Clustering Phenomena Considering the Densityof Coupled Chaotic Circuits NetworksYuji Takamaru†, Yoko Uwate†, Thomas Ott‡ 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, uwate, nishio}@ee.tokushima-u.ac.jp‡Institute of Applied Simulation, Zurich University of Applied Sciences,
Einsiedlerstrasse 31a, 8820 Waedenswil, Switzerlandphone:+41-058-934-56-84
Email:thomas.ott@zhaw.ch

Abstract— We have investigated the clustering phenomena
observed from coupled chaotic circuits networks. The networks
are globally coupled with each other by resistor using the distanceinformation. The coupling strength is depended on the distance
between the circuits.
In this study, we consider the relationship between densityand distance of each cluster that configured by chaotic circuits.
For this investigation, we study the clustering phenomena when
we change the number of circuits in a cluster. Furthermore,
we consider the clustering phenomena by changing the distance
between each cluster composed of chaotic circuits using 2dimensional place.

I. I NTRODUCTION
Nowadays, we often have to deal with huge amounts of data.
To structure and analyze such data it is useful to partition
the data set into clusters. The idea of clustering algorithms
is to find clusters consisting of similar elements. Clustering
algorithms have widespread applications in different fields,such as business data mining, image processing or the analysisof biological data. Clustering of data is helpful high-speed
information processing and observing complex relationship.
Along with the variety of applications there is a variety ofdifferent clustering algorithms. Some algorithms propose toutilize synchronization phenomena, for instance in Coupled
Map Lattics (CML), for clustering [1]-[3]. Previously, thesestudies are used discrete time model for clustering, however
analysis of using a continuous time model has not almost studies. Continuous time model can treat high-speed information
processing more than discrete time model.
Synchronization is a prominent phenomenon, widely observed and studied in the field of natural and technical sciences. One possibility to get insight into the role of synchronization phenomena is the analysis of electronic circuits.
However, there are not many studies on synchronization-based
clustering in networks of coupled chaotic circuits. We cansee the chaos synchronization when we connect chaos circuits
each other. We applied to chaos synchronization for clustering.
In a previous study, we investigated 2-dimensional networks
of complex chaotic circuits, where the coupling strength978-1-4577-1729-1/12/$26.00 ©2012 IEEE.

reflected the distance information [4]-[5]. We showed thatthe circuits arranged close to each other could synchronize
in-phase, whereas coupled circuits far from each other could
not be synchronized. We also observed that such networks ofcoupled chaotic circuits could split into different synchronizedgroups, revealing a clustering phenomenon.
In this study, we consider the relationship between densityand distance of each cluster that configured by chaotic circuits.
For this investigation, we study the clustering phenomena
when we change the number of circuits in a cluster. Furthermore, we consider the clustering phenomena affect thedistance between each cluster composed of chaotic circuits
using 2-dimensional place in continuous time model. Also, weinvestigate the relation of density and distance of each cluster.

II. C IRCUIT MODEL
Figure 1 shows the model of the used chaotic circuit called
Nishio-Inaba circuit, investigated in [6]-[8].

Fig. 1.

Chaotic Nishio-Inaba circuit.

The circuit consists of a negative resistance, a nonlinear
resistance consisting of two diodes, a capacitor and twoinductors.
The normalized equations of this circuit is described as647

follows:

III. C LUSTERING PHENOMENA𝑥˙ = 𝛼𝑥 + 𝑧𝑦˙ = 𝑧 − 𝑓 (𝑦)𝑧˙ = −𝑥 − 𝛽𝑦where 𝑓 (𝑦) is described as follows: )(1 𝛿 1  𝑓 (𝑦) =𝑦 +  − 𝑦 −  .
2 𝛿
𝛿

(1)

(2)

Figure 2 shows the chaotic attractor generated by thecircuit, obtained from a computer simulation (Fig. 2 (a)) andmeasured in a circuit experiment (Fig. 2 (b)). For the computer
simulation, we set the parameters as 𝛼 = 0.460, 𝛽 = 3.0 and𝛿 = 470. For the circuit experiment, the parameters were fixedwith 𝐿1 = 500[𝑚𝐻], 𝐿2 = 200[𝑚𝐻], 𝐶 = 0.0153[𝜇𝐹 ], and𝑟𝑑 = 1.46[𝑀 Ω].

A. Considering the density of the cluster
First, we investigate the case of the relationship between
clustering and density of a cluster. In a previous study, wehave observed that the coupled chaotic circuits arranged close
to each other could synchronize in-phase, whereas coupledchaotic circuits far from each other could not be synchronized. We study the clustering phenomena when we change
the number of circuits in a cluster. We set 2 patterns forarrangement of the circuits. The arrangement of the circuits
are shown in Fig. 3. Arrangement of Fig. 3 (a) is composed
the same number of circuits in all groups, Fig. 3 (b) is changed
the number of circuits in the middle group. Also, all circuits
are connected with each other by resistors and the couplingstrength between two circuits is determined by Eq. (4).

(a) Same number of groups.
(a) Computer simulation.

Fig. 3.
Fig. 2.

(b) Different number of groups.

(b) Circuit experiment.

Arrangement of chaotic circuits.

Chaotic attractor.

We can consider the following equations when each chaoticcircuits coupled globally with each other.
𝑑𝑥𝑖= 𝛼𝑥𝑖 + 𝑧𝑖𝑑𝜏𝑑𝑦𝑖= 𝑧𝑖 + 𝑓 (𝑦)𝑑𝜏𝑁∑𝑑𝑧𝑖= −𝑥𝑖 − 𝛽𝑦𝑖 −𝛾𝑖𝑗 (𝑧𝑖 − 𝑧𝑗 )𝑑𝜏𝑗=1

Figure 4 shows the clustering results. All circuits group aresynchronized in one cluster from the result of Fig. 4 (a), however we can see the 2 cluster from synchronization between themiddle group and the other groups from the result of Fig. 4 (b).
Figure 5 shows the phase difference between the middle groupand the other groups.

