Synchronization Phenomena of Coupled Chaotic
Circuits Network with Coupling Strength Dependingon Number of Degree
Kyohei Fujii, Shuhei Hashimoto, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Engineering,Tokushima University
2-1 Minami-Josanjima, Tokushima 7708506, Japan
Email: fujii, s-hashimoto, uwate, nishio@ee.tokushima-u.ac.jp
Abstract— We investigate the influence of direct coupling andindirect coupling for synchronization state in the network. The
node of the network is expressed by chaotic circuit. This complex
network is designed that a node has two or more links except for
one node with scale free property. The coupling strength isdetermined depending on the number of links between the nodes.
In this investigation, we find that not only the direct coupling but
also the indirect coupling by the path is strongly related forsynchronization state.
Keywords; Synchronization, Complex network, Chaotic Circuits

I.

INTRODUCTION

In our living life, complex network can be seen in various
fields such as airport network, computer network and neurons in
the brain. Furthermore, synchronization phenomena observed incomplex network have been investigated by many researchers[1], [2]. Especially synchronization phenomena of coupledoscillatory systems are very interesting. There are various
investigations of synchronization using chaotic circuits.
However, few studies focus on the difference in couplingstrength [3], [4].

Figure 1: Circuits model.

(1)

In this study, we focus on the path and coupling strength incoupled chaotic circuit network. We design a network that eachnode has different coupling strength depending the number ofdegree and the network has scale free property. In order to makethe network which the total coupling strength of node in thenetwork has nearly uniform, we apply the following rule to set
the coupling strength. When the difference of number of degree
between two nodes is large, the coupling strength between two
nodes is set to small value.

The normalized equation of this circuit is given as follows:(2)

By using the computer simulations, we investigate
synchronization between hubs when one link is opened.
Furthermore, we also investigate synchronization between hubs
when another link is opened.
II.

The nonlinear function f() corresponds to the i-v characteristicsof the nonlinear resistor consisting of the diodes and aredescribed as follows:

CIRCUIT MODEL

Figure 1 shows the chaotic circuit which is three dimensional
autonomous circuit proposed by Shinriki et al. This circuit
consists of one negative resister, two capacitors, one inductor
and dual-directional three diodes [5]. And this circuit equationis shown in Eq. (1).

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(3)

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III.

SYSTEM MODEL

From this result, we confirm that the synchronization state isaffected not only direct coupling but also indirect coupling.
Next, we focus on the link of CC18 and CC19. The
synchronization between CC18 and CC19 is measured when
only CC10 - CC19 is opened or when only CC6 – CC12 isopened.
Table 1: Synchronization rate with open one link

This complex network is designed that each node has two or
more links except for one node with scale free property. Only
three nodes are set to the hubs (CC18, CC19, CC20). All
coupling strength value are fixed with a certain rule. This rule isbased on the different degrees between nodes. The value ofcoupling strength is determined by the number of degree of thenode. Namely, the coupling strength is set to the certain value by
calculating the difference of number of degree between thenodes and add 1.0 for every links.

Location of open link
Synchronization rate ofCC18-CC19 [%]CC10-CC19

CC6-CC12

4636

The number of circuits between the two nodes is different forthe both cases of open link (CC10-CC19 and CC6-CC12).
Table 1 shows the result of synchronization rate when one linkis opened. We confirm that the synchronization rate has highervalue when the link between CC10-CC19 is opened.
V.

CONCLUSION

In this study, we have investigated influence of path by using
scale-free coupled chaotic network by changing the couplingstrength. In this result, we found an effect of the path andcoupling strength. Next, we have investigated synchronization
between hubs when one indirect link is opened. We confirm thatthe effect of the coupling strength on synchronization is larger
than the effect of the open link.
ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 16K06357.
Figure 2: Network model.
IV.

REFERENCES[1] M. Uchida, S. Shirayama, “Analysis of Network Structure and Model
Estimation for SNS” IPSJ journal, Vol. 47, No. 9, Sep. 2006.

SIMULATION RESULTIn order to analyze synchronization states, we define thesynchronization by the following equation.

[2] H. Kori, N. Masuda, “Synchronization of Coupled Oscillators on Complex
Networks” Journal of the Robotics Cociety of Japan, Vol.26, No.1, pp.6-9, 2008.

(4)[3] K. Ago, Y. Uwate, Y. Nishio, “Investigation of Synchronization in Coupled
Chaotic Circuit Network with Local Bridge”, IEEE Workshop on Nonlinear
Networks December 12-13, 2014.

First, we investigate the synchronization by changing thecoupling strength. Figure 3 shows the synchronization rate withthe coupling strength.

[4] K. Oi, Y. Uwate, Y. Nishio, “Influence of Regional Change inSynchronization of Complex Networks in Coupled Parametrically Excited
Oscillators", IEEE Workshop on Nonlinear Circuit Networks. December 9-10,
2016.
[5] K. Ago, Y. Uwate and Y. Nishio, “Synchronization of Coupled Chaotic
Circuits with Parameter Dispersion in Small World Network", Proceedings ofInternational Symposium on Nonlinear Theory and its Applications, pp. 431-434,
Dec.2015.

Figure 3: Synchronization rate of same strength link.

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