Nonuniform $(h,k,\mu,\nu)$-Dichotomy with Applications to Nonautonomous Dynamical Systems

TL;DR

This paper introduces a highly general concept of nonuniform $(h,k,mu, u)$-dichotomy that encompasses many existing dichotomies, and applies it to establish robustness, Hartman-Grobman, and stable manifold theorems for nonautonomous systems.

Contribution

It develops a broad new framework for dichotomies in dynamical systems and derives key theorems within this generalized setting.

Findings

Unified framework for various dichotomies in dynamical systems

Extended robustness results for nonuniform dichotomies

New versions of Hartman-Grobman and stable manifold theorems

Abstract

The paper develops and studies a very general notion of dichotomy, referred to as "nonuniform $(h,k,\mu,\nu)$-dichotomy". The new notion contains as special cases most versions of dichotomy existing in the literature. The paper then provides corresponding new versions of robustness, Hartman-Grobman theorem, and stable manifold theorem for nonautonomous dynamical systems in Banach spaces in term of the nonuniform $(h,k,\mu,\nu)$-dichotomy.