Abstract
In the last few years, Battelli and Fečkan have developed a functional analytic method to rigorously prove the existence of chaotic behaviors in time-perturbed piecewise smooth systems whose unperturbed part has a piecewise continuous homoclinic solution. In this paper, by applying their method, we study the appearance of chaos in time-perturbed piecewise smooth systems with discontinuities on finitely many switching manifolds whose unperturbed part has a hyperbolic saddle in each subregion and a heteroclinic orbit connecting those saddles that crosses every switching manifold transversally exactly once. We obtain a set of Melnikov type functions whose zeros correspond to the occurrence of chaos of the system. Furthermore, the Melnikov functions for planar piecewise smooth systems are explicitly given. As an application, we present an example of quasiperiodically excited three-dimensional piecewise linear system with four zones.
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