2017 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing (NCSP'17)
Guam, USA, February 28th to March 3rd, 2017

Synchronization in Two Ringsof Coupled Three van der Pol Oscillators
Daiki Nariai, Tran Minh Hai, Yoko Uwate and Yoshifumi Nishio†Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {nariai, minhhai, uwate, nishio}@ee.tokushima-u.ac.jp

2. System model

Abstract
In this study, we investigate synchronization phenomena intwo rings of van der Pol oscillators coupled by resistors. We
propose novel coupled oscillatory system such as two ringsof van der Pol oscillators coupled by resistors. We focus
on the coupling strength of coupled van der Pol oscillators.
From computer simulations, we investigate how synchronization phenomena change by changing the coupling strength. In
this results, we observe various synchronization phenomena.

iCn

ign

iLn

g

C

ign

VDP1

L

iCn
C

g

VDP2

1. Introduction
Synchronization phenomena of coupled oscillators are themost familiar phenomena. Synchronization phenomena havebeen studied in various ﬁelds since a long time ago, such as
in electrical systems, in mechanical systems, in biologicalsystems and basically everywhere. Among them, synchronization phenomena of van der Pol oscillator are similar tonatural phenomena by changing frequency. The coupled system of van der Pol oscillators is simple and easy to handle.
Many researchers have proposed various coupled oscillatory
networks of van der Pol oscillators [1] - [3]. We focus on thecoupling strength of coupled oscillatory networks consistedof two kinds of van der Pol oscillators.
The van der Pol oscillator is a simple circuit. It is consistedof resistor, inductor, capacitor and nonlinear resistor. It wasinvented by electrical engineer Balthasar van der Pol. Equation of van der Pol is second-order differential equation.
In this study, we propose a novel coupled oscillatory system such as two rings of van der Pol oscillators coupled by resistors. First ring is consisted of three van der Pol oscillatorsconnected by resistors. Second ring is consisted of three vander Pol oscillators connected by inductors and resistors. By
computer simulations, we investigate synchronization phenomena observed in the proposed circuit system by changing
the coupling strength.

Figure 1: Circuit of van der Pol oscillators.

i1acircuit 1circuit 2i2bR

VDP1i2aRiR2VDP1

i1biR1

i3b

VDP1

i3a

R

iR3

R2

R3

R1
circuit 52L

iR4circuit 4circuit 32L

i4a

VDP2
i4b

VDP2
i5bR

iR52L2L

i5a2L

R`

R`

2L

i6R
i6bcircuit 6VDP2

i6aFigure 2: Circuit model.
Figure 1 shows two circuits which were used in my research. We use six van der Pol oscillators (three VDP1 andthree VDP2). Figure 2 shows a system model with van der
Pol oscillators (VDP1 and VDP2). We use two ring circuits ofvan der Pol oscillators, three VDP1 of ﬁrst ring are connectedby resistors, three VDP2 of second ring are connected by inductors and resistors. First and second ring are connected by- 289 -resistors (R1 , R2 , R3 ). We observe synchronization phenomena of adjacent oscillators. We investigate synchronization
where n denotes the number of VDP1 and VDP2, n =phenomena how to change by changing the value of resistors. 1, 2, 3, 4, 5, 6. The parameters ε, α, β, and γ denote the couThe circuit equations of ﬁrst ring are given as follows:
pling strengh of the inductor, resistor R, resistor R and resistor Rn .
⎧−ign − icn − iLn = ian + ibn + irn⎪⎪⎪⎪⎪⎨ L diLn= vn
dt(1) 3. Simulation Resultsv−v= (ian − ib(i) )Rni⎪⎪⎪
v n − vj= (ibn − ia(j) )R⎪⎪First, we deﬁne the synchronization condition by the fol⎩ v −v
= (irn − ir(n+3) )R.
nn+3lowing equation:
The circuit equations of second ring are given as follows:⎧−ign − icn⎪⎪⎪⎪⎪⎪⎨ vn − 2L didtan=ian + ibn + irn=−(ian + ib(i) )R

|xa − xb | < 0.05.

