Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations |
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Authors: | Oleg Makarenkov Paolo Nistri |
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Institution: | a Department of Mathematics, Voronezh State University, Voronezh, Russia b Dipartimento di Ingegneria dell' Informazione, Università di Siena, 53100 Siena, Italy |
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Abstract: | In this paper we consider a class of planar autonomous systems having an isolated limit cycle x0 of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x0 as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem. |
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Keywords: | Planar autonomous systems Limit cycles Characteristic multipliers Nonsmooth periodic perturbations Periodic solutions Topological degree |
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