In this paper we present the general problem of identifying if a nonlinear dynamic system has a chaotic behavior. If the answer is positive the system will be sensitive to small perturbations in the initial conditions which will imply that there is a chaotic attractor in its state space. A particular problem would be that of identifying a chaotic oscillator. We present an example of three well known different chaotic oscillators where we have knowledge of the equations that govern the dynamical systems and from there we can obtain the corresponding time series. In a similar example we assume that we only know the time series and, finally, in another example we have to take measurements in the Chua's circuit to obtain sample points of the time series. With the knowledge about the time series the phase plane portraits are plotted and from them, by visual inspection, it is concluded whether or not the system is chaotic. This method has the problem of uncertainty and subjectivity and for that reason a different approach is needed. A quantitative approach is the computation of the Lyapunov exponents. We describe several methods for obtaining them and apply a little known method of artificial neural networks to the different examples mentioned above. We end the paper discussing the importance of the Lyapunov exponents in the interpretation of the dynamic behavior of biological neurons and biological neural networks

Espinosa, Ismael E.

Fuentes, Alberto M.

Gonzalez, J. Jesus

English

NASA Technical Reports Server

LYAPUNOV EXPONENTS FROM CHUA'S CIRCUIT TIME
SERIES USING ARTIFICIAL NEURAL NETWORKS
J.Jesfis Gonzdlez F., Ismael Espinosa E., and
Lab. de Ciberndtica, Depto. de Fisica, Fac. de Ciencias
Universidad Nacional Autdnoma de Mdxico
Alberto Fuentes M.
lnstituto de Ciencias Nucleates
Universidad National Autdnoma de Mgzico
Abstract
In this paper we present the general problem of identifying if a nonlinear dynamic sys-
tem has a chaotic bel_avior. If the answer is positive the system will be sensitive to small
perturbations in the ihitial conditions which will imply that there is a chaotic _.ttractor in
its state space. A particular problem would be that of identifying a chaotic oscillator. We
present an example of three well known different chaotic oscillators where we have knowledge
of the equations that govern the dynamical systems and from there we can obtain the corre-
sponding time series. In a similar example we assume that we only know the time series and,
finally, in another example we have to take measurements in the Chua's circuit to obtain
sample points of the time series. With the knowledge about the time series the phase plane
portraits are plotted and from them, by visual inspection, it is concluded whether or not
the system is chaotic. This method has the problem of uncertainty and subjectivity and for
that reason a different approach is needed. A quantitative approach is the computation of
the Lyapunov exponents. We describe several methods for obtaining them and apply a little
known method of artificial neural networks to the different examples mentioned above. We
end the paper discussing the importance of the Lyapunov exponents in the interpretation of
the dynamic behavior of biological neurons and biological neural networks.
1 Introduction
In the companion paper [1] we described some findings about biological oscillators that have been
presented in the recent literature. We also showed some examples of oscillator models. Here we
want to review some models of chaotic oscillators with the goal of extending the analysis to time
series (trains of action potentials) coming from biological oscillators where there are some hints
that they are chaotic.
There are many experimental situations where there is no idea of what the mathematical
model of a system could be or where the form of the equations is known but the parameters are
unknown. There is an extensive literature about methods for systems identification but they are
OF lJO011 Qulcc._-_' 263
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usually limited to linear models. Since the conditions for chaotic behavior arise from the presence
of nonlinear elements, the use of linear methods is limited.
In recent years there have been many advances in the understanding of nonlinear dynamic
systems and that has produced many methods for identifying whether or not a given system
is chaotic. For testing these methods various simple chaotic systems have been discovered or
