IEEE Workshop on Nonlinear Circuit Networks
December 12-13, 2014

Comparison of Synchronization Phenomena
in Three Network Topologies
Jun Shiomoto† , Yoko Uwate† , Thomas Ott‡ and Yoshifumi Nishio††Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {shiomoto, uwate, nishio}@ee.tokushima-u.ac.jp‡Dept. 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 couplingstrength. We investigate synchronization phenomena when thedistance between the circuits in the group is changed. Also, wemeasure the phase difference using computer simulations. Fromthe computer simulations, we could make sure of the breakdownof inter-cluster synchronization when the system is changed fromthe symmetrical system to the asymmetrical system. Additionally,
we compare the results of three systems.

(a) Ladder system.

I. I NTRODUCTION
Synchronization phenomenon is one of the typical phenomena observed in nature. Recently, many studies have beeninvestigated synchronization of chaotic circuits [1]∼[5]. It is
focused how the differences of the network structure impact
on the whole circuits. Additionally, it is applicable to the fieldsof medical science and biology and so on.
In our research group, we have observed the synchronization phenomena from symmetrical coupled chaotic circuits
and asymmetrical coupled chaotic circuits arranged in onedimensional coordinate. We used only ladder system. In theladder system, chaotic circuits are connected to only adjacentcircuits. Chaotic circuits were coupled by resister. The numberof the circuits was always ten and we investigated symmetricalsystems and asymmetrical systems. The distance between thecentral circuits was fixed. We investigated synchronization
phenomena by changing the distance between the circuits. We
have made sure of the breakdown of inter-cluster synchronization when the system was changed from the symmetricalsystem to the asymmetrical system [6].
In this study, we use three systems. The systems are laddersystem, bridge system and full coupled system. In the full
coupled system, the chaotic circuits are connected to all
chaotic circuits. In the bridge system, only the circuits in thegroup are full coupled. Figure 1 shows each system model.
We compare the results of the phase differences in the threesystems.

(b) Full coupled system.

(c) Bridge system.
Fig. 1. System models.

Fig. 2.

Circuit model.

The normalized equations of this circuit are obtained asEq. (1) by changing the variables as below.

II. C IRCUIT M ODEL
Figure 2 shows the circuit model. This is a chaotic circuit
called Nishi-Inaba circuit [7]∼[8].

- 36 -√√L1 C
i1 = LC1 V x; i2 = LV y; v = V z;
2√√
L1 C
L1r LC1 = α; L= β; rd L= δ;22√dt = L1 Cτ ; “ · ” = dτ ;
ẋ
ẏ

ż

= αx + z= z − f (y)(1)= −x − βyThe value of f (y) is described as Eq. (2).
()δ1
1f (y) =y+− y−2
δ
δ

(2)

Figure 3 shows the chaotic attractor generated from thecircuit by using computer simulation (Fig. 3(a)) and circuit
experiment (Fig. 3(b)). For the computer simulation, we set theparameters 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 ], and rd = 1.46[M Ω].

z

In full coupled system, the circuit equation is shown in Eq.
(7).
ẋi

 ẏiż i= αxi + zi
= zi − f (yi )∑N= −xi − βyi − j=1 γ{i,j} (zi − zj )(i, j = 1, 2, · · ·, N )(7)

In bridge system, the circuit equation is shown in Eq. (8).
This equation shows the case of using ten circuits.
ẋi = αxi + ziẏi = zi − f (yi )(8)żi = −xi − βyi − Γ(i = 1, 2, · · ·, N )x

CC1∼4 :(a) Computer simulation.
Fig. 3.

(b) Circuit experiment.

Γ = γ{n,k} (5 · zn −Chaotic attractor.

5∑zk )(n = 1, 2, 3, 4)

k=1

CC5 :
In the ladder system, chaotic circuits are connected to only
adjacent circuits.
When chaotic circuits are connected to only adjacent circuits, the circuit equations are shown in Eqs. (4) ∼ (6).
Where the parameter γij represents the coupling strength
between the circuits. The value of γij reflects the distance
between the circuits in an inverse way, described by thefollowing equation:g.
γ{i,j} =(3)(dij )2
dij denotes the Euclidean distance between the i-th circuit andthe j-th circuit. The parameter g is coupling coefficient that
determines the coupling strengths. In this study, we set theparameter as g = 1.0 × 10−3 .
CC1 := αx1 + z1
= z1 − f (y1 )= −x1 − βy1 − γ{1,2} (z1 − z2 )(4)

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

(5)

ẋN


ẏN
 żN



(6)

zk )

Due to the symmetry of the system, equation for from CC6
to CC10 is omitted.
III. S IMULATION M ETHOD
We use the three systems arranged in one-dimensional coordinate system. We use ten circuits in computer simulations.
We divide into the two symmetrical groups, and there are five
circuits in one side of the group.

Network structure.

In the left side and the right side groups, the distances
between the circuits are 0.3. The distance between the centralcircuits is 0.5. The symmetrical network structure is shown inFig. 5.

Fig. 5.

= αxN + zN
= zN − f (yN )= −xN − βyN−γ{N,N −1} (zN − zN −1 )6∑
k=1

Fig. 4.

 ẋ1ẏ1

ż1

CCn :
ẋnẏnżn

CCN :

Γ = γ{5,k} (6 · z5 −Symmetrical network structure.

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, the- 37 -

distances 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 andd2 , 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. Theasymmetrical network structure is shown in Fig. 6.

full coupled system, the central circuits do not become asynchronous for large number of the coupling.
From this result, we can see that the strength of synchronization is not affected by the number of the coupling of thesystems.
V. C ONCLUSIONS

Fig. 6.

Asymmetrical network structure.

We measure the phase difference between the circuits using
the computer simulation. And we investigate the change inthe phase difference when the system is changed from thesymmetrical system to the asymmetrical system. Additionally,
we compare the results of the three systems.

In this study, we have investigated the synchronization
phenomena in coupled chaotic circuits networks. We also
investigated the phase difference in the symmetrical network
system and the asymmetrical network system. Additionally, wehave compared the results of three systems. From the computer
simulation, we have confirmed the similar results in all of thesystem. Additionally, we have obtained very interesting results
to compare the central circuits of the three systems.
For the future work, we would like to confirm the same
results by using the circuit experiments.
ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Challenging Exploratory Research 26540127.

IV. S IMULATION R ESULTR EFERENCES

Figure 7 shows the graph of comparison of the resultsin each system. This figure is focused on only the phasedifference of the central circuits.

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[7] R. Stoop, P. Benner and Y. Uwate, “Real-World Existence and Origins ofthe Spiral Organization of Shrimp-Shaped Domains,” Phys. Rev. Lett.,
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Fig. 7. Comparison of the results in each system (between the centralcircuits).

From the simulation result, in the ladder system, the phasedifference is increasing gradually. And the central circuits
become asynchronous around d1 = 0.19. In the bridge system,
the phase difference is increasing gradually in the same way
as ladder system. However the central circuits in the bridgesystem becomes asynchronous around d1 = 0.22. Although
the number of coupling of the bridge system is heavier than theladder system, the bridge system become faster asynchronousthan the ladder system. In the bridge system, only the circuitsin the group are full coupled. We consider that the bridge
system becomes asynchronous faster than the ladder system,
because the coupling in the group becomes stronger. In the- 38 -