学
Chaotic dynamics in refraction galactic billiards
论文作者
论文摘要
我们证明了来自天体力学的新的机械折射台球，在高内部能量处存在拓扑混乱。给定一个平滑的封闭域$ d \ in \ mathbb {r}^2 $，中央质量在其中产生了开普勒潜力，而在$ \ mathbb {r}^2 \ setMinus \ setMinus \ edmiminus \ edmiminus \ edline {d} $中，谐波振荡器型潜在的潜在行为。在界面上，斯内尔的折射定律成立。混乱的结果是通过对域中施加渐进的假设来获得的，到达几何条件，这些几何条件通常以$ c^1 $为基础。工作流程始于符号动力学的存在，并以拓扑混乱的证明结束，通过分析性的非积分性以及不同平衡鞍点之间的多个异晶连接的存在。这项工作可以视为在Arxiv中进行的调查的最后一步：2108.11159 [Math.ds]和Arxiv：2105.02108 [Math.ds]。
We prove the presence of topological chaos at high internal energies for a new class of mechanical refraction billiards coming from Celestial Mechanics. Given a smooth closed domain $D\in\mathbb{R}^2$, a central mass generates a Keplerian potential in it, while, in $\mathbb{R}^2\setminus \overline{D}$, a harmonic oscillator-type potential acts. At the interface, Snell's law of refraction holds. The chaoticity result is obtained by imposing progressive assumptions on the domain, arriving to geometric conditions which hold generically in $C^1$. The workflow starts with the existence of a symbolic dynamics and ends with the proof of topological chaos, passing through the analytic non-integrability and the presence of multiple heteroclinic connections between different equilibrium saddle points. This work can be considered as the final step of the investigation carried on in arXiv:2108.11159 [math.DS] and arXiv:2105.02108 [math.DS].
