On a Priori Estimation of Periodic Solutions to a System of Nonlinear Ordinary Differential Equations of the Second Order

The question of a priori estimation of periodic solutions to a system of nonlinear ordinary differential equations of the second order with a distinguished main positively homogeneous part is investigated. In this question, the well-known method for deriving an a priori estimation of periodic solutions for similar systems of first-order ordinary differential equations are not directly applicable. Combining these methods with the idea of qualitative research of singularly perturbed ordinary differential equations yields the conditions that provide an a priori estimation of periodic solutions to the system of second-order equations. The conditions for the a priori estimation are formulated in terms of the properties of the main positively homogeneous part of the system of equations. The existence of periodic solutions is proved to be invariant under a continuous change of the main positively homogeneous part and the conditions of a priori estimation are preserved. Based on the results, in the future, it is possible to investigate the existence of periodic solutions.