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# The embedding theorem

In practice, it is usually impossible to measure all the components of the -dimensional vector in eq. (2.1) or (2.2). Takens [Takens, 1981] proved that one can reconstruct the attractor from a time series of a single component. To do this, the phase space may be reconstructed with the embedding vectors [Packard et al., 1980]
 (2.6)

where denotes the embedding dimension, is the sample time, is an appropriate integer (see below) and denotes one component of . The embedding theorem states that there generically exists an embedding, i.e. a diffeomorfism between the reconstructed and original phase space if
 (2.7)

where is the dimension of the compact manifold containing the attractor [Broomhead and King, 1986,Parker and Chua, 1987]. This implies that the dimension and entropy spectra of the reconstructed attractor are the same as those of the real'' attractor, since the spectra are invariant under a smooth coordinate change [Ott et al., 1984]. However, it is assumed that there are an infinite number of infinitely precise measurements of the single component. The requirement of a set of coupled differential equations implies that one component of contains information about the others. There is enough information collected in the vectors eq. (2.6) when the reconstructed trajectory is disentangled, which is generically the case for . Plausibility arguments for this reconstruction technique are given in e.g. [Parker and Chua, 1987] and [Schuster, 1988].

For infinitely long time series, the sample time and the number can be chosen almost arbitrarily. In practice however, choosing the optimal length of the time series and the optimal time delay is a difficult problem; see Chapter 6.

In the following chapters, we will discuss algorithms for estimating the generalized dimensions and entropies from a scalar time series.

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