invented. In some of them the equations that govern the system are well known but in others
only some sort of approximation is known. In the former case, it is possible to generate the
corresponding time series with very high approximation, in the latter case, the measurements
yield a sampled version of the corresponding time series. Having at hand the equations and
the time series, or at least the time series, it is possible, in very different ways, to compute the
asymptotic properties of the system. Two measures are used for this: the Lyapunov exponents
in which a positive one indicates chaotic dynamics, arid the attractor's topological dimension
which indicates topological characteristics and is directly related to the number of non-negative
Lyapunov exponents [2].
It is usual to ascertain[ the existence of chaotic dynamic by means of visual inspection of the
phase plane portrait. However, such method presents a considerable amount of t uncertainty and
subjectiveness. Taking that into account, it is important to have a quantitative method like the
one provided by the computation of the Lyapunov exponents.
2 Lyapunov Exponents
To determine if a system possess chaotic dynamics it is necessary to know if it is sensitive to
small perturbations on the initial conditions. When this occurs it is then impossible to predict the
final state of the system after a finite time. To be able of characterizing a chaotic attractor it is
necessary to establish quantitative measures concerning the sensitivity to initial conditions. The
spectrum of Lyapunov exponents gives a method of quantifying the dynamics. The Lyapunov
exponents describe the average rate of growing or shrinking of small perturbations in different
directions in the state space. When the attractor has at least one positive exponent then it has
the property of being sensitive to the initial conditions and it is called a chaotic attractor.
There are several methods for computing the Lyapunov exponents. Wolf'et al. [3} were. the
first in suggesting a method to compute them directly from the time series, without knowing the
equations that govern the system's dynamics. Kurths and Herzel [4] proposed another algorithm.
However, in these algorithms the estimations are sensitive to the number of observations, to the
sampling frequency and to the noise in the observations [3]. Trying to avoid these problems,
Gencay and Dechert [5] designed an algorithm that computes the m Lyapunov exponents from
an unknown m-dimensional dynamic system directly from a few observations on the attractor,
in such a way that the estimation is robust even for certain amount of noise. This algorithm is
based on a result by Hornik et al. [6] in which they show that the m Lyapunov exponents of a
diffeomorphism that is' topologically conjugate to the process that generates the data, are also
the m Lyapunov exponents of that process. To obtain a robust estimation considering both few
observations on the attractor and the presence of noise, Gencay and Dechert [5] applied artificial
neural networks with a cascade architecture. Such procedure is a non-parametric estimation that
Hornik et al. [6] [7] have shown to be universal approximators, that is, they can asymptotically
approximate a function and its derivatives.
264
3 Computation of the Lyapunov Exponents
The Lyapunov exponents are constants, except for a zero-measure set, and describe the direction
of nearby paths that converge or diverge in the state space of a dynamic system. The Lyapunov
exponents Xi are defined as the logarithm of the eigenvalues #i V i = 1,2,-. •, m of the symmetric
positive matrix
A_ = aim [y(x;t)_ry(x;t)] 1/2t , (1)t-.oo
where the matrix Y is dependent on the differential equation that characterizes the dynamical
system. A direct application of the above definition is not practical since the Y matrix grows
exponentially due to the fast convergence of the columns in the direction of g_eater expansion.
Using topological properties and an appropriate QR decomposition, the Lyapunov exponents are
found by computing
1 lim 1'_-1 ()_---.=- - In Rii , (2)
At t--_o_n j="-"_
where R, are the diagonal elements of the triangular matrix R.
An alternative to the aforementioned algorithm is the use of neural networks which are capable
of recovering a nonlinear map from a time series of iterates [8]. Here an unknown function is
estimated and then it is possible to compute the Lyapunov exponents using the properties of the
