2017年 NOLTAソサイエティ大会
IEICE, NOLTA
愛知, 2017年6月10日

Synchronization Phenomena ofCoupled van der Pol Oscillators Containing Two Rings
Daiki Nariai

Tran Minh Hai

Yoko Uwate

Yoshifumi Nishio

Tokushima University1Introduction
In this study, we propose a novel coupled oscillatory system such as two rings of van der Pol oscillators
coupled by resistors. We investigate synchronization
phenomena observed in the proposed circuit system bychanging the coupling strength.
2System Modelcircuit 2
i2b

Rign

circuit 1iLnvCn

LCVDP

i1ai2aR

circuit 3

i3biR2VDPVDP

VDP

i1bi3a

R

iR3

iR1
R2R3

R1iR5
circuit 42L

ign

2L

iR4vCnCNC

circuit 4i4a2L

NC
i5b2L

i5aR`

i6R
R`

2L2Li6bNC

circuit 6NC

i4bi6aR`

Figure 1 Circuit model.
Figure 1 shows the circuit model which is used inour research. We use two ring circuits of six oscillators.
Three VDP of the first ring are connected by resistors
and three NC of the 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 phenomena of adjacentoscillators by changing the values of R1 , R2 and R3 .
Nonlinear resistor is defined as follows:ign=−g1 vn + g3 vn3 .

ẏn


3

Simulation Results
The simulation results of the system model are shownin Figs. 2-5. The values of the parameters are set toε = 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 as Figs. 2 and 3. By
chaging γ1 , γ2 and γ3 , we can control synchronization
phenomena regardless of initial value. In the case of
γ1 = 0.02, γ2 = 0.005 and γ3 = 0.02, the circuit 4 andthe circuit 6 become synchronized without reference toinitial 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-phase synchronization. We observe various
synchronization phenomena by changing the couplingstrengths.

x1x2x2x2x1x2x2

Figure 4(n = 1, 2, 3)

x1

The normalized circuit equations of the second ringare given as follows:ẋn = ε(xn − x3n ) − yan−ybn + γn−3 (xn − xn−3 )



(3)
ẏan = xn − β(yan + yb(i) )





ẏ
= xn − β(ybn + ya(j) ) bn(n = 4, 5, 6)

Figure 5
4x4x6x5x2x6x5x3x4x5x4x6x5x3x3x2x1x5x6x4x6x6x5x4

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

x3x2

x1x4(2)= xn .

x4x6

Phase diﬀerence (γ1 = γ2 = γ3 = 0.02)

x1x3x6x5x4x3

Figure 3x3x2x1x5

Phase diﬀerence (γ1 = γ2 = γ3 = 0.02)

x1

x3x6x5x4x3

Figure 2x1= ε(xn − x3n ) − yn − γn (xn − xn+3 )+α(−xn + xi + xj )x1x3x2(1)

The normalized circuit equations of the first ring are
given as follows:
ẋn


where n = 1, 2, 3, 4, 5, 6. The parameters ε, α, β, andγn denote the coupling strengh of the inductor, resistor
R, resistor R′ , and resistor Rn , respectively.

x4x1x2x3x5x1x2x3x4x6x5x6x4x5x6

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

Conclusions
We have proposed a system model using two rings ofthree van der Pol oscillators coupled by resistors or inductors. We can control the synchronization phenomena by varying the coupling strengths. In the future, wewill investigate synchronization phenomena using otherparameters and analyze the proposed circuit model.

- NLS-20 -