(7)

The simulation results of the system model are shown from
Fig. 3 to Fig. 6. The value of the parameters are set to ε =(2) 0.05, α = 0.05, β = 0.05.
⎪dibn
⎪
v−2L
=−(i+i)R⎪nbn
In case of γ1 = γ2 = γ3 = 0.02, we conduct simulation,a(j)dt⎪⎪⎪⎩each one at a different initial value. In Fig. 3, synchronizationvn − vn−3= (irn − ir(n−3) )Rphenomena are observed in circuit 1 - circuit 4. However, in
where n denotes the number of circuit and n = Fig. 4, synchronization phenomena are observed in circuit 2 1, 2, 3, 4, 5, 6. i denotes the number of circuit and i = circuit 5.
2, 3, 1, 5, 6, 4. j denotes the number of circuit and j =x1
x3
x2
3, 1, 2, 6, 4, 5.
Nonlinear resistor deﬁned as follows:ign=−g1 vn + g3 vn3(3)x4

By changing the variables and parameters.
√g1
g1 Cx,
i=t = LGτ, vn = 3gn n
3g3 L yn ,31L
ε = g1 C
L,α = RC,
1L
β = R C,γ=nL
RnC.

x5

⎪⎩

ẏn

==ε(xn − x3n ) − yn − γ(xn − xn+3 )+α(−xn + xi + xj )x5x2x2x1x6x4(4)x6x4x3x3x6x5

Figure 3: Phase difference ( γ1 = γ2 = γ3 = 0.02 )The normalized equations of ﬁrst ring are given as follows:⎧ẋ⎪⎨ nx1x2(5)x4xn .

The normalized equations of second ring are given as follows:⎧ẋn = ε(xn − x3n ) − yan⎪⎪⎪⎪−ybn + γ(xn − xn−3 )⎪⎨(6)ẏan = xn − β(yan + yb(i) )
⎪⎪
⎪
⎪
⎪
⎩
ẏbn = xn − β(ybn + ya(j) )- 290 -x1x3x1x5x1x2x2x6x5x4x6x3x3x4x5x6

Figure 4: Phase difference ( γ1 = γ2 = γ3 = 0.02 )By chaging γ1 , γ2 and γ3 , we can control synchronization
phenomena regardless of initial value.
In case of γ1 = 0.001, γ2 = 0.0001, γ3 = 0.02, in-phasesynchronization phenomena are observed in oscillators ofﬁrst ring, 3-phase synchronization phenomena are observedin oscillators of second ring. In this result, when we increase
γ3 , three oscillators of ﬁrst ring become in phase synchronization phenomena, oscillators of second ring become 3-phasesynchronization phenomena.

x3x2x4x1x5x2x4x2x6x3x3x2x4x6x5x6

Figure 7: Phase difference ( γ1 = 0.02, γ2 = 0.005, γ3 = 0.02 )x3

and initial value. Figure 8 shows synchronization rate of ﬁrst
ring. Figure 9 shows synchronization rate of oscillators between oscillators of ﬁrst ring and oscillators of second ring.
Figure 10 shows synchronization rate of second ring.

x4x5x5x1x4x3x6x6x5x4x5x1x3x1x1x2x1x2x6

Figure 5: Phase difference ( γ1 = 0.001, γ2 = 0.001, γ3 = 0.02 )The time waveform of the voltage of each VDP2 after sufﬁcient time has elapsed are shown in Fig. 6. This result show
three oscillators of second ring become 3-phase synchronization phenomena.

