2019 International Symposium on Nonlinear Theory and Its Applications
NOLTA2019, Kuala Lumpur, Malaysia, December 2-6, 2019.

Clustering Phenomena in Network of Coupled Chaotic Circuits
Distributed in 3-Dimensional Space
Takumi Nara, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Email: {nara, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we investigate synchronization phenomena in coupled chaotic circuits which are connected by resistance. In addition, we investigate the diﬀerence of synchronization phenomena by distance information of coupled chaotic circuits and the diﬀerence of synchronization phenomena by increasing the number of coupled chaotic circuits. We confirm that the coupled chaoticcircuits located in the near distance are synchronized at inphase state, and the coupled chaotic circuits located in thefar distance are not synchronized. Therefore, the clustering phenomena of coupled chaotic circuits are observed in
two-dimensional space and three-dimensional space.
1. Introduction
Synchronization phenomena are the most familiar phenomena that exist in nature and they have been studiedin various fields. Synchronization phenomena can be observed everywhere in our life. For example, we can confirm
flashing firefly lights, metronome, gate patterns of four-leg
animals, beating rhythm of the heart and so on. Especially,
synchronization phenomena of oscillatory network are interesting. In addition, complex networks attract attention
from various fields. The feature of networks is the degree
distribution, the path length and the clustering coeﬃcient.
Therefore, we focus on the clustering phenomena in thisresearch.
In addition, clustering phenomena are one of interesting
nonlinear phenomena observed from coupled chaotic circuits. The clustering phenomena is to divide the set to beclassified into subsets. Previously, many of the studies forclustering have been carried out for discrete time model,
for example Coupled Map Lattics (CML) and Self Organization Map (SOM) [1]-[2] and so on. However, analysis
of using a continuous time model has not almost studied.
Therefore, we focus on research on clustering phenomenausing real electronic circuits in continuous time model.
On the other hand, coupled chaotic circuits that are realelectronic circuits can be observed various amusing phenomena. In recent years, many methods are studied to apply to clustering and synchronization phenomena observedin coupled chaotic circuits for natural sciences. At the sametime, synchronization phenomena and clustering have beenstudied associated with the chaos phenomena [3]-[4].

In this study, we focus on the clustering phenomena in
the network of coupled chaotic circuits. For this investigation, the coupling strength reflected the distance information when the chaotic circuits are placed in twodimensional and three-dimensional space. Moreover, weinvestigate from two clusters. From there, we increase thenumber of clusters. Finally, we consider the case of complex networks to extend to large scale networks and apply
to the real world data for clustering.
2. Circuit Model
Figure 1 shows the circuit model which is called
Shinriki-Mori circuit. This circuit consists of a negativeresistor, an inductor, two capacitors and dual-directional
diodes.

idiL

ig-g

v1

C1

v2

C2

L

Figure 1: Circuit model.
The circuit equation of this circuit are given as follows:diL
L
= v2

dt




dv1

(1)
C1= gv1 − idn

dt



dv2



= idn − iL .
 C2dt
The nonlinear function idn corresponds to the i − v characteristics of the nonlinear resistors consisting of the diodes
and are given as follows:- 38 -
Gd (v1 − v2 − V)idn = 0
 G (v − v + V)
d 12

(v1 − v2 > V)(|v1 − v2 | ≤ V)
(v1 − v2 < V).

(2)

Table 1: The location of chaotic circuits in the twodimensional space.

By changing the variables and parameters such that√location
1234567C2iL =V x, v1 = Vy, v2 = Vz
L
√√
C2L
L
α=, β = Gd, γ=g
C1C2C2√dt = LC2 τ, ” · ” =dτ
The normalized equations of chaos circuit are given as
follows:ẋ = zẏ= αγy − αβ f ż = β f − x.

(3)

167γ17(y − z > 1)(|y − z| ≤ 1)(y − z < 1).

y0.350.10
0.20
0.60
0.80
0.950.80

All circuits connected each other by resistors. Figure
3 shows coupling method of the first chaotic circuit as anexample.

The nonlinear function f corresponds to the characterisitics of the nonlinear resistor consisting of the diodes anddescribed as follows:


y−z−1




f =
0


 y−z+1x
0.15
0.10
0.30
0.70
0.90
0.80
0.55(4)1 γ
13γ12For the computer simulation, we set the parameters as
α = 0.50, β = 20.00 and γ = 0.50.

γ15γ1654γ14323. Simulation Result01

id

3.1. Network of seven chaotic circuits in two-dimensionaliLig

space

First, we investigate the synchronization phenomena andclustering phenomena when seven chaotic circuits are coupled in two-dimensional space. The location of sevenchaotic circuits is shown in Fig. 2 and Tab. 1.

i-g

v1

C1v2

C2

L

Figure 3: Coupling between the first chaotic circuit andothers.
We consider the coupled chaotic circuits:

Figure 2: Location of seven chaotic circuits in twodimensional space.

