Small parameter perturbations of nonlinear periodic systems**Supported by the National Research Project MIUR: ‘Feedback control and optimal control’, by RFBR grants 02-01-00189, 02-01-00307, 03-01-06308 and USCRDF—RF Ministry of Education grant VZ-010.

Abstract

In this paper we consider a class of nonlinear periodic differential systems perturbed by two nonlinear periodic terms with multiplicative different powers of a small parameter ε > 0. For such a class of systems we provide conditions that guarantee the existence of periodic solutions of given period T > 0. These conditions are expressed in terms of the behaviour on the boundary of an open bounded set U of of the solutions of suitably defined linearized systems. The approach is based on the classical theory of the topological degree for compact vector fields. An application to the existence of periodic solutions to the van der Pol equation is also presented.