Connecting a direct and a Galerkin approach to slow manifolds in infinite dimensions

TL;DR

This paper compares two methods for constructing slow manifolds in infinite-dimensional systems, showing they are asymptotically close under certain conditions, thus enabling flexible analysis of fast-slow dynamics.

Contribution

It establishes a rigorous connection between abstract evolution and Galerkin approximation approaches for slow manifolds, enhancing understanding of their relationship in infinite dimensions.

Findings

Slow manifolds from both approaches are asymptotically close.

Lyapunov-Perron methods are used for the comparison.

Main result facilitates switching between different manifold characterizations.

Abstract

In this paper, we study slow manifolds for infinite-dimensional evolution equations. We compare two approaches: an abstract evolution equation framework and a finite-dimensional spectral Galerkin approximation. We prove that the slow manifolds constructed within each approach are asymptotically close under suitable conditions. The proof is based upon Lyapunov-Perron methods and a comparison of the local graphs for the slow manifolds in scales of Banach spaces. In summary, our main result allows us to change between different characterizations of slow invariant manifolds, depending upon the technical challenges posed by particular fast-slow systems.