ON THE EXISTENCE OF PERIODIC SOLUTIONS OF A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER WITH QUASI-HOMOGENEOUS NON-LINEARITY

In the paper was investigated the a priori estimate and the existence of periodic solutions of a fixed period for a system of second-order ordinary differential equations with the main quasi-homogeneous non-linearity. It is proved that an a priori estimate of periodic solutions takes place if the corresponding unperturbed system does not have non-zero bounded solutions. Under the conditions of an a priori estimate, using methods for calculating the mapping degree of vector fields, a criterion for the existence of periodic solutions under any perturbation from a given class is formulated and proven. The results obtained differ from earlier results in that the set of zeros of the main non-linear part is not taken into account.