A piecewise contractive map on triangles

TL;DR

This paper investigates a piecewise contractive map on triangles in the plane, analyzing its dynamics, periodic orbits, and bifurcations, revealing convergence behavior and the existence of numerous periodic attractors.

Contribution

It introduces a novel piecewise contraction map on triangles and characterizes its asymptotic dynamics, including the structure and bifurcation of periodic orbits.

Findings

Orbits converge to fixed points or periodic orbits.

Existence of infinitely many periodic orbits depending on parameters.

Bifurcation behavior of periodic orbits analyzed.

Abstract

We study the dynamics of a piecewise map defined on the set of three pairwise nonparallel, nonconcurrent lines in $\mathbb{R}^2$. The geometric map of study may be analogized to the billiard map with a different reflection rule so that each iteration is a contraction over the space, thereby providing asymptotic behavior of interest. Our study emphasizes the behavior of periodic orbits generated by the map, with description of their geometry and bifurcation behavior. We establish that for any initial point in the space, the orbit will converge to a fixed point or periodic orbit, and we demonstrate that there exists an infinite variety of periodic orbits the orbits may converge to, dependent on the parameters of the underlying space.