Chaos and Homoclinic Cycles in Piecewise Nonlinear Systems

Although the classical Shilnikov criterion for chaos has been partially extended to piecewise linear systems, research on piecewise nonlinear systems remains remarkably scarce, even for those with specific structures. This paper investigates a class of three-dimensional piecewise nonlinear systems characterized by a switching mechanism between two autonomous differential equations. By employing coordinate transformations, we derive existence conditions for homoclinic cycles to a saddle-focus equilibrium within well-defined regions of the nonlinear subsystems. Furthermore, the chaotic mechanism is elucidated through an analysis of the properties of the Poincaré return map, constructed via Taylor series expansion techniques. Building upon these theoretical results, we propose a systematic approach for the construction of chaotic systems governed by nonlinear dynamics.