dynamic system [5].
,=
°=
o=
w
m
B
FIG. 1. Phase Plane Portraits for the Logistic Map, the H6non Map, and the
Lorenz System.
265
4 Examples of chaotic oscillators
To be able of testing the effectiveness of the Lyapunov exponents one has to have at hand dy-
namical systems with proved chaotic behavior. Many mathematical model systems are known,
for instance: H_non, Rossler-chaos, Lorenz, Rossler- hyperchaos, Mackey-Glass, and others [3].
Physical model systems ar k more difficult to produce, however we have the Belousov-Zhabotinsky
chemical reaction [9] and Chua's nonlinear circuit family [10]. Obviously, there lare real physical
chaotic systems but they are extremely complex as it is the case of atmospheric turbulence.
TABLE I. True Lyapunov Exponents, equations representing the chaotic systems,
initial conditions and parameters.
True Lyapunov Exponents
Logistic map H6non map Lorenz system
0.673 0.440 1.51
-1.620 0.00
-22.5
xtTi "_- #xt(1 - xt) xt = 1 - ax2t + yt ic =a(x - y)
y_ = bxt _ = x(b- z) - y
= xy - cz
xo=O.3, tt=4.0 xo=O.l,yo=O.O xo=O.O, yo= 1.1,zo=O.O
a = 1.4, b = 0.3 a = 16.0, b = 45.92, c = 4.0
TABLE II. Estimated Lyapunov Exponents. Logistic Map (q = 5, T = 100).
Hdnon Map (q = 10, T = 200). Lorenz System (q = 15, T = 1000). The error is less
than lxl0 -3. The non-spurious Lyapunov exponents are shown in boldface.
Estimated Lyapunov Exponents
p Logistic map
1 0.6794
2 0.6401
-6.7823
5
0.6350
-2.4378
-2.4930
0.6434
-1.61106
-1.7342
-5.0200
0.6790
-0.9073
i
-1.3544
-1.8468
-3.2313
H_nonmap
0.3670
-1.5673
0.4502
-1.7331
-2.8164
0.4119
-1.4803
-3.3658
-5.2263
0.4385
-1.5473
-1.4859
-1.7651
-2.5605
Lorenz system
1.7285
0.0411
-23.72
1.5910
-0.0710
-20.973
-80.325
1.4799
0.0067
-20.977
-60.702
-92.584
266
We divided this sectio_ of examples in three parts. In the first part the ordinary differential
equations for the Lorenz rhodel were integrated using the SDRIV2 subroutines [orm Kahaner et
al. [12]. For the QR decomposition use used the subroutines in [13]. In the second and third parts
we computed the Lyapunov exponents by means of the neural networks approach [5].
For the first part of the section we show the results of computing the Lyapunov exponents
for mathematical model systems with well known chaotic dynamics [3] [5] [8]. We computed the
Lyapunov exponents for the Logistic Map, for the H_non map and for the Lorenz system [3] [5].
The corresponding phase plane portraits are shown in FIG. 1, and the equations, parameters and
computed true values of the Lyapunov exponents are shown in TABLE I above.
For the second part of the section we show the results when we assume that for the same
chaotic systems than above, the equations are not known, only the time series. In TABLE II we
show the computed Lyapunov exponents where the presence of one positive exponent indicates
that the system is chaotic.
According to the established notation for neural network architectures [11], p represents the
number of nodes in the input layer and q represents the number of nodes in the hidden layer.
The output layer has one node. The error is the quadratic average summation of the differences
between the real and the estimated values for the time series, being T the total number of sample
points in the sequence.
For the third part of the section we show the results obtained when we used a nonlinear circuit
of the Chua's family with the parameters, components and initial conditions shown in FIG. 2. The
temporal series was acquired by means of a digital storage scope, the x-coordinate is the voltage
in the linear capacitor C1 and the y-coordinate is the voltage in the linear capacitor C2. A part
of the phase plane portrait is also shown in FIG. 2, from it the temporal series was obtained using
a sampling frequency of 500 Hz.
CI-IUA'S CIR£XIIT
o .o=i
o .o14
• .olt
o.oto
n. OLl
D.O
C_ua_ M_o
i
/. , .
4.in o._ .. i .
N-ol
FIG. 2. Nonlinear Circuit of the Chua's Family and a part of its phase plane
portrait that was plotted using as state variables the voltages in the capacitors C1 and
C2. The phase plane portrait changes when the parameters in the circuit are varied
between the limits denoted in the diagram.
The estimated Lyapunov exponents for the Chua's circuit are shown in TABLE III, when using
the estimation for q = 15 and T = 2500. The error was less than 5x10 -2. Notice that 11 is a
positive exponent which means that the dynamic behavior of the circuit is chaotic, as expected.
267
