Geodesic
Attractors of the Sine-Gordon Equation in the Field of a Quasiperiodic External Force
Informatics and Automation, Dynamical systems and optimization, Tome 256 (2007), pp. 219-236.

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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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A. Yu. Kolesov; E. F. Mishchenko; N. Kh. Rozov. Attractors of the Sine-Gordon Equation in the Field of a Quasiperiodic External Force. Informatics and Automation, Dynamical systems and optimization, Tome 256 (2007), pp. 219-236. http://geodesic.mathdoc.fr/item/TRSPY_2007_256_a10/

[1] Uizem Dzh., Lineinye i nelineinye volny, Mir, M., 1977

[2] Kolesov A.Yu., Rozov N.Kh., “Mnogochastotnyi parametricheskii rezonans v nelineinom volnovom uravnenii”, Izv. RAN. Ser. mat., 66:6 (2002), 49–64 | MR | Zbl

[3] Kolesov A.Yu., Rozov N.Kh., Invariantnye tory nelineinykh volnovykh uravnenii, Fizmatlit, M., 2004

[4] Kulikov A.N., “O bifurkatsiyakh rozhdeniya invariantnykh torov”, Issledovaniya po ustoichivosti i teorii kolebanii, YarGU, Yaroslavl, 1983, 112–117 | MR

[5] Bibikov Yu.N., Mnogochastotnye nelineinye kolebaniya i ikh bifurkatsii, Izd-vo LGU, L., 1991 | MR

[6] Samoilenko A.M., Elementy matematicheskoi teorii mnogochastotnykh kolebanii. Invariantnye tory, Nauka, M., 1987 | MR

[7] Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L., Metody kachestvennoi teorii v nelineinoi dinamike, ch. 1, In-t kompyut. issled., Moskva–Izhevsk, 2004

[8] Mischenko E.F., Sadovnichii V.A., Kolesov A.Yu., Rozov N.Kh., Avtovolnovye protsessy v nelineinykh sredakh s diffuziei, Fizmatlit, M., 2005

[9] http://tracer3.narod.ru

[10] http://www.math.rsu.ru/mexmat/kvm/matds

[11] Kuznetsov S.P., Dinamicheskii khaos, Kurs lektsii, Fizmatlit, M., 2001