Journal of Signal Processing, Vol.17, No.4, pp.103-106, July 2013

SELECTED PAPER

Dependence of Clustering Patterns on Density of Chaotic Circuits in Networks
Yuji Takamaru1 , Yoko Uwate1 , Thomas Ott2 and Yoshifumi Nishio1
1

Department 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
E-mail: {takamaru, uwate, nishio}@ee.tokushima-u.ac.jp
2
Institute of Applied Simulation, Zurich University of Applied Sciences
Einsiedlerstrasse 31a, 8820 Waedenswil, Switzerland
Phone: +41-058-934-56-84
E-mail: thomas.ott@zhaw.ch

Abstract

phenomenon. However, not all phenomena have been suﬃIn this study, we investigate clustering patterns generated in cou- ciently investigated. Therefore, we consider that our study
pled chaotic circuits in networks. In these networks, the coupling provides a new approach to investigating the synchronization
strength reflects the distance information, and each chaotic circuit and clustering phenomena in coupled chaotic circuits.
In a previous study, we investigated the relationship beis connected to all other chaotic circuits. We consider the relationtween
clustering and the density of coupled chaotic circuits inship between coupling strength and phase diﬀerence by changing
two-dimensional
space [4]-[6]. In our previous investigation,
the scaling parameter of coupling strength. Furthermore, we deterthe
coupling
strengthreflected the distance information andmine the various phase synchronization patterns when we changewechangedthenumberof circuits in a cluster. We showed
the number of chaotic circuits.
that clustering phenomena aﬀected other clusters when thedensity in the chaotic circuits was high. We also observed
1. Introductionthat networks of coupled chaotic circuits could be split into
Recently, it has become necessary to deal with a large
diﬀerent synchronized groups.
amount of information in our lives. Our society can be conIn this study, we investigate clustering patterns generated
sidered as an advanced information network society. Because
in coupled chaotic circuits in networks. In these networks,of this, a large amount of information is gradually processedchaotic circuits are connected to all other chaotic circuits.
day by day. Therefore, the ideas of clustering algorithms have
For this investigation, we change the parameters of coupling
been proposed and applied to information processing. Clusstrength and the density of chaotic circuits. From the results,
tering algorithms have widespread applications in diﬀerent
three clustering patterns can be observed in the networks.
fields, such as business data mining, image processing and
We consider the relationship between the scaling parameter
analysis of biological data. There are a variety of diﬀerent
and the phase diﬀerence by changing the scaling parameterclustering algorithms along with many applications. Manyof coupling strength. Additionally, we determine the various
algorithms were proposed to utilize the synchronization phephase synchronization patterns when we change the density
nomenon, for instance, in coupled map lattices (CMLs), forof chaotic circuits.
clustering [1]-[3]. Previously, many of these studies used a
discrete-time model for clustering, and hardly any analysis 2. Circuit Modelusing a continuous-time model has been carried out. ThereFigure 1 shows the model of the chaotic circuit investigatedfore, we have focused on research on clustering phenomena
in [7]-[9].
using electronic circuits using a continuous-time model.
On the other hand, the synchronization phenomenon is atypical phenomenon when we analyze coupled chaotic circuits. This phenomenon can be widely observed and has beenstudied in the fields of natural and technical sciences. In order to understand the synchronization phenomenon in detail,we analyze electronic circuits. Coupled chaotic circuits arecomposed of electronic circuits and a suitable model to anaFigure 1: Chaotic circuit
lyze the synchronization phenomenon. Moreover, we can observe various phenomena in addition to the synchronization

Journal of Signal Processing, Vol. 17, No. 4, July 2013

103

The following equations are the circuit equations when
The simulation results of clustering in these networks areeach chaotic circuit is coupled globally with all other circuits. shown in Fig. 3. From the simulation results, all chaotic circuits are synchronized in one cluster as shown in Fig. 3(a);dxihowever, we can observe two-clusters from chaos synchro= αxi + zidτ
nization between the high-density group and the low-densitydyi= zi + f (y)(1) groups in Fig. 3(b).
dτ
N
∑
dzi
= −xi − βyi −γi j (zi − z j )dτ
j=1(i, j = 1, 2, · · ·, N)

For the computer simulation, we set the parameters as α =0.460, β = 3.0 and δ = 470. The function f (y) can be described as a three-segment piecewise-linear function. The
value of γi j reflects the distance between the circuits in an
inverse manner, described by the following equation:

γi j =

g
di2j

(2)

(a) One cluster(b) Two clusters

Figure 3: Clustering results
From these results, clustering phenomena are related to thedensity of coupled chaotic circuits.

