2013 International Workshop on Nonlinear Circuits,
Communications and Signal Processing
NCSP'13, Isrand of Hawaii, Hawaii, USA. March 4-7, 2013

Clustering Patterns Depending on Density of Chaotic Circuits in Networks
Yuji 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
In this study, we investigate clustering patterns generated incoupled chaotic circuits networks. In these networks, the coupling strength is reflected the distance information and eachchaotic circuit is connected to all chaotic circuits. We consider the relationship between coupling strength and phase
diﬀerence by changing the scaling parameter of couplingstrength. Furthermore, we confirm the various phase synchronization patterns when we change the number of chaotic
circuits.
1. Introduction

not only synchronization phenomena. However, all phenomena are not really investigated. Therefore, we consider that
our study is new approach to investigate the synchronization
phenomena and clustering phenomena in coupled chaotic circuits.
In a previous study, we investigated the relationship between clustering and density of coupled chaotic circuits in 2dimensional place [4]-[6]. For this investigation, the couplingstrength reflected the distance information and we changed
the number of circuits in cluster. We showed that clustering
phenomena aﬀected other cluster when density in the chaotic
circuits was high. We also observed that networks of coupledchaotic circuits could split into diﬀerent synchronized groups.
In this study, we investigate clustering patterns generated incoupled chaotic circuits networks. In these networks, chaotic
circuits are connected to all chaotic circuits. For this investigation, we change the variable parameter of coupling strengthand density of chaotic circuits. From the results, the network
can be observed 3 clustering patterns. We consider the relationship between scaling parameter and phase diﬀerence bychanging the scaling parameter of coupling strength. Additionally, we confirm the various phase synchronization patterns when we change the density of chaotic circuits.

Recently, our lives deal with large amount of information.
Our society is called advanced information network society.
Because of this, many informations are gradually processed
day by day. Therefore, the ideas of clustering algorithms were
proposed and applied to information processing. Clustering
algorithms have widespread applications in diﬀerent fields,such as business data mining, image processing and analysisof biological data. There are variety of diﬀerent clustering algorithms along with the many applications. Many algorithms
were proposed to utilize synchronization phenomena, for instance in Coupled Map Lattics (CML), for clustering [1]-[3].
Previously, many of these studies were using discrete time 2. Circuit Model
Figure 1 shows the model of the chaotic circuit, investimodel for clustering, however analysis of using a continuous
time model has not almost studied. Therefore, we focus on gated in [7]-[9].
research of clustering phenomena using electronic circuits incontinuous time model.
On the other hand, synchronization phenomena is one of
typical phenomena when we analyze coupled chaotic circuits.
This phenomenon widely can be observed and studied in thefield of natural and technical sciences. In order to understandsynchronization phenomena in detail, we analyze electronic
circuits. Coupled chaotic circuit is composed of an electronic
Figure 1: Chaotic circuit.
circuits and one of suitable model to analyze the synchronization phenomena. Moreover, we can see various phenomena- 357 -

The following equations show circuit equations when eachchaotic circuit is coupled globally with each other.
dxi= αxi + zidτdyi
= zi + f (y)dτin one cluster from the result of Fig. 3 (a), however we cansee 2 clusters from chaos synchronization between high density group and same low density groups from the result of
Fig. 3 (b).

(1)

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 characteristic of the function
f (y) can be described 3-segment piecewise-linear function.
The value of γi j reflects the distance between the circuits in
an inverse way, described by the following equation:
γi j =

g.
(lengthi j )2(a) One cluster.

(b) Two clusters.

Figure 3: The clustering results.
(2)

