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The embedding theorem

In practice, it is usually impossible to measure all the components of the $n$-dimensional vector $\vec X$ 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]
\begin{displaymath}
\vec{x}(t_{i}) = [x(t_{i}), \; x(t_{i} + l \Delta t), \; \ldots ,\; x(t_{i} + (d-1) l \Delta t)]
\end{displaymath} (2.6)

where $d$ denotes the embedding dimension, $\Delta t$ is the sample time, $l$ is an appropriate integer (see below) and $x$ denotes one component of $\vec{X}$. The embedding theorem states that there generically exists an embedding, i.e. a diffeomorfism between the reconstructed and original phase space if
\begin{displaymath}
d \geq 2D + 1
\end{displaymath} (2.7)

where $D$ 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 $\vec X$ 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 $ d \geq 2D + 1$. 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 $l$ 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.


next up previous contents
Next: NEAREST NEIGHBOUR METHODS Up: THE CHARACTERIZATION OF CHAOTIC Previous: The generalized entropies   Contents
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