(3)(𝑖, 𝑗 = 1, 2, ⋅ ⋅ ⋅, 𝑁 )where 𝑖 in the equation represents the circuit itself, and 𝑗indicates the coupling with other circuits. The parameter 𝛾𝑖𝑗
represents the coupling strength between the circuits. The
value of 𝛾𝑖𝑗 reflects the distance between the circuits in an
inverse way, described by the following equation:(a) One cluster𝑔𝛾𝑖𝑗 =.
(𝑙𝑒𝑛𝑔𝑡ℎ𝑖𝑗 )2Fig. 4.

(b) Two clusters

The clustering results(4)

𝑙𝑒𝑛𝑔𝑡ℎ𝑖𝑗 denotes the Euclidean distance between the 𝑖 − 𝑡ℎcircuit and the 𝑗 − 𝑡ℎ circuit. The parameter 𝑔 is a weighting
or scaling parameter that determines the coupling strengths.
For our study, we set the parameter as 𝑔 = 1.0 × 10−4 .

We consider the relationship between the cluster and density
in a group. We change the density in the middle circuits groupfrom 4 to 9 as shown in Fig. 6. The results of phase difference
between the middle group and the other groups arrangementin Fig. 6 correspond to Fig. 7.

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(a) Synchronization
Fig. 5.

(b) Not synchronization(a) 4 chaotic circuits(b) 5 chaotic circuits(c) 6 chaotic circuits(d) 7 chaotic circuits(e) 8 chaotic circuits(f) 9 chaotic circuits

Phase difference between middle group and other groups.

(a) 4 chaotic circuits(b) 5 chaotic circuits(c) 6 chaotic circuits(d) 7 chaotic circuits(e) 8 chaotic circuits(f) 9 chaotic circuits

Fig. 6.

Fig. 7.

Phase difference between middle group and other groups.

Change the number of circuits in the middle cluster.

From these results in Fig. 7, we can make the following
observations. We can obtain the synchronization phenomena
between the middle group and the other groups when themiddle circuits is up to 6. A circuit in the middle group growin density from 7 to 9, the middle group is not synchronizedfrom other groups. That is the middle group of isolated circuits
from other groups more than 9. From the above results, theclustering phenomena for chaotic circuit network is related todensity of the circuits in a cluster.
B. Considering the distance for each cluster
In a next step, we investigate the clustering phenomena
obtained by changing the distance between each cluster. Eachgroups consist of 4 circuits are gradually coupled with eachother and each groups close to the middle group consistof 9 chaos circuits. Figure 8 shows the example for close
to the middle group. Now this state is clustered into 2
groups between the middle group and the other groups fromchaos synchronization. We consider the form clustering bysynchronization when arrangement the number of circuits canbe obtained clustering phenomena (the middle group consistof 7 to 9 circuits). Now, the distance 𝑑 between the middlegroup and each groups is 𝑑 = 3.54. The method for measuring
the distance is shown in Fig. 9.
Figure 10 shows the computer simulation results of thephase differences between the middle group and the othergroups when the middle group consist of 9 chaotic circuits. WeFig. 8.

Close to the middle group each cluster.

set the each distance from the middle group in Fig. 10. We canderive the following observations, exemplified by a selection
pairwise comparison. Clustering can be observed for values𝑑 ≥ 0.325. This clustering corresponds to the result shown inFig. 4 (b). Thus, we have 2 distinguishable groups. All circuits
are synchronized in one cluster if 𝑑 ≤ 0.297. Synchronization
up to 𝑑 = 0.240 is strongly. We can see from the above that
it is related to the density and distance of the chaotic circuits
for clustering.
Next, we change the number of circuits in the middle groupwhen the middle group consist of 7 chaotic circuits in caseof clustering phenomena obtained. Similarly, we simulate thephase differences between the middle group and the othergroups shown in Fig. 11. The results shown in Fig. 11 issimilar to the results when the middle group consist of 9
chaotic circuits. Thus, we found that clustering phenomena
occurs depending on the distance of the cluster and density ofthe circuits. Furthermore, all groups are synchronized when
the number of chaotic circuits in the middle group less to 7.

649

Fig. 9.

Definition of the distance 𝑑.

(a) 𝑑 = 0.325(b) 𝑑 = 0.297(c) 𝑑 = 0.269(d) 𝑑 = 0.240(e) 𝑑 = 0.212(f) 𝑑 = 0.184

Fig. 11. Phase difference when the middle group consist of 7 circuits.

ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 22500203.
(a) 𝑑 = 0.325(d) 𝑑 = 0.240(b) 𝑑 = 0.297(e) 𝑑 = 0.212

R EFERENCES(c) 𝑑 = 0.269(f) 𝑑 = 0.184

Fig. 10. Phase difference when the middle group consist of 9 circuits.

IV. C ONCLUSION
In this study, we have examined the relationship between
density and distance of each cluster that configured by chaotic
circuits. For this investigation, we have studied the clustering
phenomena when we change the number of circuits in acluster. Based on computer simulation, we have obtained theclustering phenomena from coupled chaotic circuits using
2-dimensional place in continuous time model. We showed
the clustering phenomena affected other cluster when density
in the chaotic circuits is high. Furthermore, we found thatclustering phenomena occurs depending on the distance of thecluster and density of the circuits.
In our future work, we would like to develop a more efficientmethod for determining the phase synchronization. In this casewe will research detail the relationship between the clusteringand density of coupled chaotic circuits. Besides that we apply
this method for data mining, image processing and other many
applications.

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