Nonlinear Dynamics of Electronic Systems 2012, Wolfenbüttel

Clustering Phenomena of Coupled Chaotic Circuits
for Large Scale 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.chIn a previous study, we investigated 2-dimensional networks
of complex chaotic circuits, where the coupling strength
reflected the distance information [4]. We showed that thecircuits arranged close to each other could synchronize inphase, 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 thecoupling strength and the clustering phenomenon. For thisinvestigation, we change the scaling parameter of the couplingstrength and observe the changes of the number of clusters,
using a 2-dimensional irregular network of 20 circuits. Furthermore, we also carry out the analysis for a large scale networkwith 100 circuits and we additionally consider the example ofa 3-dimensional network.

Abstract— In this study, we investigate the clustering phenomena observed from coupled chaotic circuits arranged in irregular2-dimensional and 3-dimensional networks. The networks are
fully coupled using the distance information, i.e. the couplingstrength decreases with the distance between the circuits. We
show that synchronized clusters can emerge, reflecting groupsof neighboring circuits. By means of computer simulations, westudy the clustering phenomena and confirm that the numberof clusters depends on the scaling of the coupling strength.
Furthermore, we study large scale and 3-dimensional networks ofcoupled chaotic circuits, demonstrating that clustering can also
occur in more complex situations.

I. Introduction
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. Along with the variety of applications there
is a variety of diﬀerent clustering algorithms. Some algorithms
propose to utilize synchronization phenomena, for instance in
Coupled Map Lattics (CML), for clustering [1]-[3].
Synchronization is a prominent phenomenon, widely observed and studied in the field of natural and technical sciences. For example, the synchronization of neuronal activity is
an important characteristic of brain dynamics, often considered
to play an important role for the brain’s information processing
capacity. Hence the analysis of synchronization might be a
key for the realization of a brain computer. One possibility
to get insight into the role of synchronization phenomena inbiology is the analysis of electronic circuits. In particular,
there are numerous studies about synchronization phenomena in connection with chaotic circuits, describing various
interesting aspects such as chaos synchronization, space-timechaos, chaotic wandering and so on. However, there are notmany studies on synchronization-based clustering in networks
of coupled chaotic circuits.

II. Circuit model
Figure 1 shows the model of the used chaotic circuit called
Nishio-Inaba circuit, investigated in [5]-[7].

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 following equation describes the approximatedI − V characteristic of the nonlinear resistance, where theparameter rd is the slope of the nonlinear resistance.
 rd V Vi2 +  − i2 −  .
vd (i2 ) =(1)2rdrd70

The circuit dynamics is then described by the following
piecewise-linear third-order ordinary diﬀerential equation:
di1= v + ri1
dt
di2= v − vd (i2 )L2dtdv= −i1 − i2 .
Cdt
L1(2)

Equation (2) can be normalized by changing the variables
according to√C
L1C
i1 =V x; i2 =Vy; v = Vz;
LL2 1√CL1C
L1= α;= β; rd= δ;
rL1L2L2
dt = L1Cτ; ” · ” =dτ

Fig. 3.

We consider the following globally coupled chaotic circuits:dxi= αxi + zidτdyi
= zi + f (y)
dτ
N

dzi
= −xi − βyi −γi j (zi − z j )dτ
j=1yieldingẋ = αx + zẏ = z − f (y)ż = −x − βywhere f (y) is described as follows: δ 1 1
y +  − y −  .
f (y) =2
δδ(3)Fig. 2.

(5)(i, j = 1, 2, · · ·, N)
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. The
value of γi j reflects the distance between the circuits in an
inverse way, described by the following equation:(4)

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[MΩ].

(a) Computer simulation.

Arrangement of twenty chaotic circuits.

γi j =g.
(lengthi j )2(6)lengthi j 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 first study, we set the parameter to g = 0.25 × 10−4 .
Figure 4 shows the computer simulation results of the phase
diﬀerences of several circuits. We can derive the following
observations, exemplified by a selection of pairwise comparisons: The first circuit and the second circuit, as well as
the first circuit and the third circuit are synchronized (a-b).
Actually, all the circuits within this spatial cluster or group aresynchronized. However, the first circuit is not synchronized to
the group of the seventh circuit (c) or to the group of thethirteenth circuit (d). All the circuits within the group of theseventh circuit are synchronized within the group (e-f), butthey are not synchronized to any other circuits outside thegroup (g-h). All the circuits within the group of the thirteenth
circuit are synchronized within the group (i-j), but again they
are not synchronized to any other circuits outside the group(k). The circuits in the middle (18-20) are not synchronizedwith any other circuits (l). From these results, we conclude thatthe circuits form three clusters through chaos synchronization.
Furthermore, circuits 18, 19 and 20 are isolated as they do not(b) Circuit experiment.

Chaotic attractor.

