Geodesic
Investigation of periodic solutions of a two-dimensional system of nonlinear ordinary second-order differential equations
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXV", Voronezh, April 26-30, 2024, Part 1, Tome 235 (2024), pp. 87-96.

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The problem of a priori estimate and existence of periodic solutions for a two-dimensional system of nonlinear ordinary second-order differential equations is examined. In terms of the properties of the principal nonlinear part, a theorem on a priori estimate of periodic solutions is formulated and proved. Under the conditions of the a priori estimate, a theorem on necessary and sufficient conditions for the existence of periodic solutions is proved.
Keywords: periodic solution, positive homogeneous mapping, a priori estimate, vector field, degree of a vector field, homotopic pairs of mappings 
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A. N. Naimov; M. V. Bystretskii. Investigation of periodic solutions of a two-dimensional system of nonlinear ordinary second-order differential equations. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXV", Voronezh, April 26-30, 2024, Part 1, Tome 235 (2024), pp. 87-96. http://geodesic.mathdoc.fr/item/INTO_2024_235_a7/

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[2] Mukhamadiev E., Naimov A. N., “O razreshimosti periodicheskoi zadachi dlya sistemy nelineinykh obyknovennykh differentsialnykh uravnenii vtorogo poryadka”, Differ. uravn., 60:3 (2024), 312–321 | DOI

[3] Mukhamadiev E., Naimov A. N., Kobilzoda M. M., “O razreshimosti odnogo klassa periodicheskikh zadach na ploskosti”, Differ. uravn., 57:2 (2021), 203–209 | DOI | MR | Zbl