Clustering Phenomena in Complex Networks ofChaotic Circuits
Yuji Takamaru, Hiroshige Kataoka, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan
phone:+81-88-656-7470, fax: +81-88-656-7471
Email:{takamaru, hiroshige, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— In this study, we investigate synchronization phenomena in coupled chaotic circuits which are connected withdistance information. We confirm that the chaotic circuits arranged in the near distance are synchronized at in-phase state,
and the coupled circuits with the far distance could not besynchronized. Namely, clustering phenomena of coupled chaoticcircuits is observed on two-dimensional place.

II. Circuit model
Figure 1 shows the circuit model, which is called NishioInaba circuit investigated in [4]-[6].

I. Introduction
Synchronization phenomena are very typical phenomena in
the field of natural science. Recently, many studies have been
investigated synchronization of chaotic circuits. It is applied in
the field of engineering, physics and biology and so on. Also
one of the characteristic of the brain is synchronization ofneuronal activity. In addition, this phenomenon is considered
to play an important role in brain information processing.
Then, we consider that it is very important to investigate thesynchronization phenomena of coupled chaotic circuits for thefuture engineering applications such as analysis of the brain
activity and the realization of a brain computer.
In recent years we often deal with huge amounts of data. It
is useful to partition such data sets into cluster. Also, such dataneed to divide same group for clustering. Moreover, clustering
is one of the interesting phenomena. The phenomena arealso applied to many application, such as data mining, image
processing and biological field. Many kinds of models andalgorithms by using Coupled Map Lattics (CML) are proposedfor clustering [1]-[3]. However, there are not many discussions
clustering of coupled chaotic circuits.
In this study, we investigate the clustering phenomena
resulting from the synchronization phenomena observed incoupled chaotic circuits when the chaotic circuits are placed on
2-dimensional space. In this circuit system, the chaotic circuits
are coupled with the distance information. We observe that thechaotic circuits arranged in the near distance are synchronized
at in-phase state, and the coupled circuits with the far distancecould not be synchronized. From the results we clear up therelationship between clustering and synchronization phenomena. Furthermore, we also carry out the circuit experiments
and compare with computer simulation results.
The paper is organized as follows. In Section II, we describe the coupled chaotic circuits. Section III, the observed
interesting synchronization phenomena are explained. Finally,
Section IV concludes the paper.
978-1-4673-0219-7/12/$31.00 ©2012 IEEEFig. 1.

Chaotic circuit.

The circuit consists of a negative resistance, a nonlinear
resistance consisting of two diodes, capacitor and two inductors. The approximate the I − V characteristic of the nonlinear
resistance shows the following equation and the parameter rd
is the slope of the nonlinear resistance.
 
rd V Vi2 +  − i2 −  .
(1)
vd (i2 ) =2rdrd
Then the circuit dynamics is described by the following
piecewise-linear third-order ordinary diﬀerential equation:
di1= v + ri1
dt
di2= v − vd (i2 )L2dtdv= −i1 − i2 .
Cdt
L1

By changing the variables such that
√C
L1C
i1 =V x; i2 =Vy; v = Vz;
L1L2

√
L1C
L1C
r= α;
= β; rd= δ;
L1L2L2dt = L1Cτ; ” · ” =dτ914(2)

TABLE I

That the equation (2) is normalized asẋ = αx + zẏ = z − f (y)ż = −x − βywhere f (y) is described as follows; 
1 1δ y +  − y −  .
f (y) =2δδThe location of seven circuits
location
1234567(3)

x0.15
0.10
0.30
0.70
0.90
0.80
0.55

y0.35
0.10
0.20
0.60
0.80
0.95
0.80

(4)

Figure 2 shows the chaotic attractor generated from thecircuit by using computer simulation (Fig. 2 (a)) and circuitexperiment (Fig. 2 (b)). For the computer simulation, we
set the parameters as α = 0.460, β = 3.0 and δ = 470.
For the circuit experiment, the parameters are fixed with
L1 = 500[mH], L2 = 200[mH], C = 0.0153[μF], and rd =1.46[MΩ].

(a) Computer simulation.
Fig. 2.

(b) Circuit experiment.

Chaotic attractor.

Fig. 4.

III. Synchronization phenomena

Coupling between the first chaotic circuit and the others.

We consider the globally coupled chaotic circuits:

A. Network of seven chaotic circuitsdxi= αxi + zidτdyi= zi + f (y)dτNdzi= −xi − βyi −γi j (zi − z j )dτi, j=1

First, we investigate the basic clustering phenomena whenseven chaotic circuits are coupled globally. The arrangementof seven chaotic circuits is shown in Fig. 3. Table 1 shows
the locations of the chaotic circuits. All circuits are connectedeach other by resistors. Figure 4 shows the coupling methodof the first chaotic circuit as an example.

