On the Solvability of a Periodic Problem for a Two-dimensional System of Ordinary Differential Equations of the Second Order

Abstract

In this paper we study the periodic problem with a period equal to 1 for a two-dimensional system of second-order ordinary differential equations, in which the main nonlinear part is generated by a polynomial in one complex variable. It is proven that if the convex hull of the roots of the generating polynomial does not contain numbers that are multiples of 2 i*pi, then there is an a priori estimate for solutions to the periodic problem. Under the conditions of an a priori estimate, using methods for calculating the mapping degree of vector fields, the solvability of the periodic problem for any perturbation from a given class is proven. The system of equations under consideration does not reduce to a similar system of first-order equations with the main positive homogeneous nonlinear part. For systems of first-order equations, the periodic problem was studied in the works of V.A. Pliss, M.A. Krasnoselskii and their followers using methods of a priori estimation and calculation of the mapping degree of vector fields. It is known that an a priori estimate of solutions to boundary value problems for systems of nonlinear ordinary second order differential equations is fraught with difficulties associated with an estimate of the first-order derivative of the solution when the solution itself is bounded. In this paper, using the example of a periodic problem for the considered system of second-order equations, it is established that the a priori estimate is deducible if we combine methods for studying similar systems of first-order equations and methods for qualitative research of singularly perturbed systems of equations. The results obtained can be further generalized for multidimensional systems of second-order equations, applying the idea of the directing function method.