Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones

This work is devoted to study the existence of periodic solutions for a class of s-family of discontinuous differential systems with many zones. We show that the averaged functions at any order control the existence of crossing limit cycles for systems in this class. We also provide some examples dealing with nonsmooth perturbations of nonlinear centers. (C) 2017 Elsevier B.V. All rights reserved. (AU)