Synchronization in lattices of coupled oscillators with various boundary conditions

where index i is the ordered pair given by i= (i1; i2), for 1≤ i1; i2 ≤ n; i; i are constants with i i ≤ 0; x; y are two vectors in Rn2 with components xi and yi, respectively, fi is a C2 function with fi(0; 0) = 0; gi(t) is a periodic function of t; (c1; c2) denote the coupling coe7cients. L is a diagonalizable operator on Rn2 with max{ i}= 0 ≤ 0, for all i ∈ (L) ≡ the spectrum of L; (Lx)i and (Ly)i denote the ith components of Lx and Ly, respectively. If c1 =c2 =0 then system (1.1) represents the motion of the ith oscillator in an n×n uncoupled squared lattice. The individual dynamics for the ith node in the lattice is