Chaotic Dynamics in Four-Dimensional Piecewise Affine Systems with Bifocal Heteroclinic Cycles

Abstract

The well-known Shil’nikov type theory provides an approach to proving the existence of chaotic invariant sets for some classes of smooth dynamical systems with homoclinic orbits or heteroclinic cycles. However, it cannot be applied to nonsmooth systems directly. Based on the similar ideas, this paper studies the existence of chaotic invariant sets for a class of two-zone four-dimensional piecewise affine systems with bifocal heteroclinic cycles that cross the switching manifold transversally at two points. It turns out that there exist countable infinite chaotic invariant sets in a neighborhood of the bifocal heteroclinic cycle under some eigenvalue conditions. Moreover, the horseshoes of the corresponding Poincaré map are topologically semi-conjugated to a full shift on four symbols.