How to compute multi-dimensional stable and unstable manifolds of piecewise-linear maps

Abstract

For piecewise-linear maps the stable and unstable manifolds of hyperbolic periodic solutions are themselves piecewise-linear. Hence compact subsets of these manifolds can be represented using polytopes (i.e. polygons, in the case of two-dimensional manifolds). Such representations are efficient and exact so for computational purposes are superior to representations that use a large number of points on some mesh (as is usually done in the smooth setting). We introduce a method for computing convex polytope representations of stable and unstable manifolds. For an unstable manifold we iterate a suitably small subset of the local unstable manifold and prior to each iteration subdivide polytopes where they intersect the switching manifold of the map. We prove the output converges to the (entire) unstable manifold and use it to visualise attractors and bifurcations of the three-dimensional border-collision normal form: we identify a heterodimensional-cycle, a two-dimensional unstable manifold whose closure appears to be a unique attractor, and a piecewise-linear analogue of a first homoclinic tangency where an attractor appears to be destroyed.