Here, di j denotes the Euclidean distance between the ith cir- 3.2 Investigation of clustering phenomena
cuit and the jth circuit. The parameter g is a scaling parameter
Next, we investigate the clustering result shown in Fig. 3(b)that determines the coupling strength.
in detail. This network can be divided into two-clusters withhigh density and low density as a result of chaos synchroniza3. Clustering Phenomenation. We consider the state of this network when we change
the parameter g determined by Eq. (2). Furthermore, we cal3.1 Clustering phenomena
In this section, we investigate clustering phenomena when culate the phase diﬀerence between chaotic circuits by comwe configure a network of coupled chaotic circuits in two- puter simulation.
For this simulation, the number of iterations is set to τ p =dimensional space. In our previous study [6], we investigated the relationship between clustering and the density of 10, 000 to calculate the result more precisely. Figure 4 shows
coupled chaotic circuits by changing the density of chaotic the phase diﬀerence between the two chaotic circuits whencircuits. The arrangements of chaotic circuits are shown in we change the value of parameter g.
Fig. 2. Figure 2(a) is composed of groups with the same
number of chaotic circuits; however, Fig. 2(b) is composedof a group with a high density of chaotic circuits inside andsome groups with a low density of chaotic circuits outside.
In these networks, we replace chaotic circuits with a simple
model such as a small circle.

Figure 4: Relationship between g and phase diﬀerence(a) Same number of groups(b) Diﬀerent numbers ofgroupsFigure 2: Arrangements of chaotic circuits104

We define the state of synchronization patterns from theaverage phase diﬀerence when we perform τ p = 10, 000 iterations. A synchronized state can be defined if the averagephase diﬀerence is below 40◦ . Similarly, we define an asynchronous state as having an average phase diﬀerence between
70◦ and 110◦ . In the region between g = 2.0 × 10−6 andJournal of Signal Processing, Vol. 17, No. 4, July 2013

Table 1: State of networks
Value of g (×10−4 ) In - In
In - Out0.02 ≤ g ≤ 0.3
Syn.
Not-syn.
Syn.
Not-syn.
0.4 ≤ g ≤ 2.0
Syn.
Syn.
3.0 ≤ g ≤ 4.0

Out - Out
Not-syn.
Syn.
Syn.

g = 3.0 × 10−5 , the inside group of chaotic circuits is synchronized in one cluster; however, the other chaotic circuits
composing the outside groups are not synchronized. In theregion between g = 4.0 × 10−5 and g = 2.0 × 10−4 , both of
the inside and outside groups of chaotic circuits are synchronized in one cluster. Therefore, clustering phenomena can beobserved in this region. Finally, all chaotic circuits are synchronized in one cluster in the region between g = 3.0 × 10−4
and g = 4.0 × 10−4 . Thus, three-clustering patterns can beobserved in this network from the average phase diﬀerence.

Table 2: The phase diﬀerence (in-in)
Density of circuits
Ave.
Max.
2
18.894◦102.898◦3
19.502◦109.258◦4
19.029◦107.431◦◦5
17.06786.363◦◦
6
16.27294.816◦
7
17.554◦106.801◦8
16.678◦101.588◦◦918.375118.286◦Min.
0.001◦
0.001◦
0.002◦
0.001◦
0.002◦
0.002◦
0.001◦
0.003◦

with the outside circuits when the density of the inside circuits is between 1 and 5; however, the state for other densities
is not clear. Therefore, we use the frequency distribution toreveal the synchronized state.

4. Relationship between Phase Diﬀerence and Density
In this section, we calculate the phase diﬀerence between
inside chaotic circuits, and between inside chaotic circuits
and outside chaotic circuits when we change the density ofchaotic circuits in the inside group shown in Fig. 3. Moreover, we calculate the maximum and minimum phase diﬀerence and its range.. Here, we change the number of chaotic
circuits in the inside group from 1 to 9.
First, we calculate the phase diﬀerence inside chaotic circuits shown in Tab. 2. Table 2 shows the phase diﬀerence,
maximum value and minimum value. Figure 5 shows theobtained relationship between the phase diﬀerence and thedensity of chaotic circuits. In this result, the average phase
diﬀerence is below 40◦ . Thus, the inside chaotic circuits aresynchronized regardless of the density.

Figure 5: Phase diﬀerence and range of phase diﬀerence (in-in)
Next, we calculate the phase diﬀerence between the insidechaotic circuits and outside chaotic circuits as shown in Table
3. Table 3 shows the phase diﬀerence, maximum value andminimum value. Figure 6 shows the obtained relationship between the phase diﬀerence and the density of chaotic circuits.
From this result, the inside chaotic circuits are synchronized

