ABSTRACT
The Takens embedding theorem (Takens, 1981) states that for a scalar time series X(t ) obtained from a d-dimensional deterministic system, the vector with time-delayed coordinates (X ( t ) , X ( t — r ) , . . . , X ( t — E t ), E < 2d 4 - 1, will trace out a trajectory that is a smooth coordinate transformation of the attractor of the original dynamical system. In particular, if the dynamical system has an attractor of a particular dimension, the embedded trajectory will have the same dimension. In fact, it has been recently shown (Sauer et al., 1992) that, if the attractor underlying the time series is of box-counting dimension Do, then a time delay embedding into a space whose embedding dimension E is the smallest integer greater than or equal to A), will allow accurate estimation of -Do? Al> and The dynamical system is said to be reconstructed. Reconstruction considerably simplifies the task of prediction and can lead to other more detailed analyses.
