A possibility of a
relation between the Kolmogorov-Sinai
entropy (KSE)
of a dynamical system and
the entropy rate (GPER) of a
Gaussian process isospectral to time series generated
by the dynamical system was
numerically investigated using
three
well-known chaotic
dynamical systems.
The results obtained suggest that such a
relation as a nonlinear one-to-one function may exist
when the Kolmogorov-Sinai entropy varies smoothly
with variations of system's parameters, but
is broken in critical states near bifurcation points.
Further theoretical and numerical
studies are necessary to establish
general conditions for validity
of this conclusion.
These results
could find applications in two areas
of the analysis of complex
time series:
The GPER itself could be used as a computationally cheap
tool for classification of different chaotic states
of dynamical systems;
while discrepancies
in the relation between the GPER and
the KSE/LE (or other
nonlinear entropy-rate equivalent [11])
could be applied for detecting bifurcation
onsets in structurally evolving systems.
Acknowledgements
The author would like to thank A.N. Pettitt and V.V. Anh.
The author was supported
by the Visiting Fellowship from the
Centre in Statistical Science and Industrial
Mathematics, Queensland University of Technology
in Brisbane,
and in part by the
Grant Agency of the Czech Republic
(grant No. 205/97/0921).