IEEE Workshop on Nonlinear Circuit Networks
December 13-14, 2019

Investigation of Clustering Phenomena
in Coupled Chaotic Circuits
Located in Three-Dimensional Space
Takumi Nara, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering of Tokushima University
2-1 Minami-Josanjima, Tokushima-shi, Tokushima-ken 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 byresistor. In addition, we investigate the difference of synchronization phenomena by distance information of coupled chaoticcircuits and the difference of synchronization phenomena by
increasing the number of coupled chaotic circuits. We confirm
that the coupled chaotic circuits located in the near distance aresynchronized at in-phase state, and the coupled chaotic circuits
located in the far distance are not synchronized. Therefore, theclustering phenomena of coupled chaotic circuits are observed in
two-dimensional space and three-dimensional space.

II. C IRCUIT M ODEL
Figure 1 shows the circuit model which is called ShinrikiMori circuit. This circuit consists of a negative resistor, aninductor, two capacitors and dual-directional diodes.

idiL

ig-g

v1

v2

C1

C2

LI. I NTRODUCTION
Synchronization phenomena are the most familiar phenomena that exist in nature and they have been studied invarious fields. Synchronization phenomena can be observed
everywhere in our life. For example, we can confirm flashing
firefly lights, metronome, beating rhythm of the heart andso on. Especially, synchronization phenomena of oscillatory
network are interesting. In addition, complex networks attract
attention from various fields. The feature of networks isthe degree distribution, the path length and the clustering
coefficient. Therefore, we focus on the clustering phenomenain this research.
In addition, clustering phenomena are one of interesting
nonlinear phenomena observed from coupled chaotic circuits.
The clustering phenomena are to divide the set to be classified
into subsets. Previously, many of the studies for clustering
have been carried out for discrete time model [1]-[2]. However,analysis of using a continuous time model has not almost studied. Therefore, we focus on research on clustering phenomenausing electronic circuits in continuous time model.
On the other hand, coupled chaotic circuits that are electronic circuits can be observed various amusing phenomena. In
recent years, many methods are studied to apply to clusteringand synchronization phenomena observed in coupled chaotic
circuits for natural sciences. At the same time, synchronization
phenomena and clustering have been studied associated withthe chaos phenomena [3]-[4].
In this study, we focus on the clustering phenomena in thenetwork of coupled chaotic circuits.

Fig. 1.

Circuit model.

The circuit equation of this circuit is given as follows:
 diL

= v2L


dt


dv1
C1= gv1 − idn

dt



 C dv2 = i − i .
2dnLdt(1)

The nonlinear function idn corresponds to the i − v characteristics of the nonlinear resistors consisting of the diodes
and are given as follows:(v1 − v2 > V ) Gd (v1 − v2 − V )idn =(2)0(|v1 − v2 | ≤ V )
Gd (v1 − v2 + V )(v1 − v2 < V ).
By changing the variables and parameters such that√C2V x, v1 = V y, v2 = V zL√√√
C2LLα=, β = Gd, γ=g, t = LC2 τC1C2C2- 40 -iL =The normalized equation of chaotic circuit is given asfollows:
 dx

=z

dτ

dy
= αγy − αβf
 dτ



 dz = βf − x.
dτ(3)

The nonlinear function f corresponds to the characterisitics
of the nonlinear resistor consisting of the diodes and describedas follows:f=

y−z−1(y − z > 1)

0y−z+1(|y − z| ≤ 1)(y − z < 1).

Fig. 3.

Location of seven chaotic circuits in two-dimensional space.

TABLE IT HE LOCATION OF CHAOTIC CIRCUITS IN THE TWO - DIMENSIONAL SPACE .

location
1234567(4)For the computer simulation, we set the parameters as α =0.50, β = 20.00 and γ = 0.50.
III. S IMULATION R ESULT
Definition of synchronization in this study is determined avoltage difference waveform. We define synchronization as thefollowing Eq. (5).
|yi − yj | < 0.03

(i, j = 1, 2, ..., N ).

(5)

X
0.15
0.10
0.30
0.70
0.90
0.80
0.55

Y
0.35
0.10
0.20
0.60
0.80
0.95
0.80

All circuits connected each other by resistors. Figure 4shows coupling method of the first chaotic circuit as anexample.
1670.03

γ170-0.03

1 γ
13γ12γ15
γ16

54γ14

32Fig. 2.

A difference waveform.

01idFigure 2 is a different waveform which was observed inthis simulation. The two lines in Fig. 2 correspond threshold
which is given in Eq. (5). It is determined that wave within
two lines which is the threshold is synchronization. Therefore,
we investigate the percentage of synchronization during thespecific time interval.
A. Network of seven chaotic circuits in two-dimensional spaceFirst, we investigate the synchronization phenomena andclustering phenomena when seven chaotic circuits are coupled in two-dimensional space. The location of seven chaoticcircuits is shown in Fig. 3 and Table I.

- 41 -iL

ig

iFig. 4.

-g

v1

C1

v2

C2

L

Coupling between the first chaotic circuit and others.

