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
@article{VTAMU_2024_29_147_a5,
author = {A. N. Naimov and M. V. Bystretskii},
title = {Investigation of periodic solutions of a system of ordinary differential equations with quasi-homogeneous non-linearity},
journal = {Vestnik rossijskih universitetov. Matematika},
pages = {309--324},
publisher = {mathdoc},
volume = {29},
number = {147},
year = {2024},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VTAMU_2024_29_147_a5/}
}
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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/