2017 IEEE Workshop on Nonlinear Signal Processing, Feb. 10, 2017

Synchronization in Two Ringsof Coupled van der Pol Oscillators
Daiki Nariai, Tran Minh Hai, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Eng.,
Tokushima University
Email: {nariai, minhhai, uwate, nishio}@ee.tokushima-u.ac.jp
I. I NTRODUCTION
In this study, we propose a novel coupled oscillatory systemsuch as two rings of van der Pol oscillators coupled byresistors. we investigate synchronization phenomena observedin the proposed circuit system by changing the couplingstrength.
II. S YSTEM MODEL
circuit 2i2bVDP

R

ign

circuit 1iLn
vCn

LCVDP

i1ai2aR

circuit 3

i3biR2

VDP

VDP
i1bi3a

R

iR3

iR1
R2

R3

R1iR5circuit 42L

ign

iR4vCnCNC

circuit 42L2L

NC
i5b2L

i5a

R`
i4ai6R
R`

2L

2L

i6b

NC

Fig. 1.

circuit 6
NC

i4b

R`

i6a

where n = 1, 2, 3, 4, 5, 6. The parameters ε, α, β, and γ denotethe coupling strengh of the inductor, resistor R, resistor R′ ,and resistor Rn , respectively.
III. S IMULATION RESULTS
The simulation results of the system model are shown
from Fig. 2 to Fig. 5. The value of the parameters are
set to ε = 0.05, α = 0.05, β = 0.05. In the case of
γ1 = γ2 = γ3 = 0.02, in the second ring, synchronization
phenomena change by initial value. By chaging γ1 , γ2 and γ3 ,we can control synchronization phenomena regardless of initialvalue. In the case of γ1 = 0.02, γ2 = 0.005 and γ3 = 0.02,
circuit 4 and circuit 6 become synchronized without reference
to initial value. In the case of γ1 = 0.001, γ2 = 0.0001
and γ3 = 0.02, the oscillators of the first ring become synchronized, the oscillators of the second ring become 3-phasesynchronization. we observe synchronization phenomena bychanging the coupling strengths.

Circuit model.

The normalized circuit equations of the first ring are given
as follows:ẋ= ε(xn − x3n ) − yn − γ(xn − xn+3 ) n+α(−xn + xi + xj )(2)ẏn = xn .
The normalized circuit equations of the second ring aregiven 
as follows:ẋn = ε(xn − x3n ) − yan−ybn + γ(xn − xn−3 )(3)ẏan = xn − β(yan + yb(i) )




ẏbn = xn − β(ybn + ya(j) )x1x3x2

Figure 1 shows the circuit model which is used in myresearch. we use two ring circuits of six oscillators. ThreeVDP of the first ring are connected by resistors, three NC ofthe second ring are connected by inductors and resistors. The
first and the second rings are connected by resistors (R1 , R2 ,R3 ). we investigate how to change synchronization phenomenaof adjacent oscillators by changing the value of R1 , R2 andR3 .
Nonlinear resistor is defined as follows:
ign = −g1 vn + g3 vn3 .
(1)

x1x2
Fig. 2.

x2x2x4x6x5x4x1x6x5x3

Fig. 3.

Phase difference (γ1 = γ2 = γ3 = 0.02)

x1x3x6x5x2x1x4x6

Phase difference (γ1 = γ2 = γ3 = 0.02)

x1x3x5x3x2x1x3x6x5x4x2x5x4x3x5x4x5x6x4x6x4x6x1x2

Fig. 4.

Phase difference (γ1 = 0.001, γ2 = 0.001, γ3 = 0.02)

x3x2x1Fig. 5.

x3x4x1x2x3x5x1x3x2x1x2x3x4x6x5x6x4x6x5x4x5x6

Phase difference (γ1 = 0.02, γ2 = 0.005, γ3 = 0.02)

IV. C ONCLUSIONS
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 varying thecoupling strengths.
R EFERENCES[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.

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