$C^1$ invariant, stable and inertial manifolds for non-autonomous dynamical systems

TL;DR

This paper investigates the existence and properties of invariant manifolds, including inertial and stable manifolds, for non-autonomous dynamical systems using a Lyapunov--Perron approach, establishing new characterizations and regularity results.

Contribution

It introduces a unified gap condition framework and characterizes invariant manifolds via solution growth behavior, proving $C^1$ regularity for inertial manifolds.

Findings

Characterization of invariant manifolds through solution growth behavior

Unified formulation of the gap condition for manifold existence

Proof of $C^1$ regularity for inertial manifolds

Abstract

We use the version of the Lyapunov--Perron method operating on individual solutions to investigate the existence of invariant manifolds for non-autonomous dynamical systems, focusing in particular on inertial and stable manifolds. We establish a characterization of both types of manifolds in terms of solutions exhibiting a common growth behavior, analogous to the classical characterization involving hyperbolicity. Furthermore, we introduce a unified formulation of the gap condition, from which known sharp versions are derived. Finally, we show that the constructed inertial manifolds have $C^1$ regularity.