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

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 et al (http://premat.fing.edu.uy/papers/2008/110.pdf and http://www.preprint.impa.br/Shadows/SERIE_A/2008/614.html), we numerically investigate and characterize 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.