COMMUNICATIONS ON Website: http://AIMsciences.org PURE AND APPLIED ANALYSIS Volume 6, Number 1, March 2007 pp. 103–111 PERIODIC SOLUTIONS OF NONLINEAR PERIODIC DIFFERENTIAL SYSTEMS WITH A SMALL PARAMETER Adriana Buic˘ a Department of Applied Mathematics, Babe¸ s-Bolyai University 1 Kog˘ alniceanu str., Cluj-Napoca, 400084, Romania Jean–Pierre Franc ¸oise Laboratoire J.-L. Lions, Universit´ e P.-M. Curie, Paris 6 UMR 7598, CNRS, Paris, France Jaume Llibre Departament de Matem` atiques, Universitat Aut` onoma de Barcelona 08193 Bellaterra, Barcelona, Spain (Communicated by Carmen Chicone) Abstract. 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 Poincar´ e–Andronov mapping. 1. Introduction. We consider the problem of bifurcation of T –periodic solutions for a differential system of the form, x ′ (t)= F 0 (t, x)+ εF 1 (t, x)+ ε 2 R(t, x, ε), (1) where ε is a small parameter, F 0 ,F 1 : R × Ω → R n and R : R × Ω × (−ε f ,ε f ) → R n are C 2 functions, T –periodic in the first variable, and Ω is an open subset of R n . One of the main hypotheses is that the unperturbed system x ′ (t)= F 0 (t, x), (2) has a manifold of periodic solutions. This problem was solved before by Malkin (1956) and Roseau (1966) (see [4]). We will give here a new and shorter proof (see Theorem 3.1 and its proof). In addition, we will give a series of corollaries in some particular cases. In order to describe these cases we introduce some notation. We denote the projection onto the first k coordinates by π : R k × R n−k → R k and the one onto the last (n − k) coordinates by π ⊥ : R k × R n−k → R n−k . For the 2000 Mathematics Subject Classification. Primary: 34C29, 34C25; Secondary: 58F22. Key words and phrases. Periodic solution, averaging method, Lyapunov–Schmidt reduction. A. Buic˘ a is supported by the Agence Universitaire de la Francophonie and J. Llibre is par- tially supported by a DGICYT grant number MTM2005-06098-C02-01 and by a CICYT grant number 2005SGR 00550. This joint work took place while J.-P. Fran¸ coise was visiting the CRM in Barcelona. All authors express their gratitude to the CRM for providing very nice working conditions. 103