Nonuniform dichotomic behavior: Lipschitz invariant manifolds for ODEs

TL;DR

This paper establishes new theorems on the existence of Lipschitz invariant manifolds for perturbed nonautonomous linear ODEs with nonuniform dichotomic behavior, extending classical hyperbolic results.

Contribution

It introduces global and local invariant manifold theorems under very general dichotomic conditions, broadening the scope beyond hyperbolic cases and improving existing stable manifold results.

Findings

New invariant manifold theorems for nonuniform dichotomic systems

Extension of stable manifold results to broader nonhyperbolic settings

Inclusion of cases far from classical hyperbolic assumptions

Abstract

We obtain global and local theorems on the existence of invariant manifolds for perturbations of non autonomous linear differential equations assuming a very general form of dichotomic behavior for the linear equation. Besides some new situations that are far from the hyperbolic setting, our results include, and sometimes improve, some known stable manifold theorems.