2023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

Synchronization Phenomena in Complex Oscillatory Networks with Switching Coupling
Haruka Sakohira, Kiichi Yamashita, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email:{sakohira, kiichi, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract

In this study, we focus on the synchronization rate and investigate changes in the network by changing the topology.
Complex networks, which are related to real-world networks, The network of Erdős Rényi model (ER model) using 100
have attracted much attention and have been studied in vari- van der Pol oscillators is constructed. Then the synchronizaous ﬁelds. The complex networks have common characteris- tion rate is investigated, edges with synchronization rates betics and have been studied in terms of network topology and low a set threshold are cut off, and edges are added for thatinteractions between nodes. Complex networks with circuits number of edges. Repeat this process and compare how thehave been studied, and synchronization phenomena between synchronization rate changes when the topology is changed.
circuits have been conﬁrmed.
In this study, a random network is constructed using 100
van der Pol oscillators. Then the synchronization phenom- 2. Network modelena between the circuits of the network are analyzed. TheIn this study, we use the ER model. The number of nodes
computer simulations are conducted by focusing on the synchronization rate and thereby changing the topology. As a is 100, and the average degree is set to be about 5.1. First,
result, it is conﬁrmed that the synchronization rate decreases Fig. 1 shows the model of the random network used in thisstudy. This network has the property that the number of edges
by changing the topology.
gathered at each node are small.
1. Introduction
Complex networks are huge, complex structures that exist in reality and have several properties. First, it is scalefree, in that it has hubs, which are nodes of large degree, but
most of them are connected to only a few nodes and havesmall degree. Second, it is small world, in that any number of nodes relate to only a few nodes in the middle. The
third is clustering, in which nodes are not isolated from eachother, but rather several nodes are grouped together to form asmall network of their own. These can be modeled by nodes
and edges, and have attracted various ﬁelds of attention, suchas engineering, sociology, and biology [1]-[3]. Furthermore,in the ﬁeld of engineering, complex networks using circuits
have been studied, and interesting phenomena such as synchronization have been investigated [4], [5]. In these studies,synchronization phenomena are investigated by focusing on
the topology structure of the network, and the differences inthe structure. Synchronization is phenomena in which things
around us gradually come to be in step with each other according to the inﬂuence of other things around us. It is oneof the most familiar phenomena in nature and has been thesubject of various studies [6], [7].

Figure 1: ER model.

3. Circuit model
Figure 2 shows the van der Pol oscillator used in this study.
This is a simple circuit consisting only of a capacitor, an inductor, and a nonlinear element. These van der Pol oscillators
are connected as resistors to reproduce the network.

198

2023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

nv

iLiG

LG

C

In this case, α represents the small error of each capacitor and Enk represents the adjacency matrix of the network.
In this study, the synchronization rate is investigated, edges
with synchronization rates below a set threshold are cut off,and edges are added for that number of edges. The thresholdvalue of the synchronization rate that changes the topology
is set to 50[%], and it is changed the connection 10 times.
Then, the synchronization rates are compared when the coupling strength is changed from γ = 0.3 to γ = 0.6 by 0.1.

Figure 2: van der Pol oscillator.

4. Simulation results
In order to investigate synchronization phenomena between circuits, we deﬁne synchronization as in Eq. (4).

The circuit equations are as follows:
100
∑

dvn1


C
=−i−i−(vn − vk )LgnndtR(4)|xn − xk | < 0.01.
(1)ig = −g1 v + g3 v 3 .

(2)|xn-xk|In this study, the synchronization rates are calculated in therange of τ = 10,000 to τ = 20,000, where the voltage differences settle down to steady state. Where n, k are the numberof circuits. Figure 3 shows the waveform of the differential(n, k = 1, 2, · · ·, 100).
voltage measured in this study. The two lines in the ﬁgure are
the threshold values shown in Eq. (4), and when the waveIn addition, the nonlinear element equations are as follows: form is within this range, we judge them to be synchronized
and calculate the synchronization rate.
n,k=1din
L= vn
dt


To normalize the circuit equations, the following parameters are used:√√
g1
g1 Cv=x, i =y, t = LCτ
g3
g3 L√
√
L1 L
ε = g1
, γ=.
CR C
√

0.01

-0.01

τ
Figure 3: The differential voltage waveform for determining
synchronization.

First, Table 1 shows the clustering coefﬁcient (C), averThe normalized circuit equations derived from the above age path length (L), average synchronization rate (SR),¯ andare shown as follows:
number of cutting edges (CE) when we cut edges of synchronization rate below the threshold value and changed thetopology 10 times. The coupling strength is set to γ = 0.3 for100∑
dxnthe simulation. As a result, it shows the clustering coefﬁcient2
Enk γ(xn −xk ) dτ =αεxn (1−xn )−yn −increases as the topology is changed when γ = 0.3 and then,k=1(3) threshold is set to 50[%]. However, the average synchronizadyn=xntion rate is decreased. Also, the topology is denser due to the dτ
increased clustering coefﬁcients.
(n, k=1, 2, ···, 100).
199

