Nonuniform $(h,k,\mu,\nu)$-dichotomy and Stability of Nonautonomous Discrete Dynamics

TL;DR

This paper introduces a new concept of nonuniform $(h,k,, u)$-dichotomy for linear operators, providing criteria for its existence, analyzing stability in Banach spaces, and exploring robustness and invariant manifolds under perturbations.

Contribution

It proposes the general nonuniform $(h,k,, u)$-dichotomy, establishes criteria for its existence, and studies stability, robustness, and invariant manifolds in nonautonomous discrete dynamics.

Findings

Established criteria for nonuniform $(h,k,, u)$-dichotomy existence.

Proved Lipschitz continuity of stable/unstable subspaces under perturbations.

Extended Grobman-Hartman theorem for nonlinear perturbations.

Abstract

In this paper, a new notion called the general nonuniform $(h,k,\mu,\nu)$-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 $(h,k,\mu,\nu)$-dichotomy in terms of appropriate Lyapunov exponents for the sequence of linear operators. Moreover, we investigate the stability theory of sequences of non uniformly hyperbolic linear operators in Banach spaces, which admit a nonuniform $(h,k,\mu,\nu)$-dichotomy. In the case of linear perturbations, we investigate parameter dependence of robustness or roughness of the nonuniform $(h,k,\mu,\nu)$-dichotomies and show that the stable and unstable subspaces of nonuniform $(h,k,\mu,\nu)$-dichotomies for the linear perturbed system are Lipschitz continuous for the parameters. In the case of nonlinear perturbations, we construct 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.