Chaotic dynamics in refraction galactic billiards

TL;DR

This paper demonstrates topological chaos in a novel class of celestial mechanics billiards involving refraction at an interface, using geometric, analytic, and symbolic dynamics methods.

Contribution

It introduces a new chaotic model in celestial mechanics with refraction boundary conditions and proves chaos through symbolic dynamics and heteroclinic connections.

Findings

Existence of symbolic dynamics for the system

Proof of topological chaos at high energies

Chaotic behavior holds under generic geometric conditions

Abstract

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].