平成22年度電気関係学会四国支部連合大会

328

18-29

Synchronisation in Four Coupled van der Pol
Oscillators in a Regular Tetrahedron Form
Takahiro NAGAI, Hironori KUMENO, Yoko UWATE and Yoshifumi NISHIO(Tokushima University)

1. Introduction

ε = g1High dimensional nonlinear phenomena can often be
expressed by synchronization phenomena of oscillators.
Therefore, studies of the synchronization phenomena ofcoupled oscillators are reported in many research ﬁelds.
However, synchronization phenomena of the oscillators
have not been analyzed enough yet.
In this study, we propose four coupled van der Pol oscillators in a regular tetrahedron form. We investigate
synchronisation phenomena in this circuit model.

2. Circuit Modelv3

ia3

C3

ic33L

rm

iia43L

rm

rm

R

iR43L

ia2

√

C/L , η = rm

R

rm
3L3L

v4

ib1

R3L

rm

180
1-2
1-3
1-4
2-3
2-4
3-4

ic1

140

ic2

120

v2
iR2

C/L,

We carry out computer simulations for the four coupled van der Pol oscillators in a regular tetrahedron form
circuit. We observe a synchronization phenomena thatphase diﬀerences between oscillators shift from 0◦ to 180◦or the reverse, periodically. The synchronization states
stay at the anti-phase whose sojourn time is longer thanthat of the in-phase. Fig. 2 shows examples of phase differences for the parameters ε = 1.0, η = 0.02 and γ =0.15 (Fig. 2(a)) or γ = 0.40 (Fig. 2(b)). In Fig. 2, “12” indicates a phase diﬀerence between 1st oscillator and2nd oscillator. Periods of the shifting the phase diﬀerences
become longer with increasing γ.

ib2rm 3L 3Lrm√

3. Synchronization phenomena

160

rm

