A general approach to nonautonomous shadowing for nonlinear dynamics

TL;DR

This paper establishes a general framework for nonautonomous shadowing in nonlinear differential equations on Banach spaces, encompassing various types of dichotomies and extending previous results to new settings.

Contribution

It introduces broad conditions for shadowing in nonautonomous systems with general dichotomies, including nonlinear and higher-order equations, and provides discrete-time analogs.

Findings

Established shadowing conditions for systems with general dichotomies.

Extended shadowing results to higher order differential equations.

Provided discrete-time versions of the continuous results.

Abstract

Given a nonautonomous and nonlinear differential equation \begin{equation}\label{DE} x'=A(t)x+f(t,x) \quad t\geq 0, \end{equation} on an arbitrary Banach space $X$, we formulate very general conditions for the associated linear equation $x'=A(t)x$ and for the nonlinear term $f:[0,+\infty)\times X\to X$ under which the above system satisfies an appropriate version of the shadowing property. More precisely, we require that $x'=A(t)x$ admits a very general type of dichotomy, which includes the classical hyperbolic behaviour as a very particular case. In addition, we require that $f$ is Lipschitz in the second variable with a sufficiently small Lipschitz constant. Our general framework enables us to treat various settings in which no shadowing result has been previously obtained. Moreover, we are able to recover and refine several known results. We also show how our main results can be applied to the study of the shadowing property for higher order differential equations. Finally, we conclude the paper by presenting a discrete time versions of our results.