@article{Simpson_2020, title={Chaotic attractors from border-collision bifurcations: stable border fixed points and determinant-based Lyapunov exponent bounds}, volume={50}, url={https://nzjmath.org/index.php/NZJMATH/article/view/65}, DOI={10.53733/65}, abstractNote={<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>}, journal={New Zealand Journal of Mathematics}, author={Simpson, David J.W.}, year={2020}, month={Nov.}, pages={71–91} }