Global attractors: Topology and finite-dimensional dynamics

Many dissipative evolution equations possess a global attractor script A sign with finite Hausdorff dimension d. In this paper it is shown that there is an embedding X of script A sign into ℝN, with N= [2d + 2], such that X is the global attractor of some finite-dimensional system on ℝN with trivial dynamics on X. This allows the construction of a discrete dynamical system on ℝN which reproduces the dynamics of the time T map on script A sign 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. © 1999 Plenum Publishing Corporation.