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ISSN (Print) 2079-6900

ISSN (Online) 2587-7496

(Middle Volga Mathematical Society Journal)

Scientific Journal

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Review of the works of V. N. Shchennikova on the study of the convergence of nonlinear almost periodic systems by the comparison method

A. A. Kosov1, A. V. Shchennikov2, E. V. Shchennikova3, R. V. Zhalnin4, P. A. Shamanaev5

AnnotationThe article provides an overview of the studies of V. N. Shchennikov on the problems of almost periodic convergence of nonlinear differential equations' systems. The problem of convergence established by linear or homogeneous approximation is considered. The conditions for convergence of complex systems are given, that are obtained by constructing Lyapunov vector functions and using the comparison method. It should be noted that in the course of the proof constructive estimates are made for the values of small parameters and interconnection functions. The dimensions of the region in which the limiting almost periodic mode is located are also specified. As an application, the problem of convergence in an electric circuit modeled by a second-order nonlinear differential equation with a small parameter is considered. In conclusion, possible applications and unsolved problems for new directions of research, on which V. N. Shchennikov worked in recent years, are discussed.
Keywordsconvergence, almost periodic solutions, Lyapunov vector function, nonlinear system of ordinary differential equations

1Alexander A. Kosov, Leading researcher, Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences (134, Lermontov St., Irkutsk 664033, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0003-1352-1828, kosov_idstu@mail.ru

2Aleksey V. Shchennikov, applicant of the Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Ogarev Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), ORCID: http://orcid.org/0000-0002-4734-1553, shennikov.aleksey@yandex.ru

3Elena V. Shchennikova, Professor of the Department of Fundamental Informatics, National Research Ogarev Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), D.Sc. (Physics and Mathematics), ORCID: http: //orcid.org/0000-0001-5989-3550, schennikova8000@yandex.ru

4Ruslan V. Zhalnin, Head of Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Ogarev Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-1103-3321, zhrv@mrsu.ru

5Pavel A. Shamanaev, Associate Professor, Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Ogarev Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-0135-317X, korspa@yandex.ru

Citation: A. A. Kosov, A. V. Shchennikov, E. V. Shchennikova, R. V. Zhalnin, P. A. Shamanaev, "[Review of the works of V. N. Shchennikova on the study of the convergence of nonlinear almost periodic systems by the comparison method]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,21:2 (2019) 175–186 (In Russian)