  In this paper, we investigate the existence of multiple periodic solutions
for two classes of nonlinear difference systems involving
$(\phi_1,\phi_2)$-Laplacian. First, by using an important critical point
theorem due to B. Ricceri, we establish an existence theorem of three periodic
solutions for the first nonlinear difference system with
$(\phi_1,\phi_2)$-Laplacian and two parameters. Moreover, for the second
nonlinear difference system with $(\phi_1,\phi_2)$-Laplacian, by using the
Clark's Theorem, we obtain a multiplicity result of periodic solutions under a
symmetric condition. Finally, two examples are given to verify our theorems.
