Shadowing for nonautonomous dynamics

TL;DR

This paper establishes shadowing properties for nonautonomous dynamical systems with bounded operators and differentiable perturbations, extending existing results and providing a new proof of the nonautonomous Grobman-Hartman theorem.

Contribution

It extends shadowing theory to nonautonomous systems with exponential dichotomies and differentiable maps, and offers a new proof of the nonautonomous Grobman-Hartman theorem.

Findings

Shadowing properties hold under exponential dichotomy conditions.

The results generalize previous shadowing theorems for nonautonomous systems.

A new proof of the nonautonomous Grobman-Hartman theorem is provided.

Abstract

We prove that whenever a sequence of invertible and bounded operators $(A_m)_{m\in \mathbb{Z}}$ acting on a Banach space $X$ admits an exponential dichotomy and a sequence of differentiable maps $f_m \colon X\to X$, $m\in \mathbb{Z}$, has bounded and H\"{o}lder derivatives, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in \mathbb{Z}$ has various shadowing properties. Hence, we extend recent results of Bernardes Jr. et al. in several directions. As a nontrivial application of our results, we give a new proof of the nonautonomous Grobman-Hartman theorem.