Existence of homoclinic orbits and heteroclinic cycle in a class of three-dimensional piecewise linear systems with three switching manifolds

Abstract

In this article, we construct a kind of three-dimensional piecewise linear (PWL) system with three switching manifolds and obtain four theorems with regard to the existence of a homoclinic orbit and a heteroclinic cycle in this class of PWL system. The first theorem studies the existence of a heteroclinic cycle connecting two saddle-foci. The existence of a homoclinic orbit connecting one saddle-focus is investigated in the second theorem, and the third theorem examines the existence of a homoclinic orbit connecting another saddle-focus. The last one proves the coexistence of the heteroclinic cycle and two homoclinic orbits for the same parameters. Numerical simulations are given as examples and the results are consistent with the predictions of theorems. From the Shil’nikov theorems, it is known that the existence of a homoclinic orbit and a heteroclinic cycle play a key and important role in the chaos research of dynamic systems. Moreover, chaos and piecewise linear (PWL) systems have important applications in many fields such as electronic circuits, biology, machinery, and so on. Naturally, the research on the existence of homoclinic orbits and heteroclinic cycles of piecewise linear systems is very meaningful. However, it is not easy to find the homoclinic orbit and the heteroclinic cycle of smooth dynamic systems, especially for higher-dimensional piecewise linear systems with multiple switching manifolds, which will inevitably make the exploration of this problem more complicated. Therefore, this article studies the existence of homoclinic orbits and heteroclinic cycles in a style of the piecewise linear system with three switching manifolds.