Chaotic attractors from border-collision bifurcations: stable border fixed points and determinant-based Lyapunov exponent bounds

TL;DR

This paper investigates how border-collision bifurcations in piecewise-smooth maps can lead to chaotic attractors, using stability analysis and Lyapunov exponent bounds derived from Jacobian determinants.

Contribution

It introduces combined techniques to establish chaos in border-collision bifurcations, including stability characterization and Lyapunov bounds considering orbit intersections with switching manifolds.

Findings

Chaotic attractors can form in border-collision bifurcations in $\,\mathbb{R}^d$.

A lower bound on the Lyapunov exponent is derived from Jacobian determinants.

Special considerations are made for points intersecting the switching manifold.

Abstract

The collision of a fixed point with a switching manifold (or border) in a piecewise-smooth map can create many different types of invariant sets. This paper explores two techniques that, combined, establish a chaotic attractor is created in a border-collision bifurcation in $\mathbb{R}^d$ $(d \ge 1)$. First, asymptotic stability of the fixed point at the bifurcation is characterised and shown to imply a local attractor is created. Second, a lower bound on the maximal Lyapunov exponent is obtained from the determinants of the one-sided Jacobian matrices associated with the fixed point. Special care is taken to accommodate points whose forward orbits intersect the switching manifold as such intersections can have a stabilising effect. The results are applied to the two-dimensional border-collision normal form focusing on parameter values for which the map is piecewise area-expanding.