Singular cycles connecting saddle periodic orbit and saddle equilibrium in piecewise smooth systems

TL;DR

This paper investigates the existence of singular cycles connecting saddle periodic orbits and saddle equilibria in piecewise affine systems, providing conditions for heteroclinic cycles and illustrating with examples.

Contribution

It offers new sufficient conditions for the existence of heteroclinic cycles in specific piecewise affine systems with saddle points and saddle-focus equilibria.

Findings

Conditions for one heteroclinic cycle in saddle point systems

Conditions for two heteroclinic cycles in saddle-focus systems

Examples illustrating the theoretical results

Abstract

For flows, the singular cycles connecting saddle periodic orbit and saddle equilibrium can poten- tially 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 involved in saddle periodic orbits is studied by two types of piecewise affine systems: one is the piecewise affine system having an admissible saddle point with only real eigenvalues and an admissible saddle periodic orbit, and the other is the piecewise affine system having an admissible saddle- focus and an admissible saddle periodic orbit. Precisely, 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 affine systems, respectively. In addition, some examples are presented to illustrate the results.