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

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. • Obtain a criterion for accurately detecting coexisting homoclinic cycles (degenerate and non-degenerate) in a class of piecewise systems. • Propose existence conditions of chaos, which generalizes the Shilnikov theorem of smooth systems to non-smooth systems. • Rigorously prove the coexisting cycles possibly without symmetry can give rise to topological horseshoes, which is in sharp contrast with the smooth systems that need symmetrical homoclinic cycles to produce chaos. • The new findings and theoretical results contribute to better understanding the geometry and generation mechanism of chaotic attractors. • The analysis method in this paper is also applicable to a investigation on similar objects caused by other types of coexisting singular cycles in non-smooth dynamical systems. [ABSTRACT FROM AUTHOR]