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Izv. RAN. Ser. Mat., 2011, Volume 75, Issue 3, Pages 97–126 (Mi izv4260)  

Multifrequency self-oscillations in two-dimensional lattices of coupled oscillators

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: We consider a two-dimensional lattice of coupled van der Pol oscillators obtained under a standard spatial discretization of the non-linear wave equation $u_{tt}+\varepsilon(u^2-1)u_{t}+u= a_1^2u_{xx}+a_2^2u_{yy}$, $a_1,a_2=\mathrm{const}>0$, $0<\varepsilon\ll 1$, on the unit square with the zero Dirichlet or Neumann boundary conditions. We shall prove that the corresponding system of ordinary differential equations has attractors admitting no analogues in the original boundary-value problem. These attractors are stable invariant tori of various dimensions. We also show that the number of these tori grows unboundedly as the number of equations in the lattice is increased.

Keywords: wave equation, discretization, self-oscillation, attractor, invariant torus, lattice of coupled oscillators, buffer property.

DOI: https://doi.org/10.4213/im4260

Full text: PDF file (1036 kB)
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English version:
Izvestiya: Mathematics, 2011, 75:3, 539–567

Bibliographic databases:

UDC: 517.926
MSC: Primary 35B41; Secondary 37B25, 37D45, 37L30, 65N22
Received: 17.11.2009

Citation: A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Multifrequency self-oscillations in two-dimensional lattices of coupled oscillators”, Izv. RAN. Ser. Mat., 75:3 (2011), 97–126; Izv. Math., 75:3 (2011), 539–567

Citation in format AMSBIB
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