ar X iv : 0 70 9 . 46 43 v 1 [ m at h . C A ] 2 8 Se p 20 07 PERIODIC SOLUTIONS OF PERIODICALLY PERTURBED PLANAR AUTONOMOUS SYSTEMS : A TOPOLOGICAL APPROACH
نویسنده
چکیده
Aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle x0 of least period T0 > 0 when it is perturbed by a small parameter, T1−periodic, perturbation. In the case when T0/T1 is a rational number l/k, with l, k prime numbers, we provide conditions to guarantee, for the parameter perturbation ε > 0 sufficiently small, the existence of klT0− periodic solutions xε of the perturbed system which converge to the trajectory x̃0 of the limit cycle as ε → 0. Moreover, we state conditions under which T = klT0 is the least period of the periodic solutions xε. We also suggest a simple criterion which ensures that these conditions are verified. Finally, in the case when T0/T1 is an irrational number we show the nonexistence, whenever T > 0 and ε > 0, of T−periodic solutions xε of the perturbed system converging to x̃0. The employed methods are based on the topological degree theory.
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