2014 International Symposium on Nonlinear Theory and its Applications
NOLTA2014, Luzern, Switzerland, September 14-18, 2014

Breakdown of Inter-Cluster Synchronizationof Coupled Chaotic Circuits Arranged in One-Dimensional Coordinate
Jun Shiomoto† , Yoko Uwate† , Thomas Ott‡ and Yoshifumi Nishio††Departmaent of Electrical and Electronic Enginnering, Tokushima University
2-1 Minamijosanjima, Tokushima, 770-8506, Japan
Email: {shiomoto, uwate, nishio}@ee.tokushima-u.ac.jp‡Zurich University of Applied Sciences
Einsiedlerstrasse 31a, 8820 Waedenswil, Switzerland
Email: thomas.ott@zhaw.ch
Abstract—In this study, we investigate synchronization
phenomena of coupled chaotic circuits. The chaotic circuits are combined by resisters on one-dimensional coordinate system. We change the distance between the circuits
to adapt the coupling strength. We investigate synchronization phenomena when the distance between the circuits in
the group is changed. Also, we measure the phase difference using computer simulations. From the computer simulations, we could make sure of the breakdown of intercluster synchronization when the system is changed fromthe symmetric system to the asymmetric system.

ena by changing the distance between the circuits.
In this study, we use ladder system. In the ladder system, chaotic circuits are connected to only adjacent circuits.
Figure 1 shows the system model of the ladder system. We
measure the phase difference using computer simulations.

Figure 1: Ladder system.

1. Introduction
2. Circuit Model
Synchronization phenomenon is one of the typical phenomena observed in nature. Recently, many studies
have been investigated synchronization of chaotic circuits[1]∼[5]. It is focused how the differences of the network
structure impact on the whole circuits. Additionally, it isapplicable to the fields of medical science and biology andso on.
In our research group, we have investigated the clustering phenomena resulting from the synchronization phenomena observed in coupled chaotic circuits when thechaotic circuits are arranged in two-dimensional coordinate[6], [7]. We observed that the chaotic circuits arranged in
the near distance are synchronized at in-phase state, andthe coupled circuits with the far distance could not be synchronized. From the results we confirmed the relationshipbetween clustering and synchronization phenomena.
The more detailed researches are needed for applying
to the investigation of more large scale network and general network. So in this study, we investigate the synchronization phenomena observed from symmetric coupled chaotic circuits and asymmetric coupled chaotic circuits arranged in one-dimensional coordinate. We combine
chaotic circuits by resister, and the circuits are arranged
in one-dimensional coordinate system. The number of thecircuits is always ten and we investigate symmetric systems
and asymmetric systems. The distance between the centralcircuits is fixed. We investigate synchronization phenom-

Figure 2 shows the circuit model. This is a chaotic circuit
called Nishi-Inaba circuit [8]∼[10].

Figure 2: Circuit model.
The circuit equations of this circuit are described as Eq.
(1).
di1
= v + ri1L1


dt





di2

(1)
L2= v − vd (i2 )


dt






dv


 C
= −i1 − i2dt
The characteristic of nonlinear resistance is described asEq. (2).

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VVrd
i2 +− i2 −vd (i2 ) =2rdrd!

The circuit equations are normalized as Eq.
changing the variables as below.

i1 =rqqt=√(3) by

CC1 :

√L1 C
L2 Vy; v = Vz;√L1 C
L1L2 = β; rd L2 = δ;

CL1 V x; i2 =C
L1 = α;(2)In this study, we use the ladder system. In the laddersystem, chaotic circuits are connected to only adjacent circuits.
When chaotic circuits are connected to only adjacent circuits, the circuit equations are shown in Eqs. (5) ∼ (7).

L1Cτ;ẋ 1ẏ1 ż

1d“ · ” = dτ;= αx1 + z1
= z1 − f (y1 )= −x1 − βy1 − γ{1,2} (z1 − z2 )(5)

= αxn + zn
= zn − f (yn )= −xn − βyn − γ{n,n−1} (zn − zn−1 )−γ{n,n+1} (zn − zn+1 )

(6)

CCn :

ẋ

ẏ


 ż

= αx + z= z − f (y)


ẋn


 ẏn

żn



(3)

= −x − βyThe value of f (y) is described as Eq. (4).
!
δ1
1f (y) =y+ − y−2δδFigures 3 and 4 show the chaotic attractor generated
from the circuit by using computer simulation (Fig. 3) andcircuit experiment (Fig. 4). For the computer simulation,
we set the parameters as α = 0.460, β = 3.0 and δ = 470.
For the circuit experiment, the parameters are fixed with
L1 = 500[mH], L2 = 200[mH], C = 0.0153[µF], andrd = 1.46[MΩ].

z

CC N :(4)ẋN ẏNżN= αxN + zN= zN − f (yN )= −xN − βyN−γ{N,N−1} (zN − zN−1 )Where 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 thefollowing equation:
γ{i, j} =x
Figure 3: Chaotic attractor (computer simulation).

Figure 4: Chaotic attractor (circuit experiment).

