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A Topological–Analytical Method for Proving Averaging Theorems on an Infinite Time Interval in a Degenerate Case

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Abstract

We present a topological–analytical method for proving some results of the N. N. Bogolyubov averaging method for the case of an infinite time interval. The essence of the method is to combine topological methods of proving the existence of a periodic solution applied to the averaged system with Bogolyubov’s theorem on the averaging on a finite time interval. The proposed approach allows us to dispense with the nondegeneracy condition for the Jacobi matrix from the classical theorems of the averaging method.

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Funding

The work was supported by the Russian Science Foundation under grant no. 21-71-30011, https://rscf.ru/project/21-71-30011/, and performed at the P. G. Demidov Yaroslavl State University.

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Ivan Yu. Polekhin

  2. Moscow Institute of Physics and Technology (National Research University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701, Russia

    Ivan Yu. Polekhin

  3. Lomonosov Moscow State University, Moscow, 119991, Russia

    Ivan Yu. Polekhin

  4. P. G. Demidov Yaroslavl State University, Sovetskaya ul. 14, Yaroslavl, 150003, Russia

    Ivan Yu. Polekhin

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  1. Ivan Yu. Polekhin
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Correspondence to Ivan Yu. Polekhin.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Vol. 322, pp. 195–205 https://doi.org/10.4213/tm4345.

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Polekhin, I.Y. A Topological–Analytical Method for Proving Averaging Theorems on an Infinite Time Interval in a Degenerate Case. Proc. Steklov Inst. Math. 322, 188–197 (2023). https://doi.org/10.1134/S0081543823040168

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  • DOI: https://doi.org/10.1134/S0081543823040168

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