Shadowing for Infinite Dimensional Dynamical Systems

TL;DR

This paper extends shadowing properties to infinite-dimensional dynamical systems, specifically Morse-Smale semigroups in Hilbert spaces, showing Lipschitz and Hölder shadowing under certain conditions.

Contribution

It generalizes finite-dimensional shadowing results to infinite-dimensional Hilbert space settings for Morse-Smale semigroups.

Findings

Lipschitz shadowing on the global attractor

Hölder shadowing in neighborhoods of the attractor

Structural stability results for Morse-Smale semigroups

Abstract

In this paper we extend to an infinite dimensional setting some results on the shadowing property that are known on finite dimensional compact manifolds without border and in $\mathbb{R}^n$. In fact, we show that if $\{\T(t):t\ge 0\}$ is a Morse-Smale semigroup defined in a Hilbert space with a global attractor $\mathcal{A}$ and non-wandering set given only by its equilibria, then $\T(1)|_{\mathcal{A}}:\mathcal{A}\to \mathcal{A} $ admits the Lipschitz Shadowing property. Moreover, for any positively invariant bounded neighborhood $\U\supset\mathcal{A}$ of the global attractor, the map $\T(1)|_{\U}:\U\to \U$ has the H\"{o}lder-Shadowing property. We obtain results related to the structural stability of Morse-Smale semigroups, that were only known on finite dimension and continuity of global attractors.