Border-collision bifurcations from stable fixed points to any number of coexisting chaotic attractors

TL;DR

This paper explores how border-collision bifurcations in piecewise-smooth maps can lead to complex dynamics with multiple coexisting chaotic attractors, demonstrating the generic nature of such phenomena in mathematical models.

Contribution

It introduces a method to analyze the emergence of multiple chaotic attractors from stable fixed points via border-collision bifurcations in higher-dimensional systems.

Findings

Coexisting chaotic attractors can arise from border-collision bifurcations.

The number of disjoint trapping regions can be counted using Burnside's lemma.

The transition to chaos persists under higher order perturbations.

Abstract

In diverse physical systems stable oscillatory solutions devolve into more complicated dynamical behaviour through border-collision bifurcations. Mathematically these occur when a stable fixed point of a piecewise-smooth map collides with a switching manifold as parameters are varied. The purpose of this paper is to highlight the extreme complexity possible in the subsequent dynamics. We perturb instances of the border-collision normal form in $n \ge 2$ dimensions for which the $n^{\rm th}$ iterate is a direct product of identical skew tent maps that have chaotic attractors comprised of $k \ge 2$ disjoint intervals. The resulting maps have coexisting attractors and we use Burnside's lemma to count the number of mutually disjoint trapping regions produced by taking unions of Cartesian products of slight enlargements of the disjoint intervals. The attractors are shown to be chaotic by demonstrating that some iterate of the map is piecewise-expanding. The resulting transition from a stable fixed point to many coexisting chaotic attractors is shown to occur throughout open subsets of parameter space and not destroyed by adding higher order terms to the normal form, hence can be expected to arise generically in mathematical models.