Periodic solutions of periodically perturbed planar autonomous systems: A topological approach

The aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle xo of least period T-0 > 0 when it is perturbed by a small parameter, T-1-periodic perturbation. In the case when T-0/T-1 is a rational number l/k, with l, k prime numbers, we provide conditions to guarantee, for the parameter perturbation epsilon > 0 sufficiently small, the existence of klT(0)-periodic solutions x(epsilon) of the perturbed system which converge to the trajectory (x) over tilde (0) of the limit cycle as epsilon -> 0. Moreover, we state conditions under which T = klT(0) is the least period of the periodic solutions x(E). We also suggest a simple criterion which ensures that these conditions are verified. Finally, in the case when T-0/T-1 is an irrational number we show the nonexistence, whenever T > 0 and epsilon > 0, of T-periodic solutions x(E) of the perturbed system converging to (x) over tilde (0). The employed methods are based on the topological degree.