Journal of Signal Processing, Vol. 17, No. 4, July 2013

Figure 6: Phase diﬀerence and range of phase diﬀerence (in-out)
Table 3: The phase diﬀerence (in-out)
Density of circuits
Ave.
Max.
Min.
◦
◦
1
23.64085.7730.003◦
2
17.425◦77.697◦0.009◦◦◦
3
16.44287.3980.001◦◦◦
417.50083.5910.008◦
532.938◦112.724◦ 0.006◦
6
63.203◦178.035◦ 0.135◦◦
7
90.618179.972◦ 0.032◦
8
94.031◦179.998◦ 0.043◦◦
97.988179.996◦ 0.007◦
9
Figure 7 shows the synchronization patterns obtained from
Lissajous figures. Each pattern in these figures corresponds
to the values in Fig. 6 and Table 3. We calculate the numberof phase domains as a function of each phase diﬀerence. The
number of iterations is set to τ p = 10, 000. From these figures, in the small region of the average phase diﬀerence aresynchronized with small value for phase domains. Also, for aphase diﬀerence of approximately 90◦ , are not synchronizedfrom the Lissajous figures and the phase domain. However,there are six chaotic circuits, the synchronization state is unclear. Additionally, the maximum value of the phase diﬀerence is 178.035◦ in the phase domain, and the phase domain

105

(a) 1 chaotic circuit(b) 2 chaotic circuits(c) 3 chaotic circuits(d) 4 chaotic circuits(e) 5 chaotic circuits(f) 6 chaotic circuits(g) 7 chaotic circuits(h) 8 chaotic circuits(i) 9 chaotic circuits

Figure 7: State of synchronicity (horizontal axis: phase diﬀerence, vertical axis: frequency)

is between 0.135◦ and 178.035◦ ; namely, when there are six and practical applications to enrich our lives.
chaotic circuits, they are not synchronized. Therefore, we
define the state of synchronicity or asynchronicity using the
Acknowledgment
average phase diﬀerence and phase domain.
As mentioned before, we define the synchronized state as This work was partly supported by a JSPS Grant-in-Aid forhaving a phase diﬀerence is below 40◦ . Also, we define the Scientific Research (No. 22500203).
asynchronous state as having a phase diﬀerence between 70◦
and 110◦ and a phase domain between 0◦ and 180◦ . Thus,
References
we define synchronization patterns using the average phase
Kaneko: Clustering, coding, switching, hierarchical ordering, anddiﬀerence and phase domain. Furthermore, some clustering [1] K.
control in a network of chaotic elements, Physica D, Vol. 41, pp. 137patterns can be observed in the coupled chaotic circuits in
172, 1990.
[2] T. Ott, M. Christen and R. Stoop: An unbiased clustering algorithm
networks as a result of chaos synchronization.
5. Conclusions
In this study, we have investigated clustering patterns generated in coupled chaotic circuits in networks. In these networks, the coupling strength reflects the distance information,and each chaotic circuit is connected to all other chaotic circuits. For this investigation, we have changed the scaling parameter of coupling strength and the number of chaotic circuits composing a cluster. We have observed some clustering
patterns resulting from chaos synchronization. From computer simulation results, we have confirmed that the state ofclustering patterns depends on the scaling parameter g and thedensity of chaotic circuits in networks. Furthermore, we haveclarified that it is eﬃcient to use the average phase diﬀerence
and phase domain to define synchronization patterns.
In our future work, we intend to study the clustering phenomena in large-scale networks. Additionally, we hope to apply this clustering method for data mining, image processing

106

based on self-organization processes in spiking neural networks, Proc.
NDES’06, pp. 143-146, 2006.
[3] L. Angelini, F. D. Carlo, C. Marangi, M. Pellicoro and S. Stramaglia:
Clustering data by inhomogeneous chaotic map lattice, Phys. Rev. Lett.,Vol. 85, pp. 554-557, 2000.
[4] Y. Takamaru, H. Kataoka, Y. Uwate and Y. Nishio: Clustering phenomena in complex networks of chaotic circuits, Proc. ISCAS’12, pp.
914-917, 2012.
[5] Y. Takamaru, Y. Uwate, T. Ott and Y. Nishio: Clustering phenomena of
coupled chaotic circuits for large scale networks, Proc. NDES’12, pp.
70-73, 2012.
[6] Y. Takamaru, Y. Uwate, T. Ott and Y. Nishio: Clustering phenomena considering the density of coupled chaotic circuits networks, Proc.
APCCAS’12, pp. 647-650, 2012.
[7] Y. Nishio, N. Inaba, S. Mori and T. Saito: Rigorous analyses of windows in a symmetric circuit, IEEE Trans. Circuits Syst., Vol. 37, No. 4,
pp. 473-487, 1990.
[8] C. Bonatto and J. A. C. Gallas: Periodicity hub and nested spirals in thephase diagram of a simple resistive circuit, Phys. Rev. Lett., Vol. 101,
054101, 2008.
[9] R. Stoop, P. Benner and Y. Uwate: Real-world existence and origins of

the spiral organization of shrimp-shaped domains, Phys. Rev. Lett., Vol.
105, 074102, 2010.

Journal of Signal Processing, Vol. 17, No. 4, July 2013