ECCTD’01 - European Conference on Circuit Theory and Design, August 28-31, 2001, Espoo, Finland

Markov Chain Modeling of Chaotic Wanderingin Simple Coupled Chaotic Circuits
Yoshifumi NISHIO∗†, Martin HASLER∗ and Akio USHIDA†
Abstract — In this study, chaotic wandering phenomenon observed in simple coupled chaotic circuits
is modeled by Markov chains. As coupling parameter increases, the solution of the coupled chaotic
circuits starts to wander chaotically over 6 diﬀerent
phase states which are stable for smaller value of theparameter. The wandering is modeled by ﬁrst-orderand second-order Markov chains with the 6 states
plus 3 intermediate states. Stationary probability
distribution, expected sojourn time and probability
density function of sojourn time are calculated fromtransition probabilities. These statistical quantities
are compared with those obtained from computer
simulations.

1

Introduction

Spatio-temporal phenomena observed from coupled chaotic systems attract many researchers’ attentions. For discrete-time mathematical models, there have been numerous excellent results.
Kaneko’s coupled map lattice is the most interesting and well-studied system [1]. He discovered various nonlinear spatio-temporal chaotic phenomena
such as clustering, Brownian motion of defect andso on. Also Aihara’s chaos neural network is themost important chaotic network from an engineering point of view [2]. His study indicated new possibility of engineering applications of chaotic networks, namely dynamical search of patterns embedded in neural networks utilizing chaotic wandering.
Further, application of chaos neural network to optimization problems is widely studied. However,chaotic wanderings in continuous-time circuits havenot been studied well. Therefore, in order to ﬁll thegap between studies of discrete-time mathematicalabstract and studies of continuous-time real physical systems, it is important to investigate simple
continuous-time coupled chaotic circuits generating chaotic wandering, clustering, pattern switching, and so on.
We have proposed continuous-time coupledchaotic circuits systems and have investigated generating spatial patterns and chaotic wandering of
spatial patterns [3]. In our latest studies, we havereported similar coupled circuits could generatechaotic wandering over several phase states characterized by four-phase synchronization and haveshown that dependent variables corresponding to

the angles of the solutions in subcircuit were useful to grasp the complicated phenomena [4][5]. The
important feature of our coupled circuits is theircoupling structure. Namely, adjacent four chaotic
circuits are coupled by one resistor. Because sucha coupling exhibited quasi-synchronization withphase diﬀerence [6], various spatial patterns could
be generated. It would be followed by generation of several complicated spatio-temporal chaotic
phenomena observed in discrete-time mathematicalmodels. Therefore, the network based on the coupled circuits would be a good model to make clear
physical mechanism of spatio-temporal chaotic phenomena in continuous-time systems. However,
because it is extremely diﬃcult to treat higherdimensional nonlinear phenomena in continuoustime systems theoretically, we have to develop several tools to reveal the essence of the complicatedphenomena.
In this study, chaotic wandering phenomenon observed in four simple chaotic circuits coupled by one
resistor is modeled by Markov chains. The circuits
exhibit four-phase synchronization of chaos for relatively small coupling parameter values. Becauseof the symmetry of the coupling, there coexist 6
diﬀerent phase states. As the coupling parameterincreases, the solution starts to wander chaoticallyover the 6 diﬀerent phase states which are stable
for smaller value of the parameter. The wandering
is modeled by ﬁrst-order and second-order Markovchains with the 6 states plus 3 intermediate states.
Transition probabilities between the states can beobtained by counting all of the transitions in plentyof computer simulation. Stationary probability distribution, expected sojourn time and probability
density function of sojourn time are calculated fromthe transition probabilities. These statistical quantities are compared with those obtained from computer simulations.
2Figure 1 shows four coupled chaotic circuits. Thei − v characteristics of the diodes are approximatedby a two-segment piecewise-linear function as
vd (ik ) = 0.5 (rd ik + E − | rd ik − E | ) .

∗ Laboratory of Nonlinear Systems, Department of Com-munication Systems, Swiss Federal Institute of Technology
Lausanne, EL-Ecublens, CH-1015 Lausanne, Switzerland.
E-mail: nishio@circpc.epfl.ch, martin.hasler@epfl.ch.
† Department of Electrical and Electronic Engineering,
Tokushima University, 2-1 Minami-Josanjima, Tokushima
770-8506, Japan. E-mail: nishio@ee.tokushima-u.ac.jp,
ushida@ee.tokushima-u.ac.jp, Tel: +81-88-656-7470, Fax:
+81-88-656-7471.

