A second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter

Abstract

We deal with nonlinear T–periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide the expressions of the bifurcation functions up to second order in the small parameter in order that their simple zeros are initial values of the periodic solutions that persist after the perturbation. In the end two applications are done. The key tool for proving the main result is the Lyapunov–Schmidt reduction method applied to the T–Poincaré–Andronov mapping.

The first author was also partially supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PNII- ID-PCE-2011-3-0094. The first and second authors are partially supported by the MICINN/FEDER grant number MTM2008–00694 and by a Generalitat de Catalunya grant number 2009SGR–381. The third author is partially supported by the MICINN/FEDER grant MTM2008–03437, Generalitat de Catalunya grant number 2009SGR–410 and ICREA Academia.