IEEE Workshop on Nonlinear Circuit Networks
December 9-10, 2016

Synchronization Phenomena in Two Ringsof Coupled Three van der Pol Oscillators
Daiki Nariai, Tran Minh Hai, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Eng.,
Tokushima University,
2-1 Minami-Josanjima,
E-mail: {nariai, minhhai, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we investigate synchronization phenomena in two rings of van der Pol oscillators coupled byresistors. We propose novel coupled oscillatory system such as
two rings of van der Pol oscillators coupled by resistors. We
focus on coupling strength of coupled van der Pol oscillators.
From computer simulations, we investigate how synchronization
phenomena changes by changing the coupling strength. We
observe various synchronization phenomena by changing thecoupling strengths. We demand to control the synchronization
phenomena by changing the coupling strength.

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 byinductors. First and second ring are connected by resistor (R1 ,R2 , R3 ). We observe synchronization phenomena of adjacentoscillators. We investigate synchronization phenomena howchanging by changing the value of resistors.

I. I NTRODUCTION
Synchronization phenomena of coupled oscillators is themost familiar phenomena. Among them, synchronization phenomena of van der Pol oscillators is similar to natural phenomena. Because it is easy to observe the synchronization
phenomena of van der Pol oscillators by changing frequency.
Synchronization phenomena of van der Pol oscillator has beenstudied in various fields since ancient times, such as in electrical systems, in mechanical systems, in biological systems
and basically everywhere. For example, firefly luminescence,heartbeat, and so on. We explain synchronization phenomena
with concrete descriptions in heatbeat. Heat pulsative in anormal way because all cardiac muscle cell synchronize.
Many researchers have proposed various coupled oscillatory
networks of van der Pol oscillators. We focus on couplingstrength of coupled oscillatory networks consisted of two kindsof van der Pol oscillatiors.
The van der Pol oscillator is simple ciruit. It is consistedof resistor, inductor, capacitor and nonlinear resistor. It wasinvented by electrical engineer Balthasar van der Pol. Equationof van der Pol is second-order differential equation.
In this study, we propose novel coupled oscillatory systemsuch as two rings of van der Pol oscillators coupled byresistors. First ring is consisted of three van der Pol oscillatorsconnected by resistors. Second ring is consisted of three vander Pol oscillators connected by inductors. By computer simulations, we investigate synchronization phenomena observed in
the proposed circuit system by changing the coupling strength.

II. S YSTEM MODEL
Figure 1 shows two circuits which were used in my research.
We use six van der Pol oscillators (three VDP1 and three- 15 -iCn

ign
-g

iLn
Cign

iCn
C-g

Fig. 1.

VDP1

LVDP2

Circuit of van der Pol oscillators.

The circuit equation of first ring are given as follows:−ign − icn − iLn = ian + ibn + irn L diLn= vdtvn − vi
vn − vj v −v
nn+3n= (ian − ib(i) )R= (ibn − ia(j) )R= (irn − ir(n+3) )R(1)The circuit equation of second ring are given as follows:−ign − icn= ian + ibn + irn





 vn − 2L didtan = −(ian + ib(i) )R′(2)di′
 vn − 2L dtbn = −(ibn + ia(j) )Rvn − vn−3= (irn − ir(n−3) )Rcircuit 2i2b
VDP1

i1a

circuit 1

coupling strengh of the inductor, resistor R, resistor R′ andresistor Rn .

i2aRRi3biR2

circuit 3
VDP1

VDP1
i1bi3aRIII. S IMULATION RESULTS
iR3iR1
R2R3R1iR5circuit 52L

iR42LVDP22L2L

i5bi5a2L2L

R`

circuit 4R`

i4ai6bcircuit 6
VDP2

VDP2
i4bThe simulation result of the system model are shown fromFig. 3 to Fig. 6. The value of the parameters are set to ε =0.05, α = 0.05, β = 0.05.
In case of γ1 = γ2 = γ3 = 0.02, we conduct simulation,
each one at a different initial value. In Fig. 3, synchronization
phenomena is observed in circuit 1 and circuit 4. However,in Fig. 4, synchronization phenomena is observed in circuit 2
and circuit 5.

i6a

R`

Circuit model.

x1
where n denotes the number of circuit and n =1, 2, 3, 4, 5, 6. i denotes the number of circuit and i =2, 3, 1, 5, 6, 4. j denotes the number of circuit and j =3, 1, 2, 6, 4, 5.