We consider the coupled chaotic circuits:dxi= zidτ

N
 dy
∑
i= αγyi − αβf − αri,j (yi − yj )
dτ


i,j=1



 dzi = βf − xi .
dτ(6)The nonlinear function f corresponds to the i-v characteristics of the nonlinear resistors consisting of the diodes andare given as follows: yi − z i − 1
f=0
yi − z i + 1(yi − zi > 1)
(7)(|yi − zi | ≤ 1)
(yi − zi < 1).

where, i in the equation represents the circuit itself, and j isthe coupling with other circuits. The parameter r represents
the coupling strength between the circuits. In this simulation,
we set the coupling parameter value ri,j to correspond thedistance between the circuits by the following equation:ri,j =

q.
(di,j )2x3x1x1(a)1-2(b)1-3x5x7x4(d)4-5
Fig. 5.

x1(c)1-4x6x4(e)4-6

Green
OrangeBetween clusters

x4(f)4-7

maximum minimum
99%
73%
99%
45%
8%
7%

average82.67%
65.67%
7.33%

B. Network of thirty chaotic circuits in three-dimensional
spaceNext, we investigate the case of three-dimensional networks.
Thirty chaotic circuits are located in three-dimensional, including the positional information. The location of thirty chaoticcircuits is shown in Fig. 7 and Table III. In this case, we put
the parameter q = 0.004724.
TABLE IIIT HE LOCATION OF CHAOTIC CIRCUITS IN THE THREE - DIMENSIONALSPACE .

location
123456789101112131415x4

The clustering result of seven chaotic circuits.

TABLE IIS YNCHRONIZATION RATE IN THE TWO - DIMENSIONAL SPACE .

(8)di,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 this case,
we set parameter q = 0.01.
Figure 5 shows the computer simulation results obtained
from the seven chaotic circuits located as shown in Fig. 3.
From these results, we confirm that the first, second and third
chaotic circuits are synchronized at in-phase state, and also
the fourth, fifth, sixth and seventh chaotic circuits are synchronized at in-phase state. However, the first and the fourthchaotic circuits are not synchronized. From these results, thecircuits can form two clusters defined by chaotic synchronization as shown in Fig. 6. The results of the synchronization rate
are summarized in Table II. As a result, there was a differencein the synchronization rate within and between clusters, andit was proved that the clusters could be done numerically inaddition to visual observation.

x2Fig. 6.

Phase difference between seven circuits in two-dimensional space.

- 42 -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
0.05
0.05
0.10
0.15
0.25
0.35
0.20
0.25
0.35
0.25
0.10
0.35

location
16
17
18
19
20
21
22
23
24
25
26
27
28
2930

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

Fig. 7.

Location of thirty chaotic circuits in three-dimensional space.
Fig. 9.

Figure 8 shows the computer simulation results obtained
from the thirty chaotic circuits located as shown in Fig. 7.
From these results, we confirm that the first and the second
circuits are synchronized at in-phase state. However, thefirst chaotic circuit and the thirteenth chaotic circuit are notsynchronized. Also the first chaotic circuit and the eighteenthchaotic circuit are not synchronized. Similarly, between thethirteenth and the fourteenth chaotic circuit are synchronized,
between the eighteenth and the nineteenth chaotic circuit
are synchronized. However, between the thirteenth and theeighteenth chaotic circuit are not synchronized. From theseresults, the circuits can form three clusters defined by chaotic
synchronization as shown in Fig. 9. Further, the between
clusters represent the respective synchronization rates among
the three clusters. Similarly, the results of the synchronization
rate are summarized in Table IV. As a result, there was adifference in the synchronization rate within and between
clusters, and it was proved that the clusters could be done
numerically in addition to visual observation.

x2x3x13x1(a)1-2x1(b)1-3x14x18(d)1-18(g)13-18
Fig. 8.

TABLE IVS YNCHRONIZATION RATE IN THE THREE - DIMENSIONAL SPACE .

Blue
Green
OrangeBetween clusters(e)13-14(h)18-19

average
95.79
76.60
82.87
12.56

R EFERENCES[1] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, andControl in a Network of Chaotic Elements”, Physical D, vol. 41, pp.
137-172, 1990.
[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. 554-557, 2000.
[3] Y. Takamaru, H. Kataoka, Y. Uwate and Y. Nishio, “Clustering Phenomena in Complex Networks of Chaotic Circuits”, Proceedings of IEEE
International Symposium on Circuits and Systems (ISCAS’12), pp. 914917, May 2012.
[4] T. Chikazawa, Y. Uwate, Y. Nishio. “Investigation of Spreading ChaoticBehavior in Coupled Chaotic Circuit Networks with Various Features”,Proceedings of RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing (NCSP’17), pp. 337-340, Feb.
2017.

x13(f)13-15

x20x18maximum minimum
9966
99
4394
36298

IV. C ONCLUSION
In this study, we investigated synchronization phenomena
when the chaotic circuits are located in two-dimensional andthree-dimensional space. Synchronization phenomena wereseen between circuits at near distance, and synchronization
phenomena could not be seen between circuits at far distance.
With these results, it was confirmed that the chaotic circuits
were different from synchronization phenomena by distanceinformation and the clustering phenomena were observed.
In the future works, we would like to increase the numberof chaotic circuits and the cluster. Moreover, we consider that
we would like to change of the dimensional space. We also
would like to investigate changes in the clusters by changing
the value of q.

x15x19x13(c)1-13x13

x1
x18

x1

The clustering result of thirty chaotic circuits.

x18(i)18-20

Phase difference between thirty circuits in three-dimensional space.

- 43 -