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Nelin. Dinam., 2008, Volume 4, Number 1, Pages 57–68 (Mi nd220)  

This article is cited in 6 scientific papers (total in 6 papers)

The attractors of two boundary value problems for a modifieded nonlinear telegraph equation

A. N. Kulikov

P. G. Demidov Yaroslavl State University

Abstract: Two boundary value problems for a modified nonlinear telegraph equation are considered. The problem of invariant torus bifurcation has been studied for first of them. It is shown that only the torus of a larger dimension can be asymptotically stable and also that the dimension of this attractor increases as the main bifurcation parameter decreases. The latter means that Landau's scenario of turbulence is realised in the problem under study. The existence of an infinitely dimensional attractor built up of unstable according to Lyapunov solutions has been shown for the second boundary value problem.

Keywords: attractor, bifurcation, nonlinear boundary value problems.

Full text: PDF file (143 kB)
UDC:  517.9
MSC: 35L70, 35B10, 34C23, 34C30, 34D20
Received: 14.12.2007

Citation: A. N. Kulikov, “The attractors of two boundary value problems for a modifieded nonlinear telegraph equation”, Nelin. Dinam., 4:1 (2008), 57–68

Citation in format AMSBIB
\Bibitem{Kul08}
\by A.~N.~Kulikov
\paper The attractors of two boundary value problems for a modifieded nonlinear telegraph equation
\jour Nelin. Dinam.
\yr 2008
\vol 4
\issue 1
\pages 57--68
\mathnet{http://mi.mathnet.ru/nd220}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Kulikov A.N., “Landau-Hopf Scenario of Passage to Turbulence in Some Problems of Elastic Stability Theory”, Differ. Equ., 48:9 (2012), 1258–1271  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. A. N. Kulikov, D. A. Kulikov, “Local bifurcations in the periodic boundary value problem for the generalized Kuramoto–Sivashinsky equation”, Autom. Remote Control, 78:11 (2017), 1955–1966  mathnet  crossref  isi  elib
    3. A. N. Kulikov, D. A. Kulikov, “Uravnenie Kuramoto–Sivashinskogo. Lokalnyi attraktor, zapolnennyi neustoichivymi periodicheskimi resheniyami”, Model. i analiz inform. sistem, 25:1 (2018), 92–101  mathnet  crossref  elib
    4. Kulikov A.N., Kulikov D.A., “The Kuramoto-Sivashinsky Equation. a Local Attractor Filled With Unstable Periodic Solutions”, Autom. Control Comp. Sci., 52:7 (2018), 708–713  crossref  mathscinet  isi  scopus
    5. A. N. Kulikov, “Bifurkatsii invariantnykh torov u kvazilineinykh evolyutsionnykh uravnenii vtorogo poryadka v gilbertovom prostranstve i stsenarii perekhoda k turbulentnosti”, Materialy mezhdunarodnoi konferentsii “Geometricheskie metody v teorii upravleniya i matematicheskoi fizike”, posvyaschennoi 70-letiyu S.L. Atanasyana, 70-letiyu I.S. Krasilschika, 70-letiyu A.V. Samokhina, 80-letiyu V.T. Fomenko. Ryazanskii gosudarstvennyi universitet im. S.A. Esenina, Ryazan, 25–28 sentyabrya 2018 g. Chast 1, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 168, VINITI RAN, M., 2019, 45–52  mathnet  crossref
    6. A. N. Kulikov, D. A. Kulikov, “A possibility of realizing the Landau–Hopf scenario in the problem of tube oscillations under the action of a fluid flow”, Theoret. and Math. Phys., 203:1 (2020), 501–511  mathnet  crossref  crossref  isi
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