TABLE III. Estimated Lyapunov Exponents for the Chua's Circuit.
i Estimated Lyapunov Exponents
p Chua's Circuit
A1=+4.07216
A2---0.32160
Aa---3.58462
5 Concluding Remarks
We have shown well known examples of mathematical models of chaotic attractors: Logistic,
H_non, and Lorenz. We also showed an electronic model of a chaotic attractor: a circuit of
the Chua's family. In all these examples we computed the Lyapunov exponents as a measure of
the system's sensitivity to small perturbations in the initial conditions. For the example where
we know the equations we used the standard method; for the other two examples, we used the
method of Gencay and Dechert [5] that applies a neuronal network algorithm. We went from the
easier examples to the difficult one, that is, the Chua's circuit where the time series is obtained
from direct measurements in the circuit. In the former examples the true values of the Lyapunov
exponents are known whereab in the Chua's circuit the Lyapunov exponents are estimated and
to this has to be added the differences or variations, for any reason, in the parameters for the
circuit's components. The justification for such trouble is that in a real chaotic System there is a
complexity even worse that in the electronic model. Therefore, the circuit provides a very valuable
experience that afterwards can benefit the understanding of the real chaotic attractor.
Recent experimental evidence points to biological neurons and biological neural networks as
very likely sources of chaotic attractors. As in the case of chaotic chemical reactions the biological
significance given to such behavior is speculative [9] [14]. Nevertheless, the application of the tech-
niques described in this paper might be very useful for interpreting data from neurophysiological
experiments where the electrical activity from many neurons is recorded simultaneously.
Similar to Chua's circuit situation, due to the usual difficult conditions of neurophysiologieal
experimentation, in a biological neural n,_twork we can only obtain a short duration record of the
compound time series (train of action potentials). The individual time series have to be separated
and that procedure produces and additional source of uncertainty and every tool available for
interpreting the results is welcomed [15].
As a very simple example (for details see companion paper [1]) let us consider the rhythmic
firing single neuron (1) shown in the upper plot in FIG. 3. This time series was obtained from the
simulation of a mathematical model for a single neuron and two of its phase plane portraits for
two different physiological parameters (V - f, V - h) chosen as state variables show very clearly
the possibility of chaotic behavior. In the lower plot in FIG. 3 we show another simulation of
the rhythmic firing of a neuron (2) that belongs to a recurrent-ring network composed of single
neurons like the ones given in (1). From the two phase plane portraits we cannot conclude that
the neuron (2) in the network is chaotic and the conclusion about neuron (1) required of an
expert. This ambiguity can be surmounted if the Lyapunov exponents are computed for these
268
time series and from there the importance of making such calculations and having experimental
testing circuits for imprgving the confidence in the results.
FIG. 3. (1) Rhytmically firing single neuron. (2) Rhytmically firing neuron be-
longing to a network formed with type (I) neurons.
On the other hand, in biological experiments there are problems similar to the ones present
when making measurements in the Chua's circuit. When doing an extracellular recording, the
duration of it is limited to a few minutes and afterwards a considerable amount of preprocessing is
required to get to the individual contribution of each neuron recorded [16] [l 7]. The calculation of
the Lyapunov exponents for these individual contributions could be added to help understanding
the functional role of oscillatory neurons and oscillatory networks. That is the work thai we are
about to pursue.
Acknowledgments
We thank Jorge Quiza for reading and making suggestions for improving this paper. A scholar-
ship granted by CONACYT to J.Jesds Gonz£1ez is gratefully acknowledged. This work is being
supported by DGAPA Project IN100593-UNAM.
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Lyapunov exponents from CHUA's circuit time series using artificial neural networks

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