Next: NEAREST NEIGHBOUR METHODS
Up: THE CHARACTERIZATION OF CHAOTIC
Previous: The generalized entropies
Contents
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.
Next: NEAREST NEIGHBOUR METHODS
Up: THE CHARACTERIZATION OF CHAOTIC
Previous: The generalized entropies
Contents
webmaster@rullf2.xs4all.nl