Geodesic
Investigation of periodic solutions of a system of ordinary differential equations with quasi-homogeneous non-linearity
Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 147, pp. 309-324
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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.
Keywords: quasi-homogeneous non-linearity, periodic solution, a priori estimate, invariance of the existence of periodic solutions, the mapping degree of a vector field 
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A. N. Naimov; M. V. Bystretskii. Investigation of periodic solutions of a system of ordinary differential equations with quasi-homogeneous non-linearity. Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 147, pp. 309-324. http://geodesic.mathdoc.fr/item/VTAMU_2024_29_147_a5/

[1] E. Mukhamadiev, “On the theory of periodic solutions of systems of ordinary differential equations”, Sov. Math., Dokl., 11 (1970), 1236–1239

[2] E. Mukhamadiev, A. N. Naimov, “On the solvability of a periodic problem for a system of ordinary differential equations with the main positive homogeneous nonlinearity”, Differential Equations, 59:2 (2023), 289–291 | DOI | DOI

[3] M. A. Krasnosel'skiy, P. P. Zabreiko, Geometric Methods of Nonlinear Analysis, Nauka Publ., Moscow, 1975 (In Russian)

[4] J. L. Lyons, Some Methods for Solving Nonlinear Boundary Value Problems, Nauka Publ., Moscow, 1972 (In Russian)

[5] V. G. Zvyagin, S. V. Kornev, “Method of guiding functions for existence problems for periodic solutions of differential equations”, Journal of Mathematical Sciences, 233:4 (2018), 578–601 | DOI

[6] A. I. Perov, V. K. Kaverina, “On a problem posed by Vladimir Ivanovich Zubov”, Differential Equations, 55:2 (2019), 274–278 | DOI | DOI

[7] E. Mukhamadiev, A. N. Naimov, M. M. Kobilzoda, “Solvability of a class of periodic problems on the plane”, Differential Equations, 57:2 (2021), 189–195 | DOI | DOI

[8] N. A. Bobylev, “The construction of regular guiding functions”, Sov. Math., Dokl., 9 (1968), 1353–1355