#
Chaotic Measures and Real-World Systems:

Does the Lyapunov exponent always measure chaos?

**Milan Palus **

*Institute of Computer Science,
Academy of Sciences of the Czech Republic*

*Pod vodárenskou vezí 2,
182 07 Prague 8, Czech Republic; and*

* Santa Fe Institute, 1399 Hyde Park Road
*

*Santa Fe, NM 87501, USA*

E-mail: mp@uivt.cas.cz, mp@santafe.edu

### Abstract:

Direct estimation of the largest Lyapunov exponent
as a measure of exponential divergence of nearby
trajectories is well established in the case of deterministic
dynamical systems.
Questions are naturally raised about applicability
of Lyapunov exponents and other ``chaotic measures''
when analyzing data from real-world
systems, which
are either stochastic or affected by numerous
external influences, which cannot be described in any other way
than a stochastic component in system dynamics.
In a series of numerical experiments,
Gaussian random deviates were added to
a set of chaotic time series with different Lyapunov
exponents.
It is demonstrated
that the estimated Lyapunov exponents fail to distinguish
different noisy chaotic time series when relatively small
scales are used. The distinction can be reestablished by using
larger scales. Using larger scales, however,
the estimated Lyapunov exponent is determined
by macroscopic statistical properties of the series and
provides the same information as the
autocorrelation function and/or coarse-grained mutual information.

* Nonlinear Analysis of Physiological Data *

H. Kantz, J. Kurths, G. Mayer-Kress, eds.

Springer 1998, pp. 49-66 (ISBN 3-540-63481-9)

*Milan Palus
1997, 1988*