Circuit Model(1)By changing the variables and parameters,
Ik = C/L1 E xk , ik = C/L1 E yk , vk = E zk ,t = L1 C τ, α = L1 /L2 , β = r C/L1 ,(2)
γ = R C/L1 , δ = rd C/L1 ,II-105

(a)(b)(c)

I1I2I3I4

Figure 1: Circuit model.
the normalized circuit equations are given as
4 dxk=β(x+y)−z−γxjkkk
 dτ
j=1dyk= α{β(xk + yk ) − zk − f (yk )}
dτ


 dzk = xk + yk
( k= 1, 2, 3, 4 )dτ(d)!t(e)Figure 2: Four-phase synchronization of chaos.
L1 =100.7mH, L2 =10.31mH, C=34.9nF, r=334Ω andR=198Ω. (a) I1 −I2 . (b) I1 −I3 . (c) I1 −I4 . (d) I1 −v1 .
(3) of the point (x1 (n), z1 (n)) is always π.) Figure 3
shows time evolution of ϕk (n) when chaotic wandering occurs. We can see that several switchings ofthe phase diﬀerence appear in an irregular manner.

wheref (yk ) = 0.5 (δ yk + 1 − | δ yk − 1 | ).
3

(4)

Four-Phase Synchronization and Chaotic
Wandering

Figure 2 shows an example of the observed fourphase synchronizations. Though only circuits experimental results are shown, similar results can beobtained by computer calculations. In the ﬁgures
the phase diﬀerences of I2 , I3 and I4 with respect
to I1 are almost 90◦ , 180◦ and 270◦, respectively.
However, because of the symmetry of the coupling
structure, six diﬀerent combinations of phase states
coexist.
Further, we can observe chaotic wandering of
the 6 phase states of the four-phase synchronization by tuning coupling parameter value. In order
to investigate the chaotic wandering, we deﬁne thePoincaré section as z1 = 0 and x1 < 0. Further wedeﬁne the following independent variables from thediscrete data of xk (n) and zk (n) on the Poincarémap.

−1 zk+1 (n)(xk+1 (n) > 0)
 π − tanxk+1 (n)ϕk (n) =z (n) π + tan−1 k+1(xk+1 (n) < 0)xk+1 (n)(5)( k=1, 2, 3. )Because the attractor observed from each subcircuit is strongly constrained onto the plane yk = 0
when the diode is oﬀ, these variables can correspond to the phase diﬀerences between the subcircuit 1 and the others. (Note that the argument−→ n : iteration
Figure 3: Chaotic wandering of phase states (α = 7.0,

β = 0.13, γ = 0.46 and δ = 100.0).

4

Markov Chain Modeling of Chaotic Wandering

In this study the chaotic wandering observed inour circuit is modeled by Markov chains. It gives
various statistical characteristics of the higherdimensional nonlinear phenomenon and would help
to carry out further detailed analysis.
At ﬁrst, let us deﬁne the 6 phase states ofthe four-phase synchronization concretely using thevariables in (5) as
S1S2S3
S4S5
S6::::::

ϕ1 > ϕ2 > ϕ3ϕ1 > ϕ3 > ϕ2
ϕ2 > ϕ1 > ϕ3
ϕ2 > ϕ3 > ϕ1
ϕ3 > ϕ1 > ϕ2
ϕ3 > ϕ2 > ϕ1 .

(6)

By careful investigation of the behavior of the solutions around switchings between two of the above
6 phase states, we found that the solution often
stays in intermediate phase states during a certain

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period. In the intermediate phase states, 2 of thefour circuits are almost synchronized at in-phaseand the rest are also synchronized at in-phase with
π-phase diﬀerence against the former pair (like inand opposite-phases quasi-synchronization in [3]).
These phase states can be characterized by
SI1SI2SI3: |ϕ1 − π| < θI: |ϕ2 − π| < θI: |ϕ3 − π| < θI

∩ |ϕ2 − ϕ3 | < θI
∩ |ϕ1 − ϕ3 | < θI
∩ |ϕ1 − ϕ2 | < θI(7)

where θI is a parameter deciding largeness of theregion of the intermediate phase states. Note thatthe condition of the decision of the phase states(6) should be modiﬁed because of introducing theseintermediate phase states.
Next, let us consider a state-transition diagram
representing the transitions between the abovementioned phase states S1 − S6 and SI1 − SI3 . By
virtue of the symmetry of the coupling of the original system, we have to consider only S1 and SI1 .
Figure 4 shows a part of the state-transition diagram focusing on the phase state S1 where
P1 = 1 −

7


Pi .

Figure 5: State-transition diagram II (only transitions

from SI1 ).

The stationary probability distribution describing probability of the solution being in each phasestate;
Q = [QS1 , QS2 , · · · , QSI3 ]T(11)
can be calculated from the following equations
Q = PQ and(8)6Q Si +i=13
QSIj = 1.

(12)

j=1

i=2

It is also possible to estimate expected sojourn
Note that the paths from S1 to S6 and to SI 2 do not time in each phase state by using the transition
exist, because these transitions need double switch- probabilities. For example, probability density
ing beyond one of the other phase states which was function of the sojourn time in S1 is given byvery rare in our computer simulations. Figure 5
shows a part of the state-transition diagram focus(13)PST (S1 , n) = P1n−1 (1 − P1 ).
ing on the intermediate phase state SI1 whereFrom (13) the expected sojourn time in S1 is calP8 = 1 − 2(P9 + P10 ).
(9) culated as∞
EST (S1 ) ={n × PST (S1 , n)}n=1Figure 4: State-transition diagram I (only transitions

from S1 ).

