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In diverse physical systems stable oscillatory solutions devolve into morecomplicated dynamical behaviour through border-collision bifurcations.Mathematically these occur when a stable fixed point of a piecewise-smooth mapcollides with a switching manifold as parameters are varied. The purpose ofthis paper is to highlight the extreme complexity possible in the subsequentdynamics. We perturb instances of the border-collision normal form in $n \ge 2$dimensions for which the $n^{\rm th}$ iterate is a direct product of identicalskew tent maps that have chaotic attractors comprised of $k \ge 2$ disjointintervals. The resulting maps have coexisting attractors and we use Burnside'slemma to count the number of mutually disjoint trapping regions produced bytaking unions of Cartesian products of slight enlargements of the disjointintervals. The attractors are shown to be chaotic by demonstrating that someiterate of the map is piecewise-expanding. The resulting transition from astable fixed point to many coexisting chaotic attractors is shown to occurthroughout open subsets of parameter space and not destroyed by adding higherorder terms to the normal form, hence can be expected to arise generically inmathematical models.

Border-collision bifurcations from stable fixed points to any number of coexisting chaotic attractors