A possibility of a relation between the KSE of a dynamical system and the entropy rate GPER of a Gaussian process with the same spectrum as the time series, generated by the dynamical system is investigated in this study. A formal application of formula (3) to a spectral density (periodogram) estimated from analyzed time series cannot be considered as an estimate of the KSE of an underlying dynamical system, however, if there was a one-to-one relation between the KSE and the GPER, the GPER could be used for a ``relative quantification'' [11] of dynamic processes, in particular, it could distinguish and classify different states of chaotic dynamical systems.
The numerical investigation of this hypothetical relationship was performed using these dynamical systems:
The baker transformation [5]:
for 
, or:
for 
;
, 
,
;
the logistic map [5]:
and the continuous Lorenz system [21]:
, b=4.
Each of the three systems has one positive Lyapunov exponent, equal to the system's Kolmogorov-Sinai entropy, therefore we will use the terms LE and KSE interchangeably.
Changing a parameter of a particular system
( 
, a, r in the cases of the baker,
logistic and Lorenz systems, respectively),
time series related to different system states
were generated, GPER's were estimated
and compared with LE (KSE) related to particular
system states. In each system state studied,
fifteen
time series of length 16,384 samples
(the sampling interval was 0.002 in the case of the
Lorenz system) were recorded from the first
component (x), linearly transformed in order to
have zero mean and unit
variance
and their
periodograms
computed using the fast Fourier
transform (FFT) [22]. To prevent numerical
underflow, the periodograms were
shifted
by +1,
i.e., 
was used instead of 
in Eq. (3).
For each considered dynamical state,
means and standard deviations (SD's)
of the GPER estimates,
obtained
from the 15 realizations
of 16k time series, are reported in this
paper.
The positive Lyapunov exponents were not
estimated from time series, but computed as follows.
The KSE/LE of the baker map
can be expressed analytically
as the function of the parameter [23, 24]:
For the logistic map the LE was estimated according to
its definition [5]
as the averaged logarithm of the absolute
derivative of the function (5).
A recent implementation [25] of the method
proposed by Wolf et al. [26] for estimation of
the Lyapunov exponents from equations was used
for the Lorenz system.
The LE's in these two cases were estimated using
300,000 iterations in each state.
