Asymptotic synchronization in n-dimensional second order dissipative lattices of coupled oscillators

By introducing a new norm which is equivalent to the usual norm in the phase space, we prove that for n-dimensional second order dissipative lattices of coupled oscillators with external periodic forces under Dirichlet, Neumann and periodic boundary conditions, if the system is bounded dissipative and the coupled coefficients are both large enough, the asymptotic synchronization will occur. And we give a concrete bounded dissipative second order lattices system. Our results show that the bounds of the difference between the components of any solution are directly proportional to mn/2 and inversely proportional to the coupled coefficients, where m is the mesh size and n is the space dimension of lattice points.