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Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory
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Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory, 2019, Volume 168, Pages 33–44
DOI: https://doi.org/10.36535/0233-6723-2019-168-33-44
(Mi into499)
 

Bifurcations of solutions to equations with deviating spatial arguments

A. M. Kovaleva

P.G. Demidov Yaroslavl State University
References:
Abstract: A periodic boundary-value problem for an equation with deviating spatial argument is considered. This equation describes the phase of a light wave in light resonators with distributed feedback. Optical systems of this type are used in computer technologies and in the study of laser beams. The boundary-value problem was considered for two values of spatial deviations. In the work, bifurcation problems of codimensions $1$ and $2$ were analyzed by various methods of studying dynamical systems, for example, the method of normal Poincaré–Dulac forms, the method of integral manifolds, and asymptotic formulas. The problem on the stability of certain homogeneous equilibrium states is examined. Asymptotic formulas for spatially inhomogeneous solutions and conditions for their stability are obtained.
Keywords: functional-differential equation, periodic boundary-value problem, stability, bifurcation, asymptotics, light resonator.
Funding agency Grant number
Russian Foundation for Basic Research 18-01-00672
This work was supported by the Russian Foundation for Basic Research (project No. 18-01-00672).
Document Type: Article
UDC: 517.9
MSC: 34K18, 34K21, 39A28
Language: Russian
Citation: A. M. Kovaleva, “Bifurcations of solutions to equations with deviating spatial arguments”, Proceedings   of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 1, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 168, VINITI, Moscow, 2019, 33–44
Citation in format AMSBIB
\Bibitem{Kov19}
\by A.~M.~Kovaleva
\paper Bifurcations of solutions to equations with deviating spatial arguments
\inbook Proceedings   of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25--28, 2018. Part 1
\serial Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz.
\yr 2019
\vol 168
\pages 33--44
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/into499}
\crossref{https://doi.org/10.36535/0233-6723-2019-168-33-44}
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