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The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non-autonomous -dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold of periodic solutions satisfying Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential systems, , when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the following differential system:
in , where is a small parameter, is a matrix having one pair of pure imaginary conjugate eigenvalues, zeros eigenvalues, and non-zero real eigenvalues.
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Jaume Llibre, Douglas Duarte Novaes, Iris de Oliveira Zeli, Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems. Rev. Mat. Iberoam. 36 (2020), no. 1, pp. 291–318DOI 10.4171/RMI/1131