From these results, clustering phenomena are related denlengthi j denotes the Euclidean distance between the i − th cir- sity of coupled chaotic circuits.
cuit and the j − th circuit. The parameter g is a scaling parameter that determines the coupling strengths.
3.2 Investigation of Clustering Phenomena
Next, we investigate the clustering result corresponding to
3. Clustering Phenomena
Fig. 3 (b) in detail. This network can be divided 2 clusters
between high density and low density from chaos synchro3.1 Clustering Phenomena
In this section, we investigate clustering phenomena when nization. We consider the state of this network when we
we configure network of coupled chaotic circuits in 2- change the parameter g determined by Eq. (2). Furthermore,dimensional place. In our previous study [6], we researched we calculate the phase diﬀerence between chaotic circuits usthe relationship between clustering and density of coupled ing computer simulation.
For this simulation, the iteration is set to τ p = 10, 000
chaotic circuits when we changed density of chaotic circuits.
forcalculating the result more precisely. Figure 4 shows
Arrangements of chaotic circuits are shown in Fig. 2. Figthe
phase
diﬀerence between two chaotic circuits when weure 2 (a) is composed same number of chaotic circuits, however Fig. 2 (b) is composed high density of chaotic circuits in change the value of parameter g.
inside and some low density chaotic circuits groups. In these
networks, we replace chaotic circuits with a simple model
like small circle.

(a) Same number of groups.

(b) Diﬀerent number of groups.

Figure 4: The relationship between g and phase diﬀerence.

We define state of synchronization patterns from the average of phase diﬀerence when we calculate τ p = 10, 000.
Synchronized state can be defined if the average of phase difSimulation results in these networks are shown in Fig. 3. ference below 40◦ . Similarly, we define asynchronous state
From simulation results, all chaotic circuits are synchronized that the average of phase diﬀerence is between 70◦ and 110◦ .
Figure 2: Arrangements of chaotic circuits.

- 358 -

Table 1: State of networks
The value of g
2.0 × 10−6 ≤ g ≤ 3.0 × 10−5
4.0 × 10−5 ≤ g ≤ 2.0 × 10−4
3.0 × 10−4 ≤ g ≤ 4.0 × 10−4

In - In
Syn.
Syn.
Syn.

In - Out
Not-syn.
Not-syn.
Syn.

Table 2: The phase diﬀerence (in-in)
Out - Out
Not-syn.
Syn.
Syn.

In the region between g = 2.0 × 10−6 and g = 3.0 × 10−5 inside chaotic circuits group is synchronized one cluster, however other chaotic circuits that composed outside groups are
not synchronized. In the region between g = 4.0 × 10−5 andg = 2.0 × 10−4 inside chaotic circuits group is synchronized
one cluster, also outside chaotic circuits groups are synchronized one cluster. Therefore, clustering phenomena can beobserved in this region. Finally, all chaotic circuits are synchronized in one cluster if the region between g = 3.0 × 10−4
and g = 4.0 × 10−4 . Thus, this network can be observed 3
clustering patterns from the average of phase diﬀerence.

Density of circuits
2
3
4
5
6
7
8
9

Ave.
18.894◦19.502◦19.029◦17.067◦
16.272◦17.554◦16.678◦18.375◦Max.
102.898◦
109.258◦
107.431◦86.363◦94.816◦106.801◦
101.588◦118.286◦Min.
0.001◦
0.001◦
0.002◦
0.001◦
0.002◦
0.002◦
0.001◦
0.003◦

this result, chaotic circuits between inside and outside aresynchronized when the density in inside is between 1 and 5,
however state of other density are not clear. Therefore, we use
the frequency distribution to reveal the synchronized state.

4. Relationship between Phase Diﬀerence and Density
In this section, we calculate the phase diﬀerence between
inside chaotic circuits and inside chaotic circuits, and between inside chaotic circuits and outside chaotic circuits
when we change density of chaotic circuits in inside shown in
Fig. 3. Moreover, we calculate range of phase diﬀerence between maximum and minimum. Here, we change the numberof inside chaotic circuits in inside 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 the result of relationship between phase diﬀerence and density ofchaotic circuits. In this result, the average of each phase difference is below 40◦ . Thus chaotic circuits in inside are synchronized regardless of the density.

Figure 5: Phase diﬀerence and range of phase diﬀerence (in-in).
Next, we calculate the phase diﬀerence between insidechaotic circuits and outside chaotic circuits shown in Tab. 3.
Table 3 shows the phase diﬀerence, maximum value and minimum value. Figure 6 shows the result of relationship between phase diﬀerence and density of chaotic circuits. From

Figure 6: Phase diﬀerence and range of phase diﬀerence (in-out).

