  Aim of this paper is to investigate the existence of periodic solutions of a
nonlinear planar autonomous system having a limit cycle x_0 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 e>0
sufficiently small, the existence of klT_0-periodic solutions x_e of the
perturbed system which converge to the trajectory x_1 of the limit cycle as
e->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 e>0, of T-periodic
solutions x_e of the perturbed system converging to x_1. The employed methods
are based on the topological degree theory.
