Robinson, James C. Global attractors: Topology and finite-dimensional dynamics. (English) Zbl 0953.34049 J. Dyn. Differ. Equations 11, No. 3, 557-581 (1999). 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. Reviewer: Norbert Koksch (Dresden) Cited in 28 Documents MSC: 34D45 Attractors of solutions to ordinary differential equations 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 34G20 Nonlinear differential equations in abstract spaces 34C30 Manifolds of solutions of ODE (MSC2000) 54C25 Embedding 35B41 Attractors 37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems Keywords:global attractors; inertial manifolds; exponential attractors; connectedness Citations:Zbl 0918.34048 PDFBibTeX XMLCite \textit{J. C. Robinson}, J. Dyn. Differ. Equations 11, No. 3, 557--581 (1999; Zbl 0953.34049) Full Text: DOI