<i>N</i> Heteroclinic Cycles Coexisting and Chaos in a 3D (N+1)-Zone Piecewise Smooth System

The Shil’nikov criterion shows that three-dimensional smooth autonomous differential equations can display chaos with the existence assumption of a heteroclinic cycle connecting saddle-focus equilibria. However, it is challenging to confirm this assumption in a given system, since effective methods are not available. Although it has been partially extended to two-zone piecewise-linear systems, there is very little work on piecewise systems with multiple discontinuous boundaries. This paper investigates, with detailed analysis, three types of heteroclinic cycles with a number of N in a new class of three-dimensional [Formula: see text]-zone nonsmooth systems for arbitrary integer [Formula: see text]. Moreover, the Shil’nikov criterion is extended to the system under consideration with theoretical analysis as well as numerical experiments, where the developed criteria in this paper apply not only to systems exhibiting saddle-focus behavior but also to those possessing both saddle and saddle-focus equilibria. In addition, the existence conditions established for chaos induced by N heteroclinic cycles in this work, generalize the previously reported case of [Formula: see text] to the case of arbitrary [Formula: see text].