IEEE Workshop on Nonlinear Circuit Networks
December 9-10, 2011

Clustering Phenomena of Complex
Chaotic Circuit Networks
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
Email:{takamaru, hiroshige, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— This paper presents synchronization phenomena ofcoupled chaotic circuits when chaotic circuits are connected globally with distance information. By using computer simulations
and circuit experiments, clustering phenomena of coupled chaoticcircuits is observed on two-dimensional place.

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 appliedin the field of engineering, physics, 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. By
dividing the such data into similar groups for clustering, theinformation processing can be treated in high speed. Moreover,clustering is one of the most interesting nonlinear phenomena.
The phenomena are also applied to many application, suchas data mining, image processing and biological field. Manykinds of models and algorithms by using Coupled Map Lattice(CML) are proposed for clustering [1]-[3]. The systems carriedout for discrete-time mathematical models. However, there
are not many discussions for clustering of continuous-timeelectrical circuits such as coupled chaotic circuits.
In this study, we consider the coupled chaotic circuits
networks. In order to investigate basic clustering phenomena,
seven chaotic circuits are globally coupled with the distanceinformation. The clustering phenomena are observed when thechaotic circuits are placed on 2-dimensional. We make clear
the relationship between clustering phenomena and synchronization phenomena. We observed that the synchronizationof circuit networks depends on the distance information.
Furthermore, we also carry out the circuit experiments andcompare with computer simulation results.

Fig. 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.
()
rdVV
vd (i2 ) =i2 +− i2 −.
(1)
2rdrd
Then the circuit dynamics is described by the following
piecewise-linear third-order ordinary diﬀerential equation:
di1= v + ri1dt
di2L2= v − vd (i2 )dtdv
C= −i1 − i2 .
dt
L1By changing the variables such that√√C
L1 C
i1 =V x, i2 =Vy, v = Vz;
L1L2
√
√
C
L1
L1 C
r= α,= β, rd= δ,
L1L2L2√dt = L1 C τ , ” · ” =dτ
That the equation (2) is normalized asII. Circuit modelẋ = αx + zFigure 1 shows the circuit model, which is called NishioInaba circuit investigated in [4]-[6].

ẏ = z − f (y)ż = −x − βy- 81 -(2)(3)

TABLE I

where f (y) is described as follows;()δ11f (y) =y+ − y−.
(4)
2
δ
δ
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.

The location of seven circuits
location
1234567x0.15
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

(b) Circuit experiment.

Chaotic attractor.

III. Synchronization phenomena
We investigate the clustering phenomena when sevenchaotic circuits are coupled globally. The arrangement of sevenchaotic circuits is shown in Fig. 3. Table I shows the locationsof the chaotic circuits. All the circuits are connected globallyeach other by resistors. Figure 4 shows the coupling methodof the first chaotic circuit as an example.

Fig. 4.

Coupling between the first chaotic circuit and the others.

where, i in the equation represents the circuit itself, and jis the coupling number with other circuits. The parameterγ represents the coupling strength between the circuits. In
this simulation, we set the coupling parameter value γi, j tocorrespond the distance between the circuits by the following
equation:
γi, j =

Fig. 3.

Arrangement of seven chaotic circuits.

We consider the globally coupled chaotic circuits:dxi= αxi + zidτdyi= zi + f (y)dτN∑dzi= −xi − βyi −γi j (zi − z j )dτi, j=1(5)(i, j = 1, 2, · · ·, N)- 82 -g.
(lengthi, j )3(6)The parameter of g is a variable parameter that determining
the each coupling weight. In this case, we set the parameterg = 0.00002. 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
phenomena by the circuit experiments as shown in Fig. 6.
From these results, we confirm that the first, the second andthe third chaotic circuits are synchronized with in-phase state,
and also the other chaotic circuits synchronize with in-phasestate. However, between the first and the fourth chaotic circuits
are not synchronized. Namely, the networks of chaotic circuits
can divide two groups from synchronization. Figure 7 shows
the clustering results of seven chaotic circuits.

(a) 1-2(b) 1-3(c) 1-4Fig. 7.

(d) 4-5(e) 4-6References(f) 4-7

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

Fig. 6.

(a) 1-2(b) 1-3(c) 1-4(d) 4-5(e) 4-6(f) 4-7

Clustering result for seven chaotic circuits.

Phase diﬀerence between two circuits (Circuit experiments).

IV. Conclusion
In this study, we have investigated the coupled chaotic
circuits networks. The seven chaotic circuits are globallycoupled with the distance information. We have observed thebasic clustering phenomena resulting from the synchronization
phenomena. From the computer simulations and the circuits
experiments, we have made clear that the chaotic circuits
arranged in a close distance are synchronized at in-phase state,
and when the coupled circuits with the far distance could not
be synchronized. The coupled chaotic circuits networks could
cluster into two groups. We observed the networks depend on
the distance information.
Acknowledgment
This work was partly supported by The Nakajima Foundation.

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