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

### New Zealand Journal of Mathematics

Vol. 50, (2020), Pages 71-91

David J.W. Simpson

School of Fundamental Sciences

Massey University

Palmerston North

New Zealand

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.

Keywords Piecewise-linear, asymptotic stability, topological attractor, border-collision normal form.

Classification (MSC2000) 37G35, 39A30, 37A05.