Generalized nonuniform dichotomies and local stable manifolds

TL;DR

This paper proves the existence of local stable manifolds for nonlinear perturbations of linear differential equations in Banach spaces, using a broad class of nonuniform dichotomies that include exponential and polynomial cases, even when classical Lyapunov exponents are zero.

Contribution

It introduces a generalized framework for nonuniform dichotomies, extending stable manifold theory to more complex and less restrictive dynamical systems.

Findings

Established local stable manifolds under broad nonuniform dichotomy conditions

Included cases with zero Lyapunov exponents

Provided new applications and analyzed perturbation effects

Abstract

We establish the existence of local stable manifolds for semiflows generated by nonlinear perturbations of nonautonomous ordinary linear differential equations in Banach spaces, assuming the existence of a general type of nonuniform dichotomy for the evolution operator that contains the nonuniform exponential and polynomial dichotomies as a very particular case. The family of dichotomies considered allow situations for which the classical Lyapunov exponents are zero. Additionally, we give new examples of application of our stable manifold theorem and study the behavior of the dynamics under perturbations.