(5)(i, j = 1, 2, · · ·, N)
where, i in the equation represents the circuit itself, and j isthe coupling with other circuits. The parameter γ represents
the coupling strength between the circuits. In this simulation,
we set the coupling parameter value γi, j to correspond thedistance between the circuits by the following equation:
γi, j =Fig. 3.

Arrangement of seven chaotic circuits.

g.
(lengthi, j )2(6)The parameter g is a weight parameter that determines thecoupling strengths. In this case, we set the parameter g =0.00327. And lengthi, j denotes the Euclidean distance between
the i − th and the j − th circuits.
Figure 5 shows the computer simulated results obtained
from the seven chaotic circuits arranged as shown in Fig. 3.
We also confirm the same synchronization and clustering
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phenomena by the circuit experiments as shown in Fig. 6.
From these results, we confirm that the first, the second
and the third chaotic circuits are synchronized at in-phasestate, and also the fourth, fifth, sixth and seventh chaotic
circuits synchronize with in-phase state. However, the first andthe fourth chaotic circuits are not synchronized. Namely, thecircuits form two clusters defined by chaos synchronization.
Figure 7 shows the clustering results of seven chaotic circuits.

(a) 1-2(b) 1-3

Fig. 7.

(c) 1-4

Clustering result for seven chaotic circuits.

of the chaotic circuits. Similarly, all the chaotic circuits areconnected each other by resistors, and the coupling strength
between two circuits is determined by Eq. (6). In this case,
we set the parameter g = 0.000025.
(d) 4-5(e) 4-6(f) 4-7

Fig. 5. Phase diﬀerence between two circuits (Computer simulations).

(a) 1-2(b) 1-3(c) 1-4Fig. 8.

(d) 4-5
Fig. 6.

(e) 4-6(f) 4-7

Phase diﬀerence between two circuits (Circuit experiments).

B. Network of 20 chaotic circuits
Next, we investigate the case of general networks. Twenty
chaotic circuits are placed on 2-dimensional including thepositional information. The arrangement of twenty chaoticcircuits is shown in Fig. 8. Also, Table 2 shows the locations

Arrangement of twenty chaotic circuits.

Figure 9 shows the computer simulation results of the phase
diﬀerences. From these results, we confirm that the first andthe second chaotic circuits are synchronized. However, the firstand the group of seventh chaotic circuit are not synchronized.
Also the first and the group of thirteenth circuit are notesynchronized. Similarly, between seventh circuit and group ofeight circuit are synchronized, between thirteen circuit andgroup of fourteen circuit are synchronized, however between
group of seventh circuit and group of thirteen circuit isasynchronous, From these results, the circuits can form threeclusters defined by chaos synchronization. Further, circuits 18,
19 and 20 are isolated, namely they do not synchronize to anyof the others. Figure 10 shows the clustering results of twenty

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TABLE II
The location of 20 circuits
location
1234567891011121314151617181920

x
0.13
0.10
0.10
0.16
0.20
0.20
0.80
0.80
0.85
0.85
0.90
0.90
0.30
0.30
0.35
0.36
0.40
0.30
0.55
0.65

y0.20
0.16
0.10
0.13
0.10
0.20
0.36
0.33
0.36
0.30
0.30
0.36
0.90
0.83
0.85
0.80
0.85
0.55
0.45
0.65

(a) 1-2(b) 1-3(c) 1-7(d) 1-13(e) 7-8(f) 7-11(g) 7-13(h) 7-19(i) 13-14(j) 13-16(k) 16-18(l) 19-20

chaotic circuits.
IV. Conclusion
In this study, we have investigated the clustering phenomena
resulting from the synchronization phenomena observed incoupled chaotic circuits when placed on 2-dimensional. The
chaotic circuits are coupled with the distance information. By
using the computer simulations and the circuit experiments, wehave confirmed that the chaotic circuits arranged in the near
distance are synchronized at in-phase state, and the coupledcircuits with the far distance could not be synchronized. We
could cluster the coupled chaotic circuits into several groups
depending on the circuit arrangement.
For the future works, we would like to develop the eﬃcientmethod to distinguish cluster. Applying this proposed system
to the real data is also important project to us.

Fig. 9. Phase diﬀerence between two circuits (Computer simulations).

Acknowledgment
This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 22500203.
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[5] C. Bonatto and J. A. C. Gallas, “Periodicity Hub and Nested Spirals inthe Phase Diagram of a Simple Resistive Circuit,” Phys. Rev. Lett., 101,054101, Aug. 2008.
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105, 074102, Aug. 2010.

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Fig. 10.

The clustering result of twenty chaotic circuits.