Limit cycles via higher order perturbations for some piecewise differential systems

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Buzzi, Claudio A. [UNESP]
Silva Lima, Mauricio Firmino
Torregrosa, Joan
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Elsevier B.V.
A classical perturbation problem is the polynomial perturbation of the harmonic oscillator, (x', y') = (-y + epsilon f(x, y, epsilon), x + epsilon g(x, y, epsilon)). In this paper we study the limit cycles that bifurcate from the period annulus via piecewise polynomial perturbations in two zones separated by a straight line. We prove that, for polynomial perturbations of degree n, no more than Nn-1 limit cycles appear up to a study of order N. We also show that this upper bound is reached for orders one and two. Moreover, we study this problem in some classes of piecewise Lienard differential systems providing better upper bounds for higher order perturbation in 8, showing also when they are reached. The Poincare-Pontryagin-Melnikov theory is the main technique used to prove all the results. (C) 2018 Elsevier B.V. All rights reserved.
Non-smooth differential system, Limit cycle in Melnikov higher order perturbation, Lienard piecewise differential system
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Physica D-nonlinear Phenomena. Amsterdam: Elsevier Science Bv, v. 371, p. 28-47, 2018.