IEEE Workshop on Nonlinear Circuit Networks
December 15-16, 2017

Synchronization Phenomena in Coupled Oscillators
Containing Star Structure Connected to AnotherOscillator with Different Coupling StrengthVu Minh Hien, Minh Hai Tran, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {hien,minhhai,uwate,nishio}@ee.tokushima-u.ac.jp

Abstract—In this study, synchronization phenomena in coupledoscillators containing star structure connected to another oscillator is investigated. We focus on the phase different between
two oscillators when coupling strength is changed. Furthermore,
we also investigate the effect of the initial condition. By using
computer simulations, we observe synchronization phenomenaof the system.

II. C IRCUIT MODEL
The proposed circuit is shown in Fig. 2. We used threevan der Pol oscillator coupled as star structure that connected
another oscillator via resistor r. We investigate synchronization
phenomena by changing coupling strength of resistor R.

I. I NTRODUCTION
Synchronization phenomena are observed everywhere inour world. For example, firefly luminescence, birds and frogs
crying, human aplause and so on. It has a long history ofresearch and its applications can be widely found in many
fields of science, such as chemical, physical, biological, andalso social systems. Therefore, investigation of synchronization phenomena has become an important issue.
The van der Pol oscillator is the symple circuit. It consistsof resistor, inductor, capacitor and nonlinear resistor. Synchronization phenomena of van der Pol oscillator is the oneof nolinear phenomena that can be observed similar natural
phenomena by controling frequencies.

Fig. 2. Circuit model.

The circuit equations of VDP-B are given as folows:−ig1 − ic1 − iL1 = iB2 + iB3 + iB4
diL1
L= v1
dt
v1 − v2 = r(iB2 − iA2 )v1 − v3 = r(iB3 − iA3 )v1 − v4 = r(iB4 − iA4 ),(1)

where v1 denote the voltage of oscillator VDP-B.
The circuit equations of VDP-A are given as follows:
Fig. 1. van der Pol oscillator.

In this study, we proposed a new type of coupled van der
Pol oscillators system: Star structure connected to anotheroscillator. By carring out computer simulations, relationship ofthe model between synchronization phenomena and couplingstrength is investigated.

- 69 -−igk − ick = iAk + ik4∑
vk − vLk = R
iLn .

(2)

n=2

(k = 2, 3, 4)Nonlinear resistor defined as follows:
ign = −g1 vn + g3 vn3 .

(3)

By changing the variables and parameter:√√
g1t = LCτ, vn =xn ,3g3√√
g1 CLin =yn , α = g1,3g3 LC√√
1 LC
β=,γ = R.
r CL(4)

Fig. 4. Phase difference (α = 0.01, β = 0.01, γ = 0.007).

The normalized circuit equation of VDP-B are given asfollows:
4∑
1dx1


= α(x1 − x31 ) − y1 + β(3x1 −xn )dτ3(5)n=2
 dy1 = x .
1dτNext, we decrease the values of parameter γ from 0.007 to0.005 and continue conducting computer simulation. In thiscase, all four oscillatos become synchronization with second,
third and fourth oscillator of VDP-A are in-phase and only firstoscillator of VDP-B is anti-phase. The result of simulation isshown in Fig. 5.

The normalizaed circuit equation of VDP-A are given asfollows:
1 3dxn dτ = α(xn − 3 xn ) − yn − β(x1 − xn )
4(6)
∑dyn


=x−γyn ,n dτ
n=2where parameters α, β, γ denote the coupling strengh ofthe inductor, resistors r and resistor R.
III. S YNCHRONIZATION P HENOMENAFor the computer simulations, we calculates Eqs.(2)-(5) byusing the Runge-Kutta method with the step size h = 0.05.
We show the simulation results when the parameters of thecircuit model are fixed as α = 0.01, β = 0.01 and γ = 0.007.
Figure 3 shows the attractor of each oscillator with thehorizontal axis is the voltage and the vertical axis is the electric
current of each oscillator.
First, we investigate the effect of different the initial condition. As simulation results are shown in Fig. 4, we obtainthat, by changing initial values, 2 oscillators of VDP-A canbecome in-phase.

Fig. 5. Phase difference (α = 0.01, β = 0.01, γ = 0.005).

Figure 6 shows the computer simulation results of thephase difference when parameter γ is changed from 0.001
to 0.009 with step size 0.001. In this case, we can obtainthat, when value of parameter γ is smaller than 0.005, second,
third, fourth oscillator of VDP-A are in-phase and only first
oscillator of VDP-B is anti-phase.

Fig. 6. Phase difference (α = 0.01, β = 0.01).

Fig. 3. Attractor between adjacent oscillators (α = 0.01, β = 0.01, γ =0.007).

Therefore, we can control synchronization phenomena bychanging initial values or the coupling strengths.

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IV. C ONCLUSIONSIn this study, we have investigated the synchronization
phenomena in coupled oscillators containing star structure
connected to another oscillator. By computer simulation, we
have observed the synchronization phenomena by changing
initial values and coupling strength. In the case of chaging
initial values, in three oscillators of VDP-A, two oscillators
can become in-phase. By decreasing coupling strength, allfour oscillators can become synchronization with oscillators
of VDP-A are in-phase and only oscillator of VDP-B is antiphase.
In the future, we experiment the proposed circuit model.
R EFERENCES[1] K. Ueta, Y. Uwate and Y. Nishio, “Synchronization Phenomena in
Complex Networks of van der Pol Oscillators,” Proceedings of RISPInternational Workshop on Nonlinear Circuits, Communications andSignal Processing (NCSP’17), pp. 285-288, Mar. 2017.
[2] Y. Uwate, Y. Nishio and R. Stoop, “Synchronization in Three Coupled
van der Pol Oscillators with Different Coupling Strength,” Proceedingsof RISP Inernational Workshop on Nonlinear Circuits, Communications
and Signal Processing (NCSP’10), pp. 109-112, Mar. 2010.
[3] V. Thuan, Y. Uwate and Y. Nishio , “Synchronization Phenomena invan der Pol Oscillators with Different Amplitudees,” Proceeding ofInternational Symposium on Nonlinear Theory and its Applications
NOLTA’14, Luzern, Switzerland, Sep. 2014.
[4] K. Matsumura, T. Nagai and Y. Nishio, “Analysis of Synchronization
Phenomena in Coupled Oscillator Chains” Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS’12), pp. 620-623,May. 2012.

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