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March  2007, 6(1): 103-111. doi: 10.3934/cpaa.2007.6.103

Periodic solutions of nonlinear periodic differential systems with a small parameter

Adriana Buică 1, , Jean–Pierre Françoise 2, and  Jaume Llibre 3,

1. 

Department of Applied Mathematics, Babeş-Bolyai University, 1 Kogălniceanu str., Cluj-Napoca, 400084, Romania
2. 

Laboratoire J.-L. Lions, Université P.-M. Curie, Paris 6, UMR 7598, CNRS, Paris, France
3. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia

Received  April 2006 Revised  August 2006 Published  December 2006

We deal with nonlinear periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide sufficient conditions in order that some of the periodic orbits of this invariant manifold persist after the perturbation. These conditions are not difficult to check, as we show in some applications. The key tool for proving the main result is the Lyapunov--Schmidt reduction method applied to the Poincaré--Andronov mapping.
Keywords: Periodic solution, Lyapunov--Schmidt reduction., averaging method.
Mathematics Subject Classification: Primary: 34C29, 34C25; Secondary: 58F2.
Citation: Adriana Buică, Jean–Pierre Françoise, Jaume Llibre. Periodic solutions of nonlinear periodic differential systems with a small parameter. Communications on Pure and Applied Analysis, 2007, 6 (1) : 103-111. doi: 10.3934/cpaa.2007.6.103
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