平成 29 年度電気関係学会四国支部連合大会 講演論文集 (2017 愛媛大学)
2017 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (EHIME)

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Synchronization of Coupled Six van der Pol Oscillators Containing Two Rings
Daiki NARIAI

Minh Hai TRAN Yoko UWATE(Tokushima University)

Yoshifumi NISHIO

1. Introduction

3. Results

In this study, we propose a novel coupled oscillatory system. By computer simulations and circuit experiments,
we investigate synchronization phenomena observed in theproposed circuit system by changing one of the couplingstrength.

The simulation results of the system model are shown inFig. 2 and 3. The parameters are set to ε = 0.05, α = 0.05
and β = 0.05. Figure 2 show the graph of the relationshipbetween coupling strength and phase diﬀerence. We change
the value of γ2 from 0 to 0.03 at intervals of 0.001. We
fix the value of other coupling strengths and initial value.
When γ2 is even smaller than 0.020, circuit 4 - circuit 6
exhibit in-phase synchronization.

2. System model
Figure 1 shows two circuits and a model of the system.
The circuit of a van der Pol oscillator is called circuit VDP.
The circuit with a slightly modified VDP structure is called
NC. We use six van der Pol oscillators in this study. We use
two ring circuits of van der Pol oscillators. The three VDPof the first ring are connected by resistors. The three NCof the second ring are connected by inductors and resistors.
When the two rings are not connected, the oscillators in
the first ring exhibit in-phase synchronization and 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 of adjacent oscillators. We investigate how the synchronization
phenomena change upon changing the value of R2 .

Figure 2: The graph of phase diﬀerence.

circuit 2i2bR
i1acircuit 1

VDP

i2a
R

i3bir2

circuit 3ign

VDPVDP
i1b

R

In the case γ1 = γ3 = 0.02 and γ2 = 0.03, Fig. 3
show the time waveform of the first ring and the secondring. Oscillators of first ring exhibit in-phase synchronization and oscillators of second ring exhibit synchronizationthat three-phase synchronization gravitate toward in-phasesynchronization. The time waveform of circuit experiments
is shown in Fig. 4. The element value are set to L = 10mH,
C = 22nF, R′ = 1.2kΩ, R1 = R3 = 1.56kΩ and R2 = 560Ω.
We obtained results of circuit experiments similar to simulation results.

iLn
vCn

i3aCLVDPCNC

ir3

ir1
R2

R3

R1
ir5igncircuit 52L

circuit 4

i4a2L

NC

2L

ir4

2L

i5bi5a2L

2L

R'R'

vCn

ir6i6bNC

circuit 6NC

i4bi6aR'

Figure 1: Circuit model.
The normalized equations of the first ring are given as:ẋn
=ε(xn − x3n ) − yn − γn (xn − xn+3 )+α(−xn + xi + xj )ẏ
 n=xnThe first ring(1)(n = 1, 2, 3)

and the normalized equations of the second ring are givenas:ẋn= ε(xn − x3n ) − yan−ybn + γn−3 (xn − xn−3 )



ẏan



 ẏbnThe second ring

Figure 3: The time waveform (computer simulations).

=xn − β(yan + yb(i) )=xn − β(ybn + ya(j) )The first ring

The second ring

Figure 4: The time waveform (circuit experiments).
4. Conclusions(2)

We have proposed a system model using two rings ofthree van der Pol oscillators coupled by resistors or inductors. We investigated synchronization how the synchronization phenomena change upon changing the value of γ2 . In
addition, we obtained results of circuit experiment similar
to simulation results. In the future, we will investigate synchronization phenomena using other parameters and analyze the proposed circuit model.

(n = 4, 5, 6)

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

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