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 Tr. Mat. Inst. Steklova, 2007, Volume 256, Pages 219–236 (Mi tm463)

Attractors of the Sine-Gordon Equation in the Field of a Quasiperiodic External Force

A. Yu. Kolesova, E. F. Mishchenkob, N. Kh. Rozovc

a P. G. Demidov Yaroslavl State University
b Steklov Mathematical Institute, Russian Academy of Sciences
c M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The well-known sine-Gordon equation, supplemented with small damping and small quasiperiodic external force, is studied under the zero Dirichlet boundary conditions at the endpoints of a finite interval. The main assumption is that all frequencies of the external force are in $1:1$ resonance with certain eigenfrequencies of the unperturbed equation; i.e., the so-called fundamental multifrequency resonance is observed. It is shown that in this case, by an appropriate choice of the parameters of the external force, one can make it so that the boundary value problem has a stable invariant torus of any finite dimension that bifurcates from zero on any preassigned finite set of spatial modes. It is also shown (by numerical analysis) that in a number of cases the above-mentioned torus coexists with a chaotic attractor.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2007, 256, 206–222

Bibliographic databases:

UDC: 517.926

Citation: A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Attractors of the Sine-Gordon Equation in the Field of a Quasiperiodic External Force”, Dynamical systems and optimization, Collected papers. Dedicated to the 70th birthday of academician Dmitrii Viktorovich Anosov, Tr. Mat. Inst. Steklova, 256, Nauka, MAIK «Nauka/Inteperiodika», M., 2007, 219–236; Proc. Steklov Inst. Math., 256 (2007), 206–222

Citation in format AMSBIB
\Bibitem{KolMisRoz07} \by A.~Yu.~Kolesov, E.~F.~Mishchenko, N.~Kh.~Rozov \paper Attractors of the Sine-Gordon Equation in the Field of a Quasiperiodic External Force \inbook Dynamical systems and optimization \bookinfo Collected papers. Dedicated to the 70th birthday of academician Dmitrii Viktorovich Anosov \serial Tr. Mat. Inst. Steklova \yr 2007 \vol 256 \pages 219--236 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm463} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2336901} \zmath{https://zbmath.org/?q=an:1153.37440} \elib{http://elibrary.ru/item.asp?id=9482616} \transl \jour Proc. Steklov Inst. Math. \yr 2007 \vol 256 \pages 206--222 \crossref{https://doi.org/10.1134/S0081543807010117} \elib{http://elibrary.ru/item.asp?id=13546896} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34248393909}