Chaos emerges from coexisting homoclinic cycles for a class of 3D piecewise systems

Abstract

Accurately detecting a homoclinic cycle and chaos in a concrete system takes a huge challenge. This paper gives a sufficient condition of coexisting homoclinic cycles (degenerate) with respect to a saddle-focus in a class of three-dimensional piecewise systems with two switching manifolds. Furthermore, by constructing the Poincaré map, it is rigorously shown that chaotic behavior is induced by the coexisting cycles without necessary symmetry in the considered system, which is evidently different from the usual dynamics in smooth system theory that a pair of symmetrical homoclinic cycles can generate chaos but the asymmetric ones cannot. It finally provides an example to illustrate the effectiveness of the results established.