Clustering Phenomena in Coupled Chaotic Circuitswith Different Coupling Strength
Yoko Uwate† , Thomas Ott‡ and Yoshifumi Nishio††Dept. of Electrical and Electronic Engineering,Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp‡Institute of Applied Simulation, Zurich University of Applied Sciences,
Einsiedlerstrasse 31a, 8820 Waedenswil, Switzerland
Email: thomas.ott@zhaw.ch
Abstract— In this study, we investigate synchronization phenomena in coupled chaotic circuits with different couplingstrength when chaotic circuits are arranged irregularly in one
dimensional array. Three groups are composed in this circuit
system and each group includes some chaotic circuits. The
coupling strength corresponds to the distance between the chaotic
circuits. We observe several types of clustering patterns when the
coupling strength is fixed to appropriate value.

phase difference between intergroup and intragroup circuits.
By using computer simulations, we observe several types ofsynchronization states when the coupling strength is fixed toappropriate value. Furthermore, we discuss the relationshipbetween the obtained synchronization states and clustering
phenomena.
II. C IRCUIT MODEL

I. I NTRODUCTION
Synchronization is one of the basic phenomena and it canbe observed everywhere in our living life [1]-[5]. Therefore,studies on synchronization phenomena have investigated in thevarious fields. In particular, we are interested in synchronization phenomena observed in electronic circuits. We consider
that there are many real physical phenomena exhibiting bycoupled chaotic circuits, and it is important to investigate
synchronization phenomena observed from coupled chaotic
circuits for future engineering applications.
Recently, we have investigated synchronization phenomena
observed from complex networks of the coupled chaotic circuits [6], [7]. The chaotic circuits are placed on 2-dimensional
space and the coupled parameters depend on the distance
between the circuits. We have observed interesting clustering
phenomena from this proposed system by computer simulations and circuit experimental results. In order to understandthe obtained results in detail, we need to consider more simple
circuit system.
In our previous study, we have investigated synchronization
phenomena of an array of the coupled chaotic circuits [8]. In
this circuit system, the chaotic circuits are arranged in two
groups. By using computer simulations, we have confirmed
chaos synchronization when the number of the chaotic circuitsof two groups is same. While, the breakdown of chaos
synchronization is occurred by changing from the symmetricarrangement to the asymmetric system which the number ofthe chaotic circuits of two groups is different. We consider that
it is interesting to change the synchronization states dependingon the arrangement method of the circuits.
In this study, we expand the circuit system to more complex
system. Three groups are composed in this circuit system andeach group includes some chaotic circuits. We focus on the

Figure 1 shows the chaotic circuit called Nishio-Inaba circuit, investigated in [9]-[11]. The circuit consists of a negative
resistance, a nonlinear resistance consisting of two diodes, acapacitor and two inductors.

Fig. 1.

Nishio-Inaba chaotic circuit.

Rd3
Rd2
CC

Rd1

CC

Rd1

CC

Rd1

CC

Rd2

Arrangement of chaotic circuits with globally coupling as
one dimensional array.
Fig. 2.

In this study, we consider that the chaotic circuits arearranged in one dimensional array as shown in Fig. 2. All
chaotic circuits are connected to each other by resistors.
Furthermore, the value of the coupling strength depends onthe distance btween the circuits.

By using the variables and the parameters as follows;
√
√
C
L1 C
i1 =
V x; i2 =V y; v = V z;
L1
L2
√
√
C
L1
L1 C
r
= α;
= β; rd
= δ;
L1
L2
L2
√
d
t = L1 Cτ ; ” · ” =dτWe obtain the normalized circuit equations.
dxi= αxi + zidτdyi
= zi + f (y)
dτ
N
∑
dzi
= −xi − βyi −
γij (zi − zj )dτ
j=1

asynchronous phenomena can be observed from the adjacentchaotic circuits of the intergroup circuits. We consider that it
is interesting to change the synchronization states dependingon the arrangement method of the chaotic circuits.
Group-2

CC

CC

CC

dout

din

CC

din

(a) Symmetric system (2-2 network).
Group-1(1)

CCGroup-2
CC

CC

din

(i, j = 1, 2, · · ·, N )where f (y) is described as follows:
(
)
1
1
δ
y+
− y−
.
f (y) =
2δδGroup-1

dout(b) Asymmetric system (2-1 network).
Fig. 3.

