IEEE Workshop on Nonlinear Circuit Networks
December 14-15, 2012

Clustering Patterns Generated in Coupled
Chaotic Circuits 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— This paper presents clustering patterns generated
in coupled chaotic circuits networks. In these networks, thecoupling strength is reflected the distance information. Also,
each chaotic circuit is connected globally. We investigate therelationship between coupling strength and phase diﬀerence.

I. Introduction
Synchronization phenomena is one of typical phenomena
when we analyze coupled chaotic circuits. This phenomenon
widely can observe and studying in the field of natural andtechnical sciences. In order to understand synchronization
phenomena in detail, we analyze electronic circuits. Many
researches studied on synchronization phenomena using coupled chaotic circuits. However, there are not many studies
on clustering phenomena in coupled chaotic circuits. We cansee many phenomena when we connect chaos circuits eachother, whereas all phenomena are not clear. We consider
that we can approach the new method by applying to chaos
synchronization for clustering.
In a previous study, we investigated the relationship between clustering and density of coupled chaotic circuits in 2dimensional place. For this investigation, the coupling strength
reflected the distance information [4]-[6] and we changed
the number of circuits in cluster. We showed that clustering
phenomena aﬀected other cluster when density in the chaoticcircuits was high. We also observed that networks of coupledchaotic circuits could split into diﬀerent synchronized groups.
In this study, we consider clustering patterns generated incoupled chaotic circuits networks. In these networks, chaotic
circuits are connected globally. The variable parameter ofcoupling strength and density of chaotic circuits are changed.
By using computer simulation, the network can observe 3clustering patterns.

Fig. 1.

Chaotic circuit.

The circuit consists of a negative resistance, a nonlinearresistance consisting of two diodes, a capacitor and twoinductors.
The normalized equations of this circuit are described asfollows:ẋ = αx + zẏ = z − f (y)(1)

ż = −x − βywhere f (y) is described as follows:()δ1
1f (y) =(2)y+ − y−.
2δδFor the computer simulation, we set the parameters as α =0.460, β = 3.0 and δ = 470.
We can consider the following equations when each chaotic
circuit is coupled globally with each other.

II. Circuit model
Figure 1 shows the model of the chaotic circuit, investigatedin [7]-[8].

- 138 -dxi
= αxi + zidτdyi= zi + f (y)
dτ
N
∑dzi
= −xi − βyi −γi j (zi − z j )dτ
j=1(i, j = 1, 2, · · ·, N)(3)

where i in the equation represents the circuit itself, and jindicates the coupling with other circuits. The parameter γi j
represents the coupling strength between the circuits. Thevalue of γi j reflects the distance between the circuits in an
inverse way, described by the following equation:

γi j =g.
(lengthi j )2(4)

lengthi j denotes the Euclidean distance between the i − thcircuit and the j − th circuit. The parameter g is a scaling
parameter that determines the coupling strengths.

(a) One cluster.
Fig. 3.

(b) Two clusters.

The clustering results.

III. Clustering phenomena
A. Clustering phenomena
In this section, we investigate clustering phenomena when
we configure network of coupled chaotic circuits in 2dimensional. In our previous study [6], we researched therelationship between clustering and density of coupled chaoticcircuits when we changed density of chaotic circuits. Arrangement of chaotic circuits are shown in Fig. 2. Figure 2 (a) is
composed same number of chaotic circuits, however Fig. 2 (b)is composed high density of chaotic circuits in inside and somelow density chaotic circuits groups. In these figures, we denotechaotic circuits with a simple model like small circle.

(a) Same number of groups.
Fig. 2.

the phase diﬀerence between chaotic circuits using computersimulation.
For this simulation, the iteration is set to τ p = 10, 000 forcalculating the result more precisely. Figure 4 shows the phase
diﬀerence between two chaotic circuits when we change thevalue of parameter g.
We define state of synchronization patterns from the averageof phase diﬀerence when we calculate τ p = 10, 000. The
state of synchronization can be defined if the average ofphase diﬀerence below 40◦ . Similarly, we define the state ofasynchronous that the average of phase diﬀerence is between
70◦ and 110◦ . In the region between g = 2.0 × 10−6 andg = 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 and g = 2.0×10−4 inside chaotic circuits group issynchronized one cluster, also outside chaotic circuits groups
are synchronized one cluster. Therefore, clustering phenomena
can observe in this region. Finally, all chaotic circuits aresynchronized in one cluster if the region between g = 3.0×10−4
and g = 4.0×10−4 . Thus, this network can observe 3 clustering
patterns from the average of phase diﬀerence.

(b) Diﬀerent number of groups.

Arrangement of chaotic circuits.

Simulation results of these networks are shown in Fig. 3.
From simulation results, all chaotic circuits are synchronizedin one cluster from the result of Fig. 3 (a), however we
can see 2 clusters from chaos synchronization between highdensity group and same low density groups from the result ofFig. 3 (b).
From these results, clustering phenomena are related densityof coupled chaotic circuits.
B. Investigation of clustering phenomena

Fig. 4.

Next, we investigate the clustering result corresponding toFig. 3 (b) in detail. This network can be divided 2 clusters
between high density and low density from chaos synchronization. We consider the state of this network when we change theparameter g determined by Eq. (4). Furthermore, we calculate

The relationship between g and phase diﬀerence.

IV. Conclusions
In this study, we have considered clustering patterns generated in coupled chaotic circuits networks. In these networks,- 139 -TABLE I
State of networks
The value of g

In - InIn - OutOut - Out

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

Syn.
Syn.
Syn.

Not-syn.
Not-syn.
Syn.

Not-syn.
Syn.
Syn.

the coupling strength reflected the distance information andeach chaotic circuit is connected globally. For this investigation, we have changed the scaling parameter of couplingstrength. From computer simulation results, we have confirmedthat the state of clustering patterns depend on the scaling
parameter g and density of chaotic circuits networks.
In our future work, we would like develop relationship between clustering and density of chaotic circuits. Additionally,
we hope to apply this clustering method for data mining, image
processing and something application in our lives.
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