- 39 - dxi= zidτ

N∑


 dyi
=αγy−αβf−αri, j (yi − y j )
i

dτ


i, j=1


dzi



= β f − xi .
dτ(5)The nonlinear function f corresponds to the i-v characteristics of the nonlinear resistors consisting of the diodes
and are given as follows:yi − zi − 1
f =0
 y −z +1ii

(yi − zi > 1)
(6)(|yi − zi | ≤ 1)
(yi − zi < 1).

where, i in the equation represents the circuit itself, and jis the coupling with other circuits. The parameter r represents the coupling strength between the circuits. In thissimulation, we set the coupling parameter value ri, j to correspond the distance between the circuits by the following
equation:
Figure 5: The clustering result of seven chaotic circuits.
ri, j =q.
(di, j )2(7)3.2. Network of thirty chaotic circuits in threedimensional spacedi, j represents the Euclidean distance between the i − thand the j−th circuits. Further, the parameter q is the weightparameter that determines the coupling strengths. In thiscase, we set parameter q = 0.01.
Figure 4 shows the computer simulation results obtained
from the seven chaotic circuits located as shown in Fig. 2.
From these results, we confirm that the first, second andthird chaotic circuits are synchronized at in-phase state, andalso the fourth, fifth, sixth and seventh chaotic circuits aresynchronized at in-phase state. However, the first and thefourth chaotic circuits are not synchronized. From theseresults, the circuits can form two clusters defined by chaoticsynchronization as shown in Fig. 5.

x3x2x1(c)1-4x6x7x4(d)4-5location
123456789101112131415x1(b)1-3x5

Table 2: The location of chaotic circuits in the threedimensional space.

x4x1(a)1-2

Next, we investigate the case of three-dimensional networks. Thirty chaotic circuits are located in threedimensional, including the positional information. The location of thirty chaotic circuits is shown in Fig. 6. Also,
Table 2 shows the location of the chaotic circuits. Similarly, all the chaotic circuits are connected each other byresistors, and the coupling strength between each circuits
is determined by Eq. (7). In this case, we put the parameterq = 0.004724.

x4(e)4-6x4(f)4-7

Figure 4: Phase diﬀerence between seven circuits in twodimensional space.

- 40 -x
0.15
0.20
0.35
0.25
0.30
0.05
0.15
0.05
0.20
0.25
0.35
0.25
0.75
0.80
0.95

y
0.05
0.25
0.35
0.25
0.15
0.20
0.30
0.05
0.05
0.10
0.05
0.05
0.15
0.25
0.30

z
0.15
0.30
0.25
0.050.05
0.10
0.15
0.25
0.35
0.20
0.25
0.35
0.25
0.10
0.35

location
16
1718
19
20
21
22
23
24
25
26
27
28
29
30

x
0.80
0.85
0.70
0.90
0.80
0.75
0.85
0.60
0.80
0.65
0.65
0.95
0.90
0.70
0.75

y
0.20
0.15
0.60
0.80
0.95
0.85
0.80
0.60
0.65
0.80
0.65
0.95
0.65
0.85
0.80

z
0.15
0.05
0.95
0.85
0.75
0.70
0.85
0.60
0.90
0.80
0.85
0.95
0.75
0.65
0.85

chaos circuitchaos circuit

zz

yy
x

x

Figure 6: Location of thirty chaotic circuits in threedimensional space.

Figure 8: The clustering result of thirty chaotic circuits.

Figure 7 shows the computer simulation results obtained
from the thirty chaotic circuits located as shown in Fig. 6.
From these results, we confirm that the first and the second circuits are synchronized at in-phase state. However,
the group of first chaotic circuit and the group of thirteenthchaotic circuit are not synchronized. Also the group of firstchaotic circuit and the group of eighteenth chaotic circuit
are not synchronized. Similarly, between the group of thirteenth and fourteenth chaotic circuit are synchronized, between the group of eighteenth and nineteenth chaotic circuit are synchronized. However, between the group of thirteenth and eighteenth chaotic circuit are not synchronized.
From these results, the circuits can form three clusters defined by chaotic synchronization as shown in Fig. 8.

4. Conclusionx2x3x13x1(a)1-2x14x18(d)13-18[2] L. Angelini, F. D. Carlo, C. Marangi, M. Pellicoro and
S. Stramaglia, “Clustering Data by Inhomogeneous
Chaotic Map Lattice”, Phys. Rev. Lett., 85, pp. 554557, 2000.

(c)1-13x15(e)13-14x19x13[1] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, and Control in a Network of Chaotic Elements”, Physical D, vol. 41, pp. 137-172, 1990.

x1x13x1(d)1-18

References

x1(b)1-3x18

In this study, we investigated synchronization phenomena when the chaotic circuits are located in twodimensional and three-dimensional space. Synchronization
phenomena were seen between circuits at near distance,and synchronization phenomena could not be seen between
circuits at far distance. With these results, it was confirmedthat the chaotic circuits were diﬀerent from synchronization phenomena by distance information and the clustering
phenomena were observed.
In the future works, we would like to increase the number of chaotic circuits and the cluster. Moreover, we consider that we would like to change of the dimensional
space.

[3] Y. Takamaru, H. Kataoka, Y. Uwate and Y. Nishio,“Clustering Phenomena in Complex Networks ofChaotic Circuits”, Proceedings of IEEE International
Symposium on Circuits and Systems (ISCAS’12), pp.
914-917, May 2012.

x13(f)13-15

x20x18(e)18-19[4] T. Chikazawa, Y. Uwate, Y. Nishio. “Investigation ofSpreading Chaotic Behavior in Coupled Chaotic Circuit Networks with Various Features”, Proceedings ofRISP International Workshop on Nonlinear Circuits,Communications and Signal Processing (NCSP’17),
pp. 337-340, Feb. 2017.

x18(f)18-20

Figure 7: Phase diﬀerence between thirty circuits in threedimensional space.

- 41 -