Global attractors: Topology and finite-dimensional dynamics
type: Item
instance of: scholarly article
title: Global attractors: Topology and finite-dimensional dynamics
zbMATH Open document ID: 0953.34049
DOI: 10.1023/A:1021918004832
author: James C. Robinson
published in: Journal of Dynamics and Differential Equations
publication date: 2001-01-15T00:00:00Z
review text: Many dissipative evolution equations possess a global attractor \(A\) with finite Hausdorff dimension \(d\). The existence of such an attractor naturally leads to the question of whether there is a finite-dimensional system that will adequately capture the asymptotic nature of the original flow. Using Mane's projections theorem, one can construct ordinary differential equations which reproduces the dynamics. However, these equations do not have unique solutions. The author shows that any trajectory on the attractor can be approximated for a large (but finite) length of time by the solutions to a Lipschitz continuous differential equation in \(\mathbb{R}^3\) [see also the author, Nonlinearity 11, No.~3, 529-545 (1998; Zbl 0918.34048)]. Moreover, the author shows that there is an embedding \(X\) of \(A\) into \(\mathbb{R}^N\), with \(N=[2d+2]\), such that \(X\) is the global attractor of some finite-dimensional system on \(\mathbb{R}^N\) which reproduces the dynamics of the time \(T\) map on \(A\) and has an attractor within an arbitrarily small neighborhood of \(X\). If the Hausdorff dimension is replaced by the fractal dimension, a similar construction can be shown to hold good even if one restricts to orthogonal projections rather than arbitrary embeddings.
reviewed by: Norbert Koksch
Mathematics Subject Classification ID: 34D45
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 34C30
Mathematics Subject Classification ID: 54C25
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 37L25
zbMATH DE Number: 1334593
zbMATH Keywords: global attractors
zbMATH Keywords: inertial manifolds
zbMATH Keywords: exponential attractors
zbMATH Keywords: connectedness
MaRDI profile type: Publication
full work available at URL: a:1021918004832
OpenAlex ID: W1479945628
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description: scientific article; zbMATH DE number 1334593