Robust chaos in $\mathbb{R}^n$

TL;DR

This paper establishes explicit conditions for the existence of chaotic attractors in n-dimensional piecewise-linear maps with two pieces, demonstrating their persistence under perturbations and their relevance in border-collision bifurcations.

Contribution

It provides a set of sufficient conditions for chaos in high-dimensional piecewise-linear maps, extending previous results and showing attractor persistence under nonlinear perturbations.

Findings

Chaotic attractors exist under explicit conditions in all dimensions n ≥ 2.

The attractors are persistent under nonlinear perturbations.

Chaotic attractors can be created and persist in border-collision bifurcations.

Abstract

We treat $n$-dimensional piecewise-linear continuous maps with two pieces, each of which has exactly one unstable direction, and identify an explicit set of sufficient conditions for the existence of a chaotic attractor. The conditions correspond to an open set within the space of all such maps, allow all $n \ge 2$, and allow all possible values for the unstable eigenvalues in the limit that all stable eigenvalues tend to zero. To prove an attractor exists we use the stable manifold of a fixed point to construct a trapping region; to prove the attractor is chaotic we use the unstable directions to construct an invariant expanding cone for the derivatives of the pieces of the map. We also show the chaotic attractor is persistent under nonlinear perturbations, thus when such an attractor is created locally in a border-collision bifurcation of a general piecewise-smooth system, it persists and is chaotic for an interval of parameter values beyond the bifurcation.