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

Abstract

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