Singular cycles connecting saddle periodic orbit and saddle equilibrium in piecewise smooth systems. (English) Zbl 1430.37025
Summary: For flows, the singular cycles connecting saddle periodic orbit and saddle equilibrium can potentially result in the so-called singular horseshoe, which means the existence of a non-uniformly hyperbolic chaotic invariant set. However, it is very hard to find a specific dynamical system that exhibits such singular cycles in general. In this paper, the existence of the singular cycles involving saddle periodic orbits is studied by two types of piecewise smooth systems: One is the piecewise smooth systems having an admissible saddle point with only real eigenvalues and an admissible saddle periodic orbit, and the other is the piecewise smooth systems having an admissible saddle-focus and an admissible saddle periodic orbit. Several kinds of sufficient conditions are obtained for the existence of only one heteroclinic cycle or only two heteroclinic cycles in the two types of piecewise smooth systems, respectively. In addition, some examples are presented to illustrate the results.
MSC:
| 37C27 | Periodic orbits of vector fields and flows |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 37C29 | Homoclinic and heteroclinic orbits for dynamical systems |
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