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The existence of center manifolds is closedly related to local behavior of dynamical systems. In this paper we consider center manifolds both of the discrete Toda equation and the discrete Lotka-Volterra system. Their solutions converge to eigenvalues anti singular values of certain structured matrices. A free parameter plays a key role to show the existence of a center manifold of the discrete Lotka-Volterra system. A monotone convergence of the solution of the discrete Lotka-volterra system is proved with the help of the existence of a center manifold. In contrast, a center manifold of the discrete Toda equation does not always exist.

Center manifold approach to discrete integrable systems related to eigenvalues and singular values