Chaos in the square billiard with a modified reflection law

TL;DR

This paper investigates a modified reflection law in a square billiard, revealing complex dynamics including chaos and positive topological entropy, contrasting with the integrable standard case.

Contribution

It introduces a new non-standard reflection law in square billiards and analyzes its complex dynamical behavior both numerically and analytically.

Findings

Decomposition into three invariant sets: parabolic attractor, chaotic attractor, and horseshoes

Presence of positive topological entropy indicating chaos

Contrasts with the integrable standard square billiard

Abstract

The purpose of this paper is to study the dynamics of a square billiard with a non-standard reflection law such that the angle of reflection of the particle is a linear contraction of the angle of incidence. We present numerical and analytical arguments that the nonwandering set of this billiard decomposes into three invariant sets, a parabolic attractor, a chaotic attractor and a set consisting of several horseshoes. This scenario implies the positivity of the topological entropy of the billiard, a property that is in sharp contrast with the integrability of the square billiard with the standard reflection law.