(7)g.
(di j )2(8)di j denotes the Euclidean distance between the i-th circuitand the j-th circuit. The parameter g is coupling coefficientthat determines the coupling strengths. In this study, we set
the parameter as g = 1.0 × 10−3 .
3. Simulation Method
We use the ladder system arranged in one-dimensional
coordinate system. We use ten circuits in computer simulations. We divide into the two symmetric groups, and there
are five circuits in one side of the group. In the left side andthe right side groups, the distances between the circuits are
0.3. The distance between the central circuits is 0.5. The
symmetric network structure is shown in Fig. 5(a).
We change the distance between the circuits by changing
the coupling strength. We define the distances between thecircuits in the left side group as d1 . In the same way, thedistances between the circuits in the right side group as d2 .
And we define the distance between the central circuits asdcenter . In this simulation, we fix the values of dcenter and d2 ,and the value of d1 is changed. The value of d1 is decreased
gradually, and the value of d1 is changed from 0.3 to 0.1.
The asymmetric network structure is shown in Fig. 5(b).

- 641 -Group-L
d1= 0.3
dcenter=0.5

Ave. 16.354
1-2

2-3

3-4

Phase difference
21.0834-5Group-R
d2=0.3

Ave. 15.531
6-7

7-8

5-6(center)

9-10

8-9

Figure 6: Graphics of the phase difference (computer simulations of symmetric system).

Group-L
d1=0.1dcenter=0.5

Ave. 8.263
1-2

Group-R
d2=0.32-3

3-4

Ave. 19.768

86-7

Phase difference
90.011

4-5

7-8

9-10

8-9

5-6(center)

Figure 7: Graphics of the phase difference (computer simulations of asymmetric system).

asymmetric system. From this figure, we can see that thecircuits in the groups are synchronized, however the centralcircuits become asynchronous. And the phase differencesin the right side group are larger than the symmetric system.
Additionally, the phase differences in the left side group aresmaller than the symmetric system. Table 1 shows the averages of the phase differences in each system.
Figure 8 shows the simulation result when d1 is changed.
This result shows the shift of the phase difference. We focus on the phase differences of the central circuits, between
the 4th and the 5th circuits, and between the 6th and the 7thcircuits. From the simulation result, the various results canbe obtained.

(a) Symmetric network structure.

(b) Asymmetric network structure.

Table 1: Averages of the phase differences
Group-L dcenter Group-R
Symmetric system
16.35421.08315.531
Asymmetric system
8.263
90.01119.768Figure 5: Network structures.

We measure the phase difference between the circuits
using the computer simulation. And we investigate thechange in the phase difference when the system is changed
from the symmetric system to the asymmetric system.
4. Simulation Result
Figure 6 shows the simulation result of symmetric system. From this figure, we can see that the all circuits
are synchronized. Figure 7 shows the simulation result ofIn the case of the central circuits, the phase difference
is increasing gradually. And the central circuits become
asynchronous around d1 = 0.19. In the case of between the4th and the 5th circuits, the phase difference is continued
to decrease because the distance between the circuits in theleft side group is continued to decrease. In the case of between the 6th and the 7th circuits, the phase difference isincreasing gradually, and the phase difference is decreased- 642 -[4] G. Abramson, V.M. Kenkre and A.R. Bishop, “Analytic Solutions for Nonlinear Waves in Coupled Reacting Systems,” Physica A: vol. 305, no. 3-4, pp.
427-436, 2002.
[5] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer,“Synchronization and Graph Topology,” Int. J. Bifurcation and Chaos, vol. 15, no. 11, pp. 3423-3433,
2005.
[6] Y. Takamaru, H. Kataoka, Y. Uwate and Y. Nishio,“Clustering Phenomena in Complex Networks of
Chaotic Circuits,” Proc. ISCAS12 Mar. 2012.
Figure 8: Simulation result.

from around d1 = 0.19.
In the right side group, the phase difference began todecrease with the central circuits became asynchronous.
Asynchronous state of the central circuits strengthen thecoupling strength of each groups.
5. Conclusions
In this study, we have investigated the synchronization
phenomena in coupled chaotic circuits networks. We also
investigated the phase difference in the symmetric network
system and the asymmetric network system. From the computer simulation, when using the symmetric network system, all circuits are synchronized. And when using theasymmetric network system, only central circuits become
asynchronous. From this results, we could make sure of the
breakdown of the inter-cluster synchronization.
For the future work, we would like to confirm the same
results by using the circuit experiments.

[7] Y. Takamaru, Y. Uwate, T. Ott and Y. Nishio,“Clustering Phenomena in Complex Networks Coupled with a Resisters,” Tech. Report of IEICE, no.
NLP2012-8, pp.41-44, Apr. 2012.
[8] R. Stoop, P. Benner and Y. Uwate, “Real-World
Existence and Origins of the Spiral Organizationof Shrimp-Shaped Domains,” Phys. Rev. Lett., 105,074102, Aug. 2010.
[9] C. Bonatto and J. A. C. Gallas, “Periodicity Hub andNested Spirals in the Phase Diagram of a Simple Resistive Circuit,” Phys. Rev. Lett., 101, 054101, Aug.
2008.
[10] R. Stoop, P. Benner and Y. Uwate, “Real-World
Existence and Origins of the Spiral Organizationof Shrimp-Shaped Domains,” Phys. Rev. Lett., 105,074101, Aug. 2010.

Acknowledgment
This work was partly supported by JSPS Grant-in-Aidfor Challenging Exploratory Research 26540127.
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