Horseshoes in 4-dimensional piecewise affine systems with bifocal heteroclinic cycles

By studying the Poincaré map in a neighborhood of the bifocal heteroclinic cycle (the corresponding subsystems only have conjugate complex eigenvalues), this paper provides a result on the existence of chaotic invariant sets for the two-zone 4-dimensional piecewise affine systems with bifocal heteroclinic cycles that cross the switching manifold transversally at two points. Different from Shil'nikov type theorems, the existence of chaotic invariant sets near the heteroclinic cycles depends not only on the eigenvalue conditions but also on the way of intersections of the stable manifolds and unstable manifolds of the subsystems.