Journal of Signal Processing, Vol.21, No.4, pp.121-124, July 2017

SELECTED PAPER AT NCSP'17

Synchronization in Two Rings of Three Coupled van der Pol Oscillators
Daiki Nariai,Minh Hai Tran, Yoko Uwate and Yoshifumi Nishio
Department. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
E-mail: {nariai, minhhai, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract

2. System ModelIn this study, we investigate the synchronization phenomenain two rings of van der Pol oscillators coupled by resistors.
We propose a novel coupled oscillatory system comprising
two rings of van der Pol oscillators with different coupling
schema. We focus on the coupling strengths of the coupled
van der Pol oscillators. By computer simulation, we investigate how the synchronization phenomena change by changing the coupling strengths. In these results, we observe various synchronization phenomena.

1. Introduction

ign

iLn
vCn

Figure 1: Circuit of van der Pol oscillator

circuit 2i2bR
circuit 1

i1a

VDP

i2a
R

i3b

ir2

circuit 3
VDP

VDPThe synchronization phenomena of coupled oscillators arefamiliar. Synchronization phenomena have been studied invarious fields for many years, such as in electrical systems,
mechanical systems and biological systems. Among them,the synchronization phenomena of van der Pol oscillators are
similar to natural phenomena when the frequency is changed.
A coupled system of van der Pol oscillators is simple and easy
to handle. Many researchers have proposed various coupledoscillatory networks of van der Pol oscillators [1]- [3]. We focus on the coupling strengths of coupled oscillatory networks
consisting of two van der Pol oscillators.
The van der Pol oscillator is a simple circuit. It consistsof a resistor, inductor, capacitor and nonlinear resistor. It wasinvented by the electrical engineer Balthasar van der Pol. The
equation of a van der Pol oscillator is a second-order differential equation.
In this study, we propose a novel coupled oscillatory system comprising two rings of van der Pol oscillators coupled
by resistors. The first ring consists of three van der Pol oscillators connected by resistors. The second ring consists ofthree van der Pol oscillators connected by inductors and resistors. By computer simulation, we investigate the synchronization phenomena observed in the proposed circuit system
by changing the coupling strengths.

VDPLC

i1bi3aR

ir3

ir1
R2

R3

R1
ir5

ign

circuit 52L

ir4
circuit 42Li4a2LNC

2L

i5bi5a2L

2L

R'R'

ir6
i6bNC

vCnCNC

circuit 6NC

i4b

R'

i6aFigure 2: Circuit model

Figure 1 shows the circuit of a van der Pol oscillator. We
call this circuit VDP. We use six van der Pol oscillators in thisstudy. Figure 2 shows a model of the system with six van der
Pol oscillators. We use two ring circuits of van der Pol oscillators. The three VDP of the first ring are connected by resistors. The three NC of the second ring are connected by inductors and resistors. When the two rings are not connected, theoscillators in the first ring exhibit in-phase synchronizationand oscillators of the second ring exhibit three-phase synchronization. The first and second rings are connected by resistors(R1 , R2 , R3 ). We observe the synchronization phenomena ofadjacent oscillators. We investigate how the synchronizationphenomena change upon changing the values of R1 , R2 andR3 .
The circuit equations of the first ring are given as follows:

Journal of Signal Processing, Vol. 21, No. 4, July 2017

121

−ign − icn − iLn





diLn

 L dvdt
C dtCnv n − vivn − vjvn − vn+3=

ian + ibn + irn

===
=
=vn
ICn
(ian − ib(i) )R(ibn − ia(j) )R(irn − ir(n+3) )R3

where n denotes the number of the VDP and NC (n =1, 2, 3, 4, 5, 6). The parameters ε is non-linear strength. Theparameters α, β and γ denote the coupling strengths of the(1) resistor R, resistor R′ and resistor Rn , respectively.
3. Simulation Results