Table 3: The phase diﬀerence (in-out)
Density of circuits
1
2
3
4
5
6
7
8
9

Ave.
23.640◦17.425◦16.442◦17.500◦32.938◦63.203◦90.618◦94.031◦97.988◦

Max.
85.773◦77.697◦87.398◦83.591◦112.724◦178.035◦
179.972◦
179.998◦
179.996◦Min.
0.003◦
0.009◦
0.001◦
0.008◦
0.006◦
0.135◦
0.032◦
0.043◦
0.007◦

Figure 7 shows the synchronization patterns from Lissajous figures. Each pattern in these figures correspond to
Fig. 6 and Tab. 3. We calculate the number of phase domain for the phase diﬀerence. The number of counts is set
to τ p = 10, 000. From these figures, in the small region of
the average of phase diﬀerence are synchronized with smallvalue for phase domain. Also, in the region near value of 90◦
are not synchronized from Lissajous figures and phase domain. However, in the region of number of 6 chaotic circuits,
synchronization state is not sure. Additionally, the maximum
value is 178.035◦ in the phase domain. In this region, thephase domain is between 0.135◦ and 178.035◦ , namely thisregion composed of 6 chaotic circuits is not synchronized.

- 359 -

(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 synchronization patterns (horizontal axis: phase diﬀerence, vertical axis: frequency distribution).

Therefore, we define the state of synchronization or asynchronous by using the average of phase diﬀerence and phasedomain.
As we mentioned before, we define synchronized state thatphase diﬀerence is below 40◦ . Also, we define asynchronous
state that the phase diﬀerence is between 70◦ and 110◦ andthe phase domain is between 0◦ and 180◦ . Thus, we define
synchronization patterns by using phase diﬀerence and phasedomain. Furthermore, the coupled chaotic circuits networks
can be observed some clustering patterns from chaos synchronization.
5. Conclusions
In this study, we have investigated clustering patterns generated in coupled chaotic circuits networks. In these networks, the coupling strength reflected the distance information and each chaotic circuit is connected to all chaotic circuits. For this investigation, we have changed the scaling parameter of coupling strength and number of chaotic circuits
composed of cluster. We have observed some clustering patterns from chaos synchronization. From computer simulation
results, we have confirmed that the state of clustering patterns
depend on the scaling parameter g and density of chaotic circuits networks. Furthermore, we have made clear that it iseﬃcient for using the average of phase diﬀerence and phasedomain.
In our future work, we would like to study the clustering
phenomena in the case of large scale networks. Additionally,

we hope to apply this clustering method for data mining, image processing and something application in our lives.
Acknowledgment
This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 22500203.
References[1] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, andControl in a Network of Chaotic Elements,” Physical D, vol. 41, pp.
137-172, 1990.
[2] T. Ott, M. Christen and R. Stoop, “An Unbiased Clustering Algorithm
Based on Self-organization Processes in Spiking Neural Networks”,
Proc. of 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., 85, pp. 554-557, 2000.
[4] Y. Takamaru, H. Kataoka, Y. Uwate and Y. Nishio, “Clustering Phenomena in Complex Networks of Chaotic Circuits”, Proc. of ISCAS’12,
pp. 914-917, Mar. 2012.
[5] Y. Takamaru, Y. Uwate, T. Ott and Y. Nishio, “Clustering Phenomena of Coupled Chaotic Circuits for Large Scale Networks”, Proc. of
NDES’12, pp. 70-73, Jul. 2012.
[6] Y. Takamaru, Y. Uwate, T. Ott and Y. Nishio, “Clustering Phenomena
Considering the Density of Coupled Chaotic Circuits Networks”, Proc.
of APCCAS’12, pp. 647-650, Dec. 2012.
[7] Y. Nishio, N. Inaba, S. Mori and T. Saito, “Rigorous Analyses of Windows in a Symmetric Circuit,” IEEE Transactions on Circuits and Systems, vol. 37, no. 4, pp. 473-487, Apr. 1990.
[8] C. Bonatto and J. A. C. Gallas, “Periodicity Hub and Nested Spirals in
the Phase Diagram of a Simple Resistive Circuit,” Phys. Rev. Lett., 101,
054101, Aug. 2008.
[9] R. Stoop, P. Benner and Y Uwate, “Real-World Existence and Ori-

- 360 -

gins of the Spiral Organization of Shrimp-Shaped Domains,” Phys. Rev.
Lett., 105, 074102, Aug. 2010.