xntimeFigure 6: Time waveform of the voltage.
Figure 8: Synchronization rate.
In case of γ1 = 0.02, γ2 = 0.005, γ3 = 0.02, synchronization phenomena are observed in circuit 4 - circuit 6 without
reference to initial value. In this result, when we increase twoof γ1 and γ3 , three oscillators of ﬁrst ring and two of three oscillators of second ring become synchronization phenomena.
We investigate various synchronization phenomena bychanging the coupling strengths. In this result, when wedemand two oscillators of second ring to synchronize, westrengthen up two of γ1 , γ2 and γ3 . When we demand threeoscillators of second ring to become 3-phase synchronization
phenomena, we strengthen up two of γ1 , γ2 and γ3 . Therefore, we can control synchronization phenomena by couplingstrengths.
Next, we investigate the synchronization rate this circuit
model. We change the value of γ2 from 0 to 0.03 at intervals of 0.001. We ﬁx the value of other coupling strengths

In Fig. 8, when γ2 is from 0 to 0.007, circuit 1 - circuit 2,circuit 2 - circuit 3 and circuit 3 - circuit 1 become in-phasesynchronization phenomena. When γ2 is 0.008, synchronization rate of circuit 1 - circuit 2 and circuit 2 - circuit 3 begin todecrease. When γ2 is 0.02, synchronization rate of synchronization rate of synchronization rate of circuit 1 - circuit 2 isdecrease sharply. When γ2 is from 0.021 to 0.03, synchronization rate of circuit 1 - circuit 2, circuit 2 - circuit 3 andcircuit 3 - circuit 1 are steady - state value.
In Fig. 9, when γ2 is from 0.012, synchronization rate ofcircuit 1 - circuit 4 and circuit 3 - circuit 6 begin to decrease.
When γ2 is from 0.015, synchronization rate of circuit 2 circuit 5 begin to increase. When γ2 is from 0.021 to 0.03,synchronization rate of circuit 1 - circuit 4 and circuit 3 -- 291 -we strengthened up one of γ1 , γ2 and γ3 , oscillators of ﬁrst
ring become in-phase synchronization phenomena, oscillators of second ring become 3-phase synchronization phenomena. when we increase two of γ1 , γ2 and γ3 , three oscillatorsof ﬁrst ring and two oscillators of second ring become synchronization phenomena. In the future, we investigate synchronization phenomena using other parameters and analyze
the proposed circuit model.
Acknowledgment
This work was partly supported by JSPS Grant-in-Aid for
Challenging Exploratory Research 26540127.
References

Figure 9: Synchronization rate.

[1] Y. Uwate, Y. Nishio, R. Stoop, “Synchronization in
Three Coupled van der Pol Oscillators with Different
Coupling Strength”, Proceedings of RISP International
Workshop on Nonlinear Circuits, Communications andSignal Processing (NCSP’10), pp. 109-112, Mar. 2010.

circuit 6 are steady - state value.

[2] K. Matsumura, T. Nagai, Y. Uwate, Y. Nishio, “Analysisof Synchronization Phenomenon in Coupled OscillatorChains”, Proceedings of IEEE International Symposium
on Circuits and Systems (ISCAS’12), pp. 620-623, May
2012.
[3] H. Kume, Y. Uwate, Y. Nishio, “Synchronization Phenomena in Different Sizes of Rings of Coupled Oscillators”, Proceedings of International Symposium onNonlinear Theory and its Applications (NOLTA’14), pp.
369-372, Sep. 2014.
Figure 10: Synchronization rate.
In Fig. 10, when γ2 is even slightly smaller than γ1 andγ3 , circuit 4 - circuit 5 become in-phase synchronization phenomena. However when γ1 , γ2 , γ3 equal, circuit 4 - circuit 5
do not become in-phase synchronization. When γ2 is evenbigger than γ1 and γ3 , any oscillators do not become in-phasesynchronization phenomena.

4. Conclusion
We have proposed a system model using two rings ofcoupled three van der Pol oscillators coupled by resistors
or inductors. We can control the synchronization phenomena by changing the coupling strengths. When three coupling strengths (γ1 , γ2 , γ3 ) equal, synchronization phenomena are observed by changing initial value. However when- 292 -