= −g1 vn + g3 vn3

By changing the variables and parameters.
√√√
g1
g1 Ct = LGτ, vn = 3gx,i=n n3g3 L yn ,3√√1Lε = g1 CL,α = RC,√√1L
β = R′ C,γ=nLRnCx5(3)

(4)



ẏn

==ε(xn − x3n ) − yn − γ(xn − xn+3 )+α(−xn + xi + xj )x2

x1

where n denotes the number of VDP1 and VDP2, n =1, 2, 3, 4, 5, 6. The parameters ε, α, β, and γ denote thex2

x6x4x3x6x5x4

Fig. 4.

x6x1x3x5

The normalized equations of second ring are given asfollows:ẋn = ε(xn − x3n ) − yan−ybn + γ(xn − xn−3 )(6)ẏan = xn − β(yan + yb(i) )




ẏbn = xn − β(ybn + ya(j) )x4x5x1xnx3

Phase difference ( γ1 = γ2 = γ3 = 0.02 )

x2

(5)

x6x6x4

Fig. 3.

x3x2x1

The normalized equations of first ring are given as follows:ẋ nx2x5x4

Nonlinear resistor defined as follows:
ignx1x3x2
Fig. 2.

x3x4x5x6

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 = 0.02, γ2 = 0.005, γ3 = 0.02, synchronization phenomena is observed in circuit 4 and 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 threeoscillators of second ring are synchronized.

- 16 -

x2x1x2x5x4x5x6x3x4x6

R EFERENCESx6x5x4Fig. 5.

x3x2x1inductors. We can control the synchronization phenomenon byvarying the coupling strengths. When three coupling strengths(γ1 , γ2 , γ3 ) equal, synchronization phenomena are observedby changing initial value. However when we increase two of
γ1 , γ2 and γ3 , three oscillators of first ring and two oscillatorsof second ring are synchronized. When we strengthened up
one of γ1 , γ2 and γ3 , oscillators of first ring become inphase synchronization phenomena, oscillators of second ring
become 3-phase synchronization phenomena. In the future, weinvestigate synchronization phenomena using other parameters
and analyze my circuit model.

x1x3

Phase difference ( γ1 = 0.02, γ2 = 0.005, γ3 = 0.02 )In case of γ1 = 0.001, γ2 = 0.0001, γ3 = 0.02, inphase synchronization phenomena is observed in oscillatorsof first ring, 3-phase synchronization phenomena is observedin oscillators of second ring. In this result, when we increase
one of γ3 , three oscillators of first ring become in phasesynchronization phenomena, oscillators of second ring become
3-phase synchronization phenomena.

x3x2x1x4x2x5x6x4x3x4x6x5x3x2x1

Fig. 6.

x1[1] Y. Uwate, Y. Nishio, ”Two van der Pol Oscillators Coupled by Chaotically
Varying Resistor” Proceedings of International Workshop on Nonlinear
Dynamics of Electronic Systems (NDES’06), pp. 189-192, Jun. 2006.
[2] K. Marsumura, T. Nagai, Y. Uwate, Y. Nishio, ”Analysis of Synchronization Phenomenon in Coupled Oscillator Chains” Proceedings of IEEE
International Symposium on Circuits and Systems (ISCAS’12), pp. 620623, May 2012.
[3] K. Oi, Y. Uwate, Y. Nishio, ”Synchronization Phenomena in Coupledvan der Pol Oscillators by Adding Oscillators with Different Frequencies” Proceedings of IEEE Workshop on Nonlinear Circuit Networks(NCN’15), pp. 1-3, Dec. 2015.
[4] K. Oi, Y. Uwate, Y. Nishio,”Synchronization Phenomena of Parametrically Excited Oscillators with Small Mismatch in Random Network” Proceedings of IEEE Workshop on Nonlinear Circuit Networks (NCN’14),
pp. 49-51, Dec. 2014.
[5] K. Oi, K. Ago, Y. Uwate, Y. Nishio,”Effect of the Hub in Complex
Networks of Coupled Parametrically Excited Oscillators with Dispersion” Proceedings of IEEE Workshop on Nonlinear Circuit Networks(NCN’15), pp. 11-14, Dec. 2015.

x5x6

Phase difference ( γ1 = 0.001, γ2 = 0.001, γ3 = 0.02 )We observe various synchronization phenomena by changing the coupling strengths. In this results, when we demand
two oscillators of second ring to synchronize, we strengthened
up two of γ1 , γ2 and γ3 . When we demand three oscillatorsof second ring to become 3-phase synchronization phenomena,
we strengthened up two of γ1 , γ2 and γ3 . Therefore we cancontrol synchronization phenomena by coupling strengths.
IV. C ONCLUSIONS
We have proposed a system model using two rings ofcoupled three van der Pol oscillators coupled by resistors or- 17 -