title: On the Solvability of a Periodic Problem for a Two-dimensional System of Ordinary Differential Equations of the Second Order
metadata.name: dc.contributor.author
metadata.value: Mukhamadiev, Ergashboy
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Naimov, Alizhon Nabidjanovich
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2024-12-02T13:35:14Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2024-12-02T13:35:14Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2024-06
metadata.lang: -
metadata.name: dc.identifier.other
metadata.value: https://doi.org/10.21638/11701/spbu35.2024.204
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/48048
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: 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.
metadata.lang: en_GB
metadata.name: dc.description.sponsorship
metadata.value: The research was supported by the grant Russian
  Science Foundation No. 23-21-00032 (https://rscf.ru/project/23-21-00032/).
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Differential Equations and Control Processes;Issue 1
metadata.lang: -
metadata.name: dc.subject
metadata.value: periodic problem
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: a priori estimate
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: the mapping degree of vector field
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: On the Solvability of a Periodic Problem for a Two-dimensional System of Ordinary Differential Equations of the Second Order
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 24204-jdecp-naimov.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/48048/1/24204-jdecp-naimov.pdf
resources.format: Adobe PDF
resources.size: 222,33 kB