On the Existence of Periodic Solutions of a System of Second-Order Ordinary Differential Equations with a Quasihomogeneous Nonlinearity

In the present paper, we study an a priori estimate and the existence of periodic solutionsof a given period for a system of second-order ordinary differential equations with the mainquasihomogeneous nonlinearity. It is proved that an a priori estimate of periodic solutions takesplace if the corresponding unperturbed system does not have nonzero bounded solutions. Underthe conditions of the a priori estimate, using methods for calculating the mapping degree of vectorfields, a criterion for the existence of periodic solutions is stated and proved for any perturbationin a given class. The results obtained differ from earlier results in that the set of zeros of the mainnonlinearity is not taken into account.