III. Synchronization phenomena
A. Network of 20 chaotic circuits
First, we investigate the case of a general network, where20 chaotic circuits are placed irregularly on a 2-dimensional
grid. The arrangement is shown in Fig. 3. All chaotic circuits
are connected to each other by resistors.

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(a) 1-2(b) 1-3(c) 1-7(d) 1-13(e) 7-8(f) 7-11(g) 7-13(h) 7-19(i) 13-14Fig. 5.

The clustering result of twenty chaotic circuits.

The relationship between the coupling strength and thenumber of distinguishable groups.

Fig. 6.
(j) 13-16(k) 16-18(l) 19-20

Fig. 4. Phase diﬀerence between two circuits (computer simulation).

B. Network of 100 chaotic circuits
Next, we investigate the case of large scale networks, bystudying a 2-dimensional irregular network of 100 chaoticcircuits. The arrangement of the circuits is shown in Fig. 7.
Again, all the circuits are connected to each other by resistors
and the coupling strength between two circuits is determinedby Eq. (6). In this case, we set the parameter g = 0.85 × 10−6 .

synchronize to any other circuit. Figure 5 summarizes theseclustering results.
In a next step, we consider the relationship between thecoupling strength, controlled by the parameter g, and thenumber of clusters that occur. For the analysis we introduce thefollowing group numbering: Group 1 consists of the circuits
1-6, group 2 consists of the circuits 7-12, group 3 consistsof the circuits 13-17 and the residual circuits are group 4.
Figure 6 shows the number of synchronized clusters that occur
depending on the value of g. For small values g ≤ 0.05 × 10−4
no synchronization occurs (no distinguishable groups) In theregion between g = 0.06 × 10−4 and g = 0.32 × 10−4 clustering
can be observed. The clustering in this region corresponds
to the result shown in Fig. 4 and Fig. 5. So, we have 4
distinguishable groups. For values between g = 0.33 × 10−4and g = 0.62 × 10−4 group 1 and 2 are synchronized in
one cluster. Hence we have two synchronized clusters andone group of isolated circuits (3 distinguishable groups).
Between g = 0.63 × 10−4 and g = 1.10 × 10−4 all thecircuits of the groups 1, 2 and 3 are synchronized. Hence wehave one synchronized cluster and still one group of isolated
circuits (2 distinguishable groups). Finally, all the circuits aresynchronized in one cluster if g ≥ 1.20 × 10−4 .

Fig. 7.

Arrangement of 100 chaotic circuits.

Figure 8 shows the corresponding clustering results. A
cluster consists of 30 chaotic circuits. The networks can divide
3 cluster from synchronization.

72with a nontrivial shape such as a ring .

Fig. 10.

Fig. 8.

Irregular 3-dimensional network of 100 chaotic circuits.

The clustering result of 100 chaotic circuits.

Fig. 11.

Corresponding clustering result.

V. Conclusion
In this study, we examined clustering phenomena in 2- and3-dimensional irregular networks of coupled chaotic circuits
in dependence on the coupling strength. Based on computer
simulations, we showed that the circuits could cluster into
several groups of synchronized units. We found that the number of clusters changed depending on the coupling strength.
Interesting clustering results only occurred for intermediate
values of the coupling strength, whereas for weak and strong
couplings no clustering could be observed.
In our future work, we would like to develop a more eﬃcientmethod for determining the phase synchronization. This willallow us to examine more complex networks or to use themethod for data clustering.

The relationship between the coupling strength and thenumber of distinguishable groups.

Fig. 9.

Figure 9 shows the relationship between the couplingstrength and the number of occurring groups in the network.
We define the blue group (in Fig. 8) as group 1, the red groupas group 2 and the yellow group as group 3. All the othercircuits belong to group 4. There is again no synchronization
for small values g ≤ 0.74 × 10−6 . Between g = 0.75 × 10−6 andg = 1.37 × 10−6 clustering with 3 synchronized clusters andone group of isolated circuits can be observed (see example).
Between g = 1.38 × 10−6 and g = 1.57 × 10−6 group 1 and
2 are synchronized. Thus the network can be divided into
3 distinguishable groups. If g ≥ 1.58 × 10−6 this network
becomes divergent.
From these examples we can make the following observations. The networks are not synchronized if the couplingstrength is weak. For intermediate values of the couplingstrength we can observe synchronized clusters. Finally, forstrong couplings the networks are fully synchronized or divergent.

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IV. Chaotic circuits in 3-dimensional network
Finally, we study a 3-dimensional network of coupledchaotic circuits. The arrangement of the circuits is shown inFig. 10. It consists of a plane ring with 50 circuits and asphere of radius 0.1 placed in the center, containing another50 circuits. Again, the circuits are globally coupled according
to equation (6). The result for this network is shown in Fig.
11. We see that clustering can also occur in 3D for clusters

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