Sequences of Periodic Solutions and Infinitely Many Coexisting Attractors in the Border-Collision Normal Form

Abstract

The border-collision normal form is a piecewise-linear continuous map on $\mathbb{R}^N$ that describes dynamics near border-collision bifurcations of nonsmooth maps. This paper studies a codimension-three scenario at which the border-collision normal form with $N=2$ exhibits infinitely many attracting periodic solutions. In this scenario there is a saddle-type periodic solution with branches of stable and unstable manifolds that are coincident, and an infinite sequence of attracting periodic solutions that converges to an orbit homoclinic to the saddle-type solution. Several important features of the scenario are shown to be universal, and three examples are given. For one of these examples infinite coexistence is proved directly by explicitly computing periodic solutions in the infinite sequence.