Arrangement of chaotic circuits with two groups.

(2)

In this equation, i in the equation represents the circuit itself,
and j indicates the coupling with other circuits. The parameterγij represents the coupling strength between the circuits. The
value of γij reflects the distance between the circuits in an
inverse way, described by the following equation:
g
γij =
(3)
.
(dij )2
dij denotes the Euclidean distance between the i − th circuitand the j−th circuit. The parameter g is a weighting or scaling
parameter that determines the coupling strengths.

(a) Group1-Group1.

(b) Group1-Group2.

Phase difference of intragroup and intergroup circuits forsymmetric arrangement. ((a): 9.44◦ , (b): 28.19◦ )Fig. 4.

III. BASIC S YNCHRONIZATION PHENOMENAFirst, we explain the basic synchronization phenomena
when the chaotic circuits are coupled at irregular intervals.
The two types arrangements of chaotic circuits (symmetric andasymmetric models) are considered as shown in Fig 3. In thesecircuit models, two groups are arranged in one dimensional
space. The nearest distance between the chaotic circuits within
the group is set to din = 0.01. The distance between theintergroup circuits is set to dout = 0.1. In this simulation, we
set the scaling parameter to g = 1.0 × 10−5 and the couplingstrength between two circuits is determined by Eq. (3).
Figure 4 shows the simulation results of the symmetricsystem (Fig. 3 (a)). The chaotic circuits within the group aresynchronized with in-phase state as shown in Fig. 4 (a). Also,
the in-phase synchronization is observed from the intergroup
circuits (see Fig. 4 (b)). Similarly, Fig. 5 shows the results ofthe phase differences between the chaotic circuits of the asymmetric system (Fig. 3 (b)). In this case, the asynchronization
can be observed from the intergroup circuits (see Fig. 5 (b)).
From these results, all chaotic circuits are synchronizedwhen the chaotic circuits are coupled by symmetry. However,the case of that the chaotic circuits are coupled by asymmetry,(a) Group1-Group1.

(b) Group1-Group2.

Phase difference of intragroup and intergroup circuits forasymmetric arrangement. ((a): 9.54◦ , (b): 90.00◦ )Fig. 5.

IV. S YNCHRONIZATION PHENOMENA
In this section, several types of the circuit arrangement areconsidered as follows.
A. Synchronization phenomena I
Here, we investigate synchronization phenomena when the
chaotic circuits are arranged in three groups. The two types
arrangements of chaotic circuits (symmetric and asymmetricmodels) are considered as shown in Fig 6. In this circuit

Group-1

Group-2

Group-3

CC CC CC CC

CC CC CC CC CC CC CC CC CC

CC CC CC CC

din

dout

din

din

dout

(a) Symmetric system (4-9-4 network).
Group-1

Group-2

CC CC CC CC

CC CC CC CC CC CC CC CC CC

din

dout

Group-3din

CC... CC

dindout(b) Asymmetric system (4-9-x network).
Fig. 6.

Arrangement of chaotic circuit with three groups.

system, the distance parameters are fixed with din = 0.01 anddout = 0.1, and the coupling strength is set to g = 1.0×10−5 .
For the asymmetric model, the number of the chaotic circuits
in group3 is changed from 1 to 3.
Figures 7-10 show the simulation results of the phase difference of the intergroup circuits. In the case of symmetric 4-9-4
network, all intergroup circuits are synchronized with in-phasestate as shown in Fig. 7. Figure 8 shows the phase differencesof asymmetric 4-9-3 network. The chaotic circuits between
group1 and group3 are synchronized with in-phase. While, theother intergroup circuits are not synchronized. By decreasing
the number of chaotic circuits in group3, the different typeof synchronization is observed. In the case of 4-9-2 and 4-91 networks, the chaotic circuits between group1 and group2
are synchronized with in-phase state. Asynchronization can beobtained from the other intergroup circuits (see Figs. 9, 10).

(a) Group1-Group2.

