Spherical wedge billiard: from chaos to fractals and Talbot carpets

TL;DR

This paper introduces the spherical wedge billiard, a non-Euclidean dynamical system exhibiting complex, fractal, and regular behaviors, with detailed analysis of its Poincaré map and phase space structures.

Contribution

It provides the first analytic form of the Poincaré map for the spherical wedge billiard and explores its rich dynamical regimes, including chaos, fractals, and Talbot carpets.

Findings

Complex dynamics ranging from chaos to regularity.

Fractal features in the phase space.

Merging of fixed points due to spherical aberration.

Abstract

We introduce the spherical wedge billiard, a dynamical system consisting of a particle moving along a geodesic on a closed non-Euclidean surface of a spherical wedge. We derive the analytic form of the corresponding Poincar\'e map and find very complex dynamics, ranging from completely chaotic to very regular, exhibiting fractal features. Further, we show that upon changing the billiard parameter, the fixed points of the Poincar\'e map merge in complex ways, which has origin in the spherical aberration of the billiard mapping. We also analyze in detail the regular regime when phase space diagram is closely related to Talbot carpets.