Bifurcations of periodic and chaotic attractors in pinball billiards with focusing boundaries

TL;DR

This paper investigates how modifying collision rules in pinball billiards with focusing boundaries leads to various bifurcations, resulting in the emergence of periodic and chaotic attractors as the contraction parameter varies.

Contribution

It introduces a dissipative billiard model with a contraction parameter and numerically characterizes the bifurcations of attractors, extending previous theoretical results.

Findings

Some billiards exhibit only periodic attractors

Others exhibit only chaotic attractors

Some have coexistence of periodic and chaotic attractors

Abstract

We study the dynamics of billiard models with a modified collision rule: the outgoing angle from a collision is a uniform contraction, by a factor lambda, of the incident angle. These pinball billiards interpolate between a one-dimensional map when lambda=0 and the classical Hamiltonian case of elastic collisions when lambda=1. For all lambda<1, the dynamics is dissipative, and thus gives rise to attractors, which may be periodic or chaotic. Motivated by recent rigorous results of Markarian, Pujals and Sambarino, we numerically investigate and characterise the bifurcations of the resulting attractors as the contraction parameter is varied. Some billiards exhibit only periodic attractors, some only chaotic attractors, and others have coexistence of the two types.