2023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

¯ and number of cutting edges (CE)
Table 1: The clustering coefﬁcient (C), average path length (L), average synchronization rate (SR),when we cut edges of synchronization rate below the threshold value and changed the topology 10 times(γ = 0.3).
CL
¯
SRCE

0th0.04901
2.954
60.58
-1st0.03756
2.934
55.54
122

2nd0.05835
2.993
58.21
145

3rd0.07064
3.035
54.97
130

4th
0.05984
3.058
58.32
135

5th
0.04059
3.005
52.58
129

6th
0.06277
2.970
57.34
150

7th
0.04902
2.977
60.45
138

8th
0.05695
2.987
55.95
125

9th
0.05524
3.018
57.14
135

10th
0.08747
3.012
50.69
128

Finally, Figs. 6-7 show the relationship between the average synchronization rate and the number of pair nodes when
the parameter γ is set to 0.3 and 0.6, respectively. They showthat the number of pair nodes with 90-100[%] is the highest
both for γ = 0.3 and γ = 0.6, and that the number is higher
for γ = 0.6, which has stronger coupling strength. Also, weAlso, Fig. 5 shows the relationship between the average can see that the percentage of 90-100[%] is decreasing andsynchronization rate and the number of cutting edges. It is the percentage around 0-30[%] is increasing by changing theshown that the synchronization rate is higher when the num- topology. These are similar results were obtained for othercoupling strengths.
ber of cutting edges is smaller.

Average synchronization rate[%]The number of pair nodes

Figure 4 shows the transition of the average synchronization rate at each coupling strength. It shows that the averagesynchronization rate becomes higher as the coupling strength
increases. It can be seen that the synchronization rate decreases when the topology is changed by cutting the edges
below the threshold value.

synchronization rate[%]Count

Figure 6: The relationship between the average synchronization rate and the number of paired nodes (γ = 0.3).

The number of pair nodes

Average synchronization rate[%]Figure 4: The transition of the average synchronization rate
at each coupling strength.

synchronization rate[%]Number of cutting edges

Figure 5: The relationship between the average synchronization rate and the number of cutting edges.

Figure 7: The relationship between the average synchronization rate and the number of paired nodes (γ = 0.6).

2002023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

5. Conclusions

works Consisting of Oscillators ”, ISOCC 2021, pp.181182, 2021.

In this study, we investigated how the average synchronization rate changes when the topology is changed by cut- [6] Chikayasu Higashi and Yoshifumi Nishio, “Investigation of Synchronization Phenomena in Coupled Simulting edges below a threshold value. The results showed thattaneous Oscillators ”, 2010 International Workshop on
the clustering coefﬁcient increased and the network became
Nonlinear Circuits,Communication and Signal Processdenser, but the average synchronization rate decreased. In
ing (NCSP’10), pp.113-116, March 2010.
addition, when the number of edges to be cut were smaller,
the average synchronization rate became larger, and when the[7] Shuhei Hashimoto, Yoko Uwate and Yoshifumi Nishio,
number of edges to be cut were larger, the average synchro“Synchronization of Coupled van der Pol Oscillators withnization rate became smaller. This is considered to be due
Distributed Hub Nodes ”, 2018 International Symposium
to the fact that the smaller the number of edges to be cut, theon Nonlinear Theory and Its Applications (NOLTA2018),
stronger the coupling strength and the higher the synchronizapp.542-545, September 2018.
tion rate per edge.
In the future, the simulations were investigated with a loopat τ = 10,000 to τ = 20,000 in this study, but we would
like to conduct dynamic simulations by running the simulation time. Also, the results of this study showed that the average synchronization rate decreased, but we would like toinvestigate a more detailed study and investigate how the average synchronization rate changes. In addition, we would
like to compare how the average synchronization rate changes
by different parameters apart from the coupling strength andby different network models.
References[1] Luciano da Fontoura Costa, Osvaldo N. Oliveira Jr., Gonzalo Travieso, Francisco Aparecido Rodrigues, Paulino
Ribeiro Villas Boas, Lucas Antiqueira, Matheus Palhares
Viana and Luis Enrique Correa da Rocha, “Analyzing andModeling Real-World Phenomena with Complex Networks: A Survey of Applications”, Advances In Physics,
May 2011.
[2] 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 circuits and Systems-I: Regular Papers, vol. 60,No. 12, December 2013.
[3] Yongxiang Xia and David J. Hill, “Attack Vulnerabilityof Complex Communication Networks”, IEEE Transactions on circuits and Systems-II: Express Briefs, VOL.55,
No.1, January, 2008.
[4] Tsuyoshi Isozaki, Takumi Nara, Yoko Uwate and Yoshihumi Nishio, “Research on the Relationship between
Network Structure and Synchronization Phenomena in
Complex Networks by Using Oscillators”, Proceedings of IEEE Workshop on Nonlinear Circuit Networks(NCN’20), pp.11-14, December 2020.
[5] Tsuyoshi Isozaki, Yoko Uwate and Yoshihumi Nishio,“Information Transmission Focusing on Complex Net201