Robust Devaney chaos in the two-dimensional border-collision normal form

TL;DR

This paper demonstrates that a broad class of two-dimensional border-collision maps exhibit robust Devaney chaos, with stable manifolds densely filling regions and identifying a new bifurcation leading to attractor destruction.

Contribution

It proves the existence of robust chaos in the two-dimensional border-collision normal form and describes a novel heteroclinic bifurcation affecting the attractor.

Findings

Chaos is robust in an open parameter region.

Stable manifolds densely fill regions containing the attractor.

A new heteroclinic bifurcation can destroy the attractor.

Abstract

The collection of all non-degenerate, continuous, two-piece, piecewise-linear maps on $\mathbb{R}^2$ can be reduced to a four-parameter family known as the two-dimensional border-collision normal form. We prove that throughout an open region of parameter space this family has an attractor satisfying Devaney's definition of chaos. This strengthens existing results on the robustness of chaos in piecewise-linear maps. We further show that the stable manifold of a saddle fixed point, despite being a one-dimensional object, densely fills an open region containing the attractor. Finally we identify a heteroclinic bifurcation, not described previously, at which the attractor undergoes a crisis and may be destroyed.