In this paper, we consider the existence of periodic solutions for a class of nonlinear difference systems involving classical $({\varphi }_1,{\varphi }_2)$-Laplacian. By using the least action principle, we obtain that the system with classical $({\varphi }_1,{\varphi }_2)$-Laplacian has at least one periodic solution when potential function is $(p,q)$-sublinear growth condition, subconvex condition. The results obtained generalize and extend some known works.