The circuit equations of the second ring are given as follows:−ign − C dvdtCn






 vn − 2L didtanvn − 2L didtbnvn − vn−3=

ian + ibn + irn=−(ian + ib(i) )R′=−(ibn + ia(j) )R′=(irn − ir(n−3) )R3The simulation results of the system model are shown from
Figs.3-6. The parameters are set to ε = 0.05, α = 0.05 andβ = 0.05.
In the case of γ1 = γ2 = γ3 = 0.02, we conducted thesimulation using different initial values. In Fig.3, synchronization phenomenon is observed in circuits 1-4. However, in(2)Fig.4, synchronization phenomenon is observed in circuit 25.

where n, i and j is denotes the number of the circuit (n =1, 2, 3, 4, 5, 6, i i = 2, 3, 1, 5, 6, 4 and j j = 3, 1, 2, 6, 4, 5.)
A nonlinear resistor defined as follows:ign=−g1 vn + g3 vn3x5


ẏ
 n

=

ε(xn − x3n ) − yn − γn (xn − xn+3 )+α(−xn + xi + xj )

=

xn .

122

=

xn − β(yan + yb(i) )=xn − β(ybn + ya(j) )(n = 4, 5, 6)x2x3x6x5(5)x1x2x6x5ε(xn − x3n ) − yan−ybn + γn−3 (xn − xn−3 )x6

x1x3x4(n = 1, 2, 3)=x4x5x1

and the normalized equations of the second ring are given
as:
ẋn



ẏan
 ẏbn

x3Figure 3: Phase difference (γ1 = γ2 = γ3 = 0.02)

x2

the normalized equations of the first ring are given as:x6x6x4(4)x3x2x1

t=

ẋn





x2x5x4

By changing the variables and parameters as follows.
√√
√
g1
g1 C
LCτ, vn = 3gx,
i=nn
3g3 L yn ,3√√1L
ε = g1 C,
α=LRC,√√
1L
β = R′ C,γ=nL
RnCx1(3)x1x3x2x4x3x4x5x6

Figure 4: Phase difference (γ1 = γ2 = γ3 = 0.02)(6)

By changing γ1 , γ2 and γ3 , we can control the synchronization phenomena regardless of initial value.
In the case of γ1 = 0.001, γ2 = 0.001 and γ3 = 0.02,
in-phase synchronization phenomenon is observed in the os-

Journal of Signal Processing, Vol. 21, No. 4, July 2017

cillators of the first ring and three-phase synchronization phenomenon is observed in the oscillators of the second ring.
Hear, when we increase γ3 , the three oscillators of the firstring exhibit in-phase synchronization and the oscillators ofthe second ring exhibit three-phase synchronization.

x3x2x1x4x5x4x4x5x3x6x3x2x4x6x3x5x6

Figure 7: Phase difference (γ1 = 0.02, γ2 = 0.005, γ3 =0.02)

x4x6x5x2x5x5x6x2x1x1x1x3x1x3x4x1x2x2x6

P=N
Number of trials × 100[%](8)

Figure 5: Phase difference (γ1 = 0.001, γ2 = 0.001, γ3 =0.02)

where the number of trials is 1,000,000. N denotes the number of oscillator being synchronized.
We change the value of γ2 from 0 to 0.03 at intervals ofThe time waveform of the voltage of each NC after suf- 0.001. We fix the other coupling strengths and the initial
ficient time has elapsed is shown in Fig.6. This result shows value. Figure 8 shows the synchronization rate of the firstthat the three oscillators of the second ring exhibit three-phase ring. Figure 9 shows the synchronization rate of the oscillasynchronization.
tors of the first ring with these of the second ring. Figure 10
shows the synchronization rate of the second ring.