(b) Group2-Group3.

(c) Group3-Group1.

Phase difference of intergroup circuits for 4-9-2 network.
((a): 27.66◦ , (b): 90.00◦ , (c): 62.77◦ )Fig. 9.

(a) Group1-Group2.

(b) Group2-Group3.

(c) Group3-Group1.

Phase difference of intergroup circuits for 4-9-1 network.
((a): 29.16◦ , (b): 90.00◦ , (c): 90.00◦ )Fig. 10.

(a) Group1-Group2.

(b) Group2-Group3.

(c) Group3-Group1.

Phase difference of intergroup circuits for 4-9-4 network.
((a): 23.66◦ , (b): 24.48◦ , (c): 14.26◦ )Fig. 7.

Next, we consider the relationship between the obtained
synchronization states and clustering phenomena. We define
the synchronized circuits as one cluster. Table I summarizes
the clustering phenomena for all types of the networks. We
observe one cluster in the symmetric system. In the case of
the asymmetric system, two clusters are observed. From theseresults, we confirm that several types of clustering phenomena
can be observed from the asymmetric circuit systems.
B. Synchronization phenomena II(a) Group1-Group2.

(b) Group2-Group3.

(c) Group3-Group1.

Phase difference of intergroup circuits for 4-9-3 network.
((a): 40.13◦ , (b): 58.39◦ , (c): 17.57◦ )Fig. 8.

Next, we investigate the synchronization state of symmetric
4-9-4 network when the coupling strength is changed. Figure
11 shows the simulation results of the phase differences.
When the coupling strength is small value, we confirm theasynchrous state for whole circuit system. First, the phasedifference of the intergroup circuits decreases around g =TABLE I

TABLE II: SUMMARY OF SYNCHRONIZATION STATE.

S UMMARY OF CLUSTERING PHENOMENA .
Network
4-9-4
4-9-3
4-9-2
4-9-2

Clustering number
1 cluster
2 clusters
2 clusters
2 clusters

Clustering type(groups 1-2-3)(groups 1-3), (group 2)(groups 1-2), (group 3)(groups 1-2), (group 3)

1.0 × 10−6 . At this region, the synchronization of the chaotic
circuits within the group is occurred. Next, the phase difference between the group1 and the group3 shows in-phasestate. However, the chaotic circuits between the other groups
are not synchronized (see Fig. 12). Finally, we confirm fully
synchronization state around g = 1.0 × 10−5 .
Table II summarizes the relationship between the obtained
synchronization state and the clustering phenomena. We confirm that the several types of clustering phenomena can beobtained by setting the coupling strength.

phase difference [deg.]

Synchronization States

Clustering type

Fully asynchronization

4

9

4

0.8e-7 : 0.3e-6 Synchronization of chaotic circuits in same group

4

9

4

0.4e-6 : 0.9e-6 Synchronization between Group1 and Group3

4

9

4

1e-5 : 0.3e-4

4

9

4

Fully synchronization

V. C ONCLUSION
In this study, we have investigated synchronization phenomena in coupled chaotic circuits with different couplingstrength when chaotic circuits are arranged irregularly in
one dimensional array. Three groups are composed in thiscircuit system and each group includes some chaotic circuits.
The coupling strength corresponds to the distance between
the chaotic circuits. We have observed several types of synchronization states when the coupling strength is fixed toappropriate value.
ACKNOWLEDGMENT

140

Group1-Group1
Group1-Group2
Group1-Group3

120
100
80
60
40
20
0
1e-08

Fig. 11.

g
1e-8 : 0.6e-7

1e-07

g

1e-06

1e-05

Average of phase difference with coupling strength.

(a) Group1-Group1.

(b) Group2-Group2.

(c) Group3-Group3.

(d) Group1-Group2.

(e) Group2-Group3.

(f) Group3-Group1.

Fig. 12. Synchronization state of two clusters. ((a): 10.09◦ , (b): 9.37◦ ,(c): 9.96◦ , (d): 90.00◦ , (e): 90.00◦ , (f): 29.68◦ )This work was partly supported by JSPS Grant-in-Aid forYoung Scientists 23700269.
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