TY - JOUR AU - Simpson, David J.W. PY - 2020/11/14 Y2 - 2024/03/28 TI - Chaotic attractors from border-collision bifurcations: stable border fixed points and determinant-based Lyapunov exponent bounds JF - New Zealand Journal of Mathematics JA - NZ J Math VL - 50 IS - 0 SE - Articles DO - 10.53733/65 UR - https://nzjmath.org/index.php/NZJMATH/article/view/65 SP - 71--91 AB - <p>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.</p> ER -