Embedding Nonlinear Dynamical Systems: A Guide to Takens' Theorem

The embedding theorem forms a bridge between the theory of nonlinear dynamical systems and the analysis of experimental time series. This memorandum describes the theorem and gives a detailed account of its proof. The necessary differential topology is briefly reviewed, and then a proof of the theorem is presented; this proof follows broadly the argument of Takens, although it differs in some details. Some extensions to the theorem, which facilitate its use in applications, are described. The memo concludes with a brief discussion of what the theorem implies about time series, viewed as the raw material for signal processing algorithms.