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

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.