xntimeFigure 6: Time waveform of the voltage of each NC
In the case of γ1 = 0.02, γ2 = 0.005 and γ3 = 0.02, synchronization phenomenon is observed in circuit 4-6 regardless of the initial value. Here, when we increase two of γ1 ,γ2 and γ3 , the three oscillators of the first ring and two of thethree oscillators of the second ring exhibit synchronization.
We investigate various synchronization phenomena bychanging the coupling strengths. Here, when we require twooscillators of the second ring to synchronize, we increase twoof γ1 , γ2 and γ3 . When we require three oscillators of the secFigure 8: Synchronization rate (first ring)
ond ring to three-phase synchronization, we increase two of
γ1 , γ2 and γ3 . Therefore, we can control the synchronization
phenomena via the coupling strengths.
In Fig.8, when γ2 is from 0 to 0.007, circuit 1 - circuit 2,
Next, we define the synchronization condition by the fol- circuit 2 - circuit 3 and circuit 3 - circuit 1 exhibit in-phaselowing equation:synchronization. When γ2 is 0.008, the synchronization ratesof circuit 1 - circuit 2 and circuit 2 - circuit 3 begin to decrease. When γ2 is 0.02, the synchronization rate of circuit 1|xa − xb | < 0.05(7) - circuit 2 decreases sharply. When γ is from 0.021 to 0.03,
2
we denote the synchronization rate. We define the synchro- the synchronization rates of circuit 1 - circuit 2, circuit 2 circuit 3 and circuit 3 - circuit 1 reach steady-state value.
nization rate (P ) by the following equation:

Journal of Signal Processing, Vol. 21, No. 4, July 2017

123

coupling strengths. When the three coupling strengths (γ1 ,γ2 , γ3 ) are equal, synchronization phenomenon was observed
after changing the initial value. However, when we increased
one of γ1 , γ2 and γ3 , the oscillators of the first ring exhibit inphase synchronization and the oscillators of the second ringexhibited three-phase synchronization. When we increased
two of γ1 , γ2 and γ3 , the three oscillators of the first ring andtwo oscillators of the second ring exhibited synchronization
phenomena. In the future, we will investigate synchronization
phenomena using other parameters and analyze the proposed
circuit model.
Acknowledgment
Figure 9: Synchronization rate (second ring)
In Fig.9, when γ2 is from 0.012, the synchronization ratesof circuit 1 - circuit 4 and circuit 3 - circuit 6 begin to decrease. When γ2 is from 0.015, synchronization rate of circuit 2 - circuit 5 begins to increase. When γ2 is from 0.021 to0.03, synchronization rates of circuit 1 - circuit 4 and circuit 3- circuit 6 reach steady-state values.

This work was partly supported by JSPS Grant-in-Aid for
Challenging Exploratory Research No.26540127.
References[1] Y. Uwate, Y. Nishio and R. Stoop: Synchronization inthree coupled van der Pol oscillators with different coupling strength, Proceedings of RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP’10), pp. 109-112, Mar. 2010.
[2] K. Matsumura, T. Nagai, Y. Uwate and Y. Nishio: Analysis of synchronization phenomenon in coupled oscillator chains, Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS’12), pp. 620-623,May 2012.
[3] H. Kume, Y. Uwate and Y. Nishio: Synchronization
phenomena in different sizes of rings of coupled oscillators, Proceedings of International Symposium on
Nonlinear Theory and its Applications (NOLTA’14), pp.
369-372, Sep. 2014.

Figure 10: Synchronization rate (between first and secondrings)
In Fig.10, when γ2 is even slightly smaller than γ1 and γ3 ,circuit 4 - circuit 5 exhibits in-phase synchronization. However, when γ1 , γ2 and γ3 are equal, circuit 4 - circuit 5 do notexhibit in-phase synchronization. When γ2 is larger than γ1and γ3 , none of the oscillators do exhibit in-phase synchronization.
4. Conclusion
We have proposed a system model using two rings of threevan der Pol oscillators coupled by resistors or inductors. We
can control the synchronization phenomena by changing the124

Journal of Signal Processing, Vol. 21, No. 4, July 2017