Analysis of Synchronization Phenomenain Complex Networks
Consisting of van der Pol Oscillators
Tsuyoshi Isozaki, Takumi Nara, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima 7708506, Japan
E-mail: {isozaki, nara, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Complex networks have been the subject of greatinterest. Some characteristics of complex networks are closely
related to the real world network and have been studied in termsof network topology and interactions between nodes. In thisstudy, we build three networks which are composed of van der
Pol oscillator and investigate synchronization phenomena in eachnetwork. As a result, it was confirmed that the synchronization
states are different depending on the network structure.

1.

Introduction

Complex networks have attracted a lot of attention from various
fields such as sociology, biology and engineering [1]. Further in thefield of engineering, research on complex networks using circuits
has been studied, and interesting phenomena such assynchronization between circuits have been observed. For
example, studies have focused on the structure and degreedistribution of networks [2],[3]. However, there are still many
unclear points, and studies in networks close to the real world will
become even more important in the future.

(c)Figure 2: Network model.
(a) Regular. (b) ER random. (c) BA scale-free.
Table 1: Node number 𝑵, Average node degree 𝒅𝒂𝒗𝒈 , Maximum
node degree 𝒅𝒎𝒂𝒙 , Minimum node degree 𝒅𝒎𝒊𝒏 and Average
distance 𝑫𝒂𝒗𝒈 of three network models.

In this study, we build three network models by using oscillatorsand investigate synchronization phenomena between nodes innetworks.

2.

(b)

(a)System ModelFigure 1 shows van der Pol oscillator. This oscillator consists of acapacitor, an inductor and a nonlinear element. Further we use
three network models in this study. Figure 2(a) shows regularlattice network, Fig. 2(b) shows Erdos-Renyi model (ER random)and Fig. 2(c) shows Barabasi-Albert model (BA scale-free). Table
1 shows each network properties.

networkRegular

ER random

BA scale-free𝑵

100

100100

𝒅𝒂𝒗𝒈

4.0

3.74

3.92

𝒅𝒎𝒂𝒙

4.0

9.0

32.0

𝒅𝒎𝒊𝒏

4.0

1.0

2.0

𝑫𝒂𝒗𝒈

12.87

3.68

2.98

In this study, we set the average node degree to be close to 4.0
each network model. Further as features of the network, themaximum node degree is largest for BA scale-free, the average
distance is largest for Regular and smallest for BA scale-free.

The normalized circuit equations are described as follows:
𝒅𝒙𝒊𝒅𝝉
𝒅𝒚𝒊𝒅𝝉

= 𝜶{𝜺𝒙𝒊 (𝟏 − 𝒙𝒊 𝟐 ) − 𝒚𝒊 − ∑𝟏𝟎𝟎
𝒊,𝒋=𝟏 𝜸𝒊𝒋 (𝒙𝒊 − 𝒙𝒋 )}
= 𝒙𝒊

(1)
(𝒊, 𝒋 = 𝟏, 𝟐,∙∙∙, 𝟏𝟎𝟎).

Figure 1: van der Pol oscillator.

978-1-7281-8331-2/20/$31.00 ©2020 IEEE187

ISOCC 2020

3.

Results

4.

In this study, we built three network models by using oscillatorsand investigated synchronization phenomena between nodes innetworks. It is not confirmed that synchronization rates andmismatch differences are heavily related in regular network.
Further it is confirmed that synchronization rates and mismatch
differences are related in ER random and BA scale-free.

We set the parameters of van der Pol oscillator as 𝛆 = 𝟎. 𝟏 and
𝛂 = 𝟏. 𝟎. Here, the mismatch is added to 𝛂. The mismatch isgenerated by random and the range of the mismatch is set to [0.5:0.5]. In this study, we vary the coupling strength 𝛄 in each
network model and show results on the relation between mismatch
differences and synchronization rates in the three network models.

5.

(ⅰ) 𝛄 = 𝟏. 𝟎(ⅱ) 𝛄 = 𝟐. 𝟎(ⅲ) 𝛄 = 𝟑. 𝟎6.

(ⅱ) 𝛄 = 𝟐. 𝟎

Future work

In the future, we will investigate how the synchronization ratechanges when the coupling strength of ER random is furtherincreased. Further it was confirmed the number of pair of nodes
decreased when the coupling strength was increased in BA scalefree. Therefore it is assumed that the optimal coupling strength
exists around 0.5 and will investigate in detail.

(a) Regular.

(ⅰ) 𝛄 = 𝟏. 𝟎Conclusion

Reference[1] Jiajing Wu, Chi K. Tse, Francis C. M. Lau and Ivan W. H. Ho“Analysis of Communication Network Performance From a Complex
Network Perspective”, IEEE TRANSACTIONS ON CIRCUITSANDSYSTEMS-Ⅰ: REGULAR PAPERS, vol. 60, NO. 12,
DECEMBER 2013.

(ⅲ) 𝛄 = 𝟑. 𝟎(b) ER random.

[2] Kenta Ago, Yoko Uwate and Yoshifumi Nishio “Synchronization ofCoupled Circuits with Parameter Dispersion in Small-WorldNetwork”, 2015 International Symposium on Nonlinear Theory andIts Applications NOLTA2015, Kowloon, Hong Kong, China,December 1-4, 2015.
(ⅰ) 𝛄 = 𝟎. 𝟓(ⅱ) 𝛄 = 𝟏. 𝟎(ⅲ) 𝛄 = 𝟏. 𝟐

[3] Shuhei Hashimoto, Takahiro Chikazawa, Yoko Uwate, andYoshifumi Nishio “Synchronization Phenomena in Complex
Networks of Coupled Chaotic Circuits with Different DegreeDistribution”, 2017 RISP International Workshop on Nonlinear
Circuits, Communications and Signal Processing (NCSP'17) Guam,USA, February 28th to March 3rd, 2017(c) BA scale-free.
Figure 3: The relation between mismatch differences,synchronization rates and coupling strength 𝛄.
(The vertical axis is synchronization rates and the horizontal
axis is mismatch differences.)

First, when the coupling strength is varied, the synchronization
rates do not change significantly in Fig. 3(a). Therefore it is notconfirmed that synchronization rates and mismatch differences
are heavily related in regular network. Second, when the couplingstrength is higher, the number of pair of nodes with higher
synchronization rates increases as the mismatch difference issmaller in Fig. 3(b). Therefore it is confirmed that synchronization
rates and mismatch differences are related in ER random. Third,
when the coupling strength is higher, the number of pair of nodes
with higher synchronization rates decreases, and thesynchronization rates vary even when the coupling strength issmall compared to other networks in Fig. 3(c). Therefore it isconfirmed that synchronization rates and mismatch differences
are related in BA scale-free. These results confirm that the optimal
coupling strength in synchronization between circuits depends onthe characteristics of the network structure.

978-1-7281-8331-2/20/$31.00 ©2020 IEEE

188

ISOCC 2020