Global attractors: Topology and finite-dimensional dynamics
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.
scientific article; zbMATH DE number 1334593