Abstract
We study a normal periodic system of ordinary differential equations with a small parameter, which is quasilinear in a neighborhood of infinity, under the assumption that the right-hand side of the system has a critical linear approximation. In terms of the properties of the first homogeneous nonlinear approximation of the monodromy operator, we obtain conditions for the existence of a periodic solution whose initial value is infinitely large for an infinitesimal value of the parameter.
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V. V. Abramov, “Stability of a small periodic solution,” Vestn. Ross. Akad. Estestv. Nauk, 13, No. 4, 3–5 (2013).
V. V. Abramov, “Branching of periodic solutions with positive initial values,” Vestn. Ross. Akad. Estestv. Nauk, 17, No. 4, 4–7 (2017).
M. A. Krasnosel’sky, Operator of Shift along Trajectories of Differential Equations, Nauka, Moscow (1966).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 168, Proceedings of the International Conference “Geometric Methods in the Control Theory and Mathematical Physics” Dedicated to the 70th Anniversary of Prof. S. L. Atanasyan, 70th Anniversary of Prof. I. S. Krasil’shchik, 70th Anniversary of Prof. A. V. Samokhin, and 80th Anniversary of Prof. V. T. Fomenko. Ryazan State University named for S. Yesenin, Ryazan, September 25–28, 2018. Part I, 2019.
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Abramov, V.V. On the Branching of a Large Periodic Solution of a System of Differential Equations with a Parameter. J Math Sci 262, 767–772 (2022). https://doi.org/10.1007/s10958-022-05854-4
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DOI: https://doi.org/10.1007/s10958-022-05854-4