Stability Analysis Of Non-autonomous Discrete Dynamic Systems

In this paper, we systematically analyze the stability of nonautonomous dis-crete systems. A new notion called the general nonuniform (h, k,μ,v)-dichotomy for a sequence of linear operators is proposed, which occurs in a more natural way and is related to nonuniform hyperbolicity. Then, sufficient criteria are established for the existence of nonuniform (h, k,μ,v)-dichotomy in terms of appropriate Lya-punov exponents for the sequence of linear operators. Moreover, we investigate the stability theory of sequences of nonuniformly hyperbolic linear operators in Banach spaces, which admit a nonuniform (h,k,μ, v)-dichotomy. In the case of linear per-turbations, we investigate parameter dependence of robustness or roughness of the nonuniform (h, k,μ,v)-dichotomies and show that the stable and unstable subspaces of nonuniform (h, k,μ,v)-dichotomies for the linear perturbed system are Lipschitz continuous for the parameters. In the case of nonlinear perturbations, we construc-t a new version of the Grobman-Hartman theorem and explore the existence of parameter dependence of stable Lipschitz invariant manifolds when the nonlinear perturbation is of Lipschitz type.