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

Abstract

In diverse physical systems stable oscillatory solutions devolve into more complicated solutions 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. By perturbing instances of the n-dimensional border-collision normal form for which the n th iterate is a direct product of chaotic skew tent maps, it is shown that many chaotic attractors can arise. Burnside's lemma is used to count the attractors; chaoticity is proved by demonstrating that some iterate of the map is piecewise-expanding. The resulting transition from a stable fixed point to many coexisting chaotic attractors occurs throughout open subsets of parameter space and is not destroyed by higher order terms, hence can be expected to occur generically in mathematical models.