58 research outputs found
Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations
In this paper we consider a class of planar autonomous systems having an
isolated limit cycle x_0 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 x_0 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.Comment: To appear in J. Math. Anal. App
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