平成25年度電気関係学会四国支部連合大会 講演論文集 （2013 徳島大学）
2013 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (TOKUSHIMA)

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Synchronization Phenomena of Complex Networks
Arranged in One-dimensional Coordinate
Jun SHIOMOTO†
Yoko UWATE†
Thomas OTT‡
Yoshifumi NISHIO††‡( Tokushima University) ( Zurich University of Applied Sciences)

1. Introduction

In addition, we use the model shown in Fig. 2 in thisstudy. The number of coupling circuits is N , and we doresearch in N = 4, 6, 8, 10. And we measured the phasedifference between the circuits of the central circuits.Thenumber of circuit always used an even number, we investigate system which is symmetrical system. We change
the vary of distance between circuits and adjust the coupling strength. The distance between the circuit in thecenter of the system fixed, and after they are group of twosymmetrically, we do research.

In this study, we investigate synchronization phenomena observed from coupled chaotic circuits. First, we combine chaotic circuits by resister which are arranged in onedimensional coordinate system. We investigated synchronization phenomena by changing the distance between thecircuit and the number of coupling circuits. Also, we measured the phase difference between the central circuits using computer simulations.

2. Circuit model

3. Simulation results

Figure 1 shows the circuit model. This is a circuit model
called nishio circuit.

In this study, we carried out the simulation in d =0.08 ∼ 0.5. We show the simulation results in Table 1 andFig. 3. We obtained the results that the phase difference
becomes the minimum in d = 0.1 ∼ 0.2. In N = 6, 8, 10,
the phase difference becomes the minimum in d = 0.1,
and the phase difference also increases as the distanceincreases. However, in N = 4, the phase difference become the minimum in d = 0.2, and the phase difference issmaller again in d = 0.4 as the distance increases.

d d d d1

Table 1: Simulation result [degree]{{{{{{{{{

Figure 1: Circuit model.

d d d dFigure 2: System model.
We can consider the following equations, when all circuits are coupled globally each other.
dxi= αxi + zidτdyi= zi − f (yi )dτd

0.08

0.1

0.2

0.3

0.4

0.5

N=444.7

35.7

26.9

36.2

29.4

30.3

N=628.2

21.2

22.1

24.4

29.3

32.9

N=819.3

14.3

20.3

23.5

26.2

33.2

N=1015.9

13.1

17.8

20.8

22.3

26.1

(1)
NX
dzi
= −xi − βyi −
γij (zi − zj )dτ
j=1(i, j = 1, 2, · · ·, N )f (y) in the formula can be expressed as follows:f (y) =δ2



y+1
1− y−
δδ

(2)

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:
γij =g(dij )2.

Figure 3: Simulation result.

4. Conclusions
In this study, we have researched the synchronization
phenomena in coupled chaos circuits networks. We confirmed that the various results are obtained by changing
the distance between the circuit and the number of coupling circuits.

(3)

dij denotes the Euclidean distance between the i-th circuit and the j-th circuit. The parameter g is coupling
coefficient that determines the coupling strengths. In thisstudy, we set the parameter as g = 1.0 × 10−2 .

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