Investigation of periodic solutions of a system of ordinary differential equations with quasi-homogeneous non-linearity

The article considers a system of ordinary differential equations in which the main nonlinear part, which is a quasi-homogeneous mapping, is distinguished. The question of the existence of periodic solutions is investigated. Consideration of a quasi-homogeneous mapping allows us to generalize previously known results on the existence of periodic solutions for a system of ordinary differential equations with the main positively homogeneous non-linearity. An a priori estimate for periodic solutions is proved under the condition that the corresponding unperturbed system of equations with a quasi-homogeneous right-hand side does not have non-zero bounded solutions. Under the conditions of an a priori estimate, the following results were obtained: 1) the invariance of the existence of periodic solutions under continuous change (homotopy) of the main quasi-homogeneous non-linear part was proved; 2) the problem of homotopy classification of two-dimensional quasi-homogeneous mappings satisfying the a priori estimation condition has been solved; 3) a criterion for the existence of periodic solutions for a two-dimensional system of ordinary differential equations with the main quasi-homogeneous non-linearity is proved.