P1P2
 P
 3
 P4
 P5
 0
 P60P7P2P1P50P3P4P6P70P3P4P1P20P50P6P7P50P2P1P4P3P7P60P4P30P5P1P20P7P60P5P4P3P2P1P70P6P9P90P100P10P8000P10P9P9P1000P80P100P10 0 P9 .
P9 0 0P8(10)(1 − P1 )=n=11.
1 − P1nP1n−1(14)In order to get more accurate statistical information of the phenomena, we also model the phenomenon by second-order Markov chain. In thiscase, the transition probability matrix becomes
57×57 and the probability density function of thesojourn time becomes a little bit complicated.
5From the whole state-transition diagram, we canget the transition probability matrix P as∞=Results and Discussions

Table 1 shows the stationary probabilities andthe expected sojourn times obtained from computer simulation, ﬁrst-order Markov chain andsecond-order Markov chain. Computer simulations were carried out over 10,000,000 iterations ofthe Poincaré map. Transition probabilities of theMarkov chains are obtained by counting all of transitions during the computer simulations. We cansee the results obtained from both of the ﬁrst-orderMarkov chain and the second-order one agree very
well with computer simulated results.

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(1)

(2)

(a)

(b)

(c)Figure 6: Probability density functions of sojourn time (α = 7.0, β = 0.13, γ = 0.50 and δ = 100.0). (a) θI =π/12. (b) θI = π/6. (c) θI = π/4. (1) S1 − S6 . (2) SI1 − SI3 .
Table 1: Stationary probabilities Q and expected sojourn times EST (α = 7.0, β = 0.13, γ = 0.50 andδ = 100.0).
Markov Markov
θI
Sim.
(1st)
(2nd)
QS1 −S60.1633
0.16330.1631QSI1 −SI30.0068
0.00680.0071
πE(S)
7.009
7.002
7.050
ST1−612EST (SI1−I3 )
1.170
1.170
1.170
QS1 −S60.1536
0.1536
0.1536
QSI1 −SI3
0.02610.02610.0262π
EST (S1−6 )
7.326
7.319
7.324
6
EST (SI1−I3 )
1.778
1.778
1.779
QS1 −S6
0.13570.13570.1357
QSI1 −SI3
0.06190.06190.0619π
EST (S1−6 )
7.139
7.135
7.137
4
EST (SI1−I3 )
2.566
2.566
2.566

Figure 6 shows the probability density functionsof the sojourn time. From the ﬁgures, we can saythat the ﬁrst-order Markov chain could not explainthe chaotic wandering phenomenon in our systemcorrectly. On the other hand, the second-orderMarkov chain displays better agreement especiallyfor the intermediate states SI1 − SI3 (Fig. 6(2)).
We have to mention that error for S1 − S6 aroundn = 2, 3 increases as θI increases (Fig. 6(1c)). Thiserror is considered to be caused by unneglectablehigher-order Markov property. Investigating the relationship between the error and the order of theMarkov chains is our important future problem.
6Conclusions

In this study, chaotic wandering phenomenon observed in four simple chaotic circuits coupled by

one resistor has been modeled by ﬁrst-order andsecond-order Markov chains with 6 original states
plus 3 intermediate states. Stationary probability
distribution, expected sojourn time and probability
density function of sojourn time have been calculated from the transition probabilities. By comparing these statistical quantities with those obtained from computer simulations, We have theresults that the ﬁrst-order Markov chain can notexplain the phenomenon correctly and that thesecond-order Markov chain could explain the phenomenon eﬃciently. We consider that the result of
this study would help to analysis chaotic wanderingin continuous-time circuits in detail.
References[1] K. Kaneko (ed.), Theory and Applications of Coupled Map Lattices, John Wiley & Sons, Chichester,
1993.
[2] K. Aihara, T. Takabe and M. Toyoda, “Chaotic
Neural Networks,” Phys. Lett. A, vol. 144, no. 6&7,pp. 333–340, 1990.
[3] Y. Nishio and A. Ushida, “Spatio-Temporal Chaos
in Simple Coupled Chaotic Circuits,” IEEE Trans.
Circuits Syst. I, vol. 42, no. 10, pp. 678–686, Oct.
1995.
[4] Y. Nishio and A. Ushida, “On Phase Pattern ina Two-Dimensional Coupled Chaotic Circuits Network,” Proc. NOLTA’98, vol. 1, pp. 31-34, Sep.
1998.
[5] Y. Nishio and A. Ushida, “Analysis of Chaotic Wandering of Phase Patterns in a Two-Dimensional
Coupled Chaotic Circuits Network,” Proc. ISCAS’99, vol. 5, pp. 422-425, May 1999.
[6] Y. Nishio and A. Ushida, “Quasi-Synchronization
Phenomena in Chaotic Circuits Coupled by One Resistor,” IEEE Trans. Circuits Syst. I, vol. 43, no. 6,
pp. 491–496, Jun. 1996.

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