Invariant tori, topological horseshoes, and their coexistence in piecewise smooth hybrid systems

This paper investigates the dynamics of tori and chaos in some classes of three-dimensional piecewise smooth hybrid systems, which consist of ordinary differential equations (ODEs) and maps. The existence of invariant tori is proved in a class of piecewise smooth hybrid systems consisting of quadratic ODEs and linear reset maps. Furthermore, it is demonstrated that the invariant torus can be filled with periodic orbits or serve as the positive limit set of a quasi-periodic orbit, depending on different conditions. Additionally, the existence of topological horseshoes is established in another class of piecewise smooth impacting hybrid systems consisting of linear ODEs and nonlinear reset maps. Finally, by appropriately coupling the aforementioned two types of piecewise smooth hybrid systems, the coexistence of tori and chaos is achieved. Several examples are provided to illustrate the main results.