Structure and evolution of strange attractors in non-elastic triangular billiards

TL;DR

This paper investigates the complex fractal structures and evolution of strange attractors in a non-elastic triangular billiard system, revealing how attractor properties change with the contraction parameter.

Contribution

It provides a rigorous analysis of the structure and bifurcation of strange attractors in non-elastic billiards, including the formation of Cantor sets and basin boundary fractality.

Findings

For small contraction, attractor is a Cantor set times an interval.

Larger contraction leads to nonaccessible phase space regions.

Near elastic limit, attractors split into multiple transitive components.

Abstract

We study pinball billiard dynamics in an equilateral triangular table. In such dynamics, collisions with the walls are non-elastic: the outgoing angle with the normal vector to the boundary is a uniform factor $\lambda < 1$ smaller than the incoming angle. This leads to contraction in phase space for the discrete-time dynamics between consecutive collisions, and hence to attractors of zero Lebesgue measure, which are almost always fractal strange attractors with chaotic dynamics, due to the presence of an expansion mechanism. We study the structure of these strange attractors and their evolution as the contraction parameter $\lambda$ is varied. For $\lambda$ in the interval (0, 1/3), we prove rigorously that the attractor has the structure of a Cantor set times an interval, whereas for larger values of $\lambda$ the billiard dynamics gives rise to nonaccessible regions in phase space. For $\lambda$ close to 1, the attractor splits into three transitive components, the basins of attraction of which have fractal basin boundaries.