Heteroclinic cycles in a new class of four-dimensional discontinuous piecewise affine systems

Abstract

It is a huge challenge to give an existence theorem for heteroclinic cycles in the high-dimensional discontinuous piecewise systems (DPSs). This paper first provides a new class of four-dimensional (4D) two-zone discontinuous piecewise affine systems (DPASs), and then gives a useful criterion to ensure the existence of heteroclinic cycles in the systems by rigorous mathematical analysis. To illustrate the feasibility and efficiency of the theory, two numerical examples, exhibiting chaotic behaviors in a small neighborhood of heteroclinic cycles, are discussed.

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

“…For PWL systems with low-dimensional or less switching manifolds, great progress has been made in the study of the existence of homoclinic orbits or heteroclinic cycles. [19][20][21][22][23][24][25][26] Some researchers have also studied the existence of homoclinic orbits, heteroclinic cycles, and chaos in higher-dimensional PWL systems, and obtained some results [27][28][29][30][31] by constructing a Poincaré map and proving the existence of topological horseshoes.…”