State variables scaling to solve the Malkin's problem on periodic oscillations in perturbed autonomous systems
By means of a version of the implicit function theorem for
directionally continuous functions we establish the existence,
uniqueness and the asymptotic stability of periodic solutions of a T-periodically perturbed autonomous system bifurcating from a T-periodic limit cycle of the autonomous unperturbed system (Malkin's problem). The main point of this method is the scaling of the state variables in a suitably defined map whose zeros are T-periodic solutions of the perturbed system. In order to define this map we introduce a projector by means of a convenient change of the state variables of the unperturbed system. Finally, by applying to this map the implicit function theorem mentioned before we solve the Malkin's problem without any reduction of the dimension of the state space as it is done in the literature.