平成 28 年度電気関係学会四国支部連合大会 講演論文集 (2016 徳島大学)
2016 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (TOKUSHIMA)

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Synchronization Phenomena in Two Rings of van der Pol Oscillators
Daiki NARIAI1

Minh Hai TRAN1 Yoko UWATE1(1 Tokushima University)

1. Introduction

where n denotes the number of VDP1 and VDP2, n =1, 2, 3. The parameters ε, α, β, and γ denote the couplingstrengh of the inductor, resistor R, resistor R′ and resistor
Rn .

Synchronization phenomena is the most familiar phenomena that exist in nature and it has been studied in various
fields since ancient times, such as in electrical systems, inmechanical systems, in biological systems, basically every
where.
In this study, we propose novel coupled oscillatory systemsuch as two rings of van der Pol oscillators coupled by resistors. We investigate synchronization phenomena observedin the proposed circuit system by changing the couplingstrength.

3. Simulation results
The simulation result of the system model are shown fromFig 2 to Fig 5. 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, In
second ring, synchronization phenomena change by initialvalue. 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, circuit 4 and circuit 6 become synchronization phenomena without reference to initial value. In case of γ1 = 0.001, γ2 = 0.0001, γ3 = 0.02,
oscillators of first ring become synchronization phenomena,
oscillators of second ring become 3-phase synchronization
phenomena. We observe synchronization phenomena bychanging the coupling strengths.

2. System modelFigure 1 shows a system model constituted van der Poloscillators (VDP1 and VDP2). We use two rings of vander Pol oscillators, three VDP1 of first ring are connectedby resistor, three VDP2 of second ring are connected byinductor. First and second ring are connected by resistor(R1 , R2 , R3 ). We investigate synchronization phenomena
how changing by changing the value of resistors.

x13x12x11

circuit 2x11x12

VDP1circuit 1RRVDP1x12

VDP1-gLC2Lx12x11x13x12x112L

R`

R`

2L2LR`-gC x˙n= y˙ε(x1n − x31n ) − y1n+α(−x1n + x1n+1 + x1n−1 ) − γ(x1 n − x2 n)

=xn .

=

x23

x21

x13

x11x12x21x13x22x23x12x21x13x22x11x22x23x12x13x21x23x22x21x23x13x11x12x23x22x23
x13x21x21x22x23

4. Conclusions
We have proposed a system model using two rings ofcoupled three van der Pol oscillators coupled by resistors orinductors. We can control the synchronization phenomenon
by varying the coupling strengths. When three couplingstrengths (γ1 , γ2 , γ3 ) equal, synchronization phenomena
are observed by changing initial value. However when westrengthened up two of γ1 , γ2 and γ3 , three oscillators offirst ring and two oscillators of second ring are synchronized.
In the future, we investigate synchronization phenomenausing other parameters.

(1)

ε(xn − x3n ) − yan − ybn + γ(x2n − x1n )x2n − β(ya2n + yb2n+1 )x2n − β(yb2n + ya2n−1 ).

x12x22x21x11

Figure 5: γ1 = 0.001, γ2 = 0.001, γ3 = 0.02.

The normalized equations of VDP2 circuit are given as
follows:

yb 2nx22x23x22x21x11x13x12

The normalized equations of VDP1 circuit are given as
follows:y ˙2n =
 a˙x23x22

Figure 4: γ1 = 0.02, γ2 = 0.005, γ3 = 0.02.

Figure 1: Circuit model.

=x21

circuit 6x11 x2˙ nx13

VDP2

VDP2

circuit 4x12x21

Figure 3: γ1 = γ2 = γ3 = 0.02.

2LVDP2nx11x13R3circuit 52LVDP2x11

x13x23x22x23

VDP1R2
R1x22x21

Figure 2: γ1 = γ2 = γ3 = 0.02.

circuit 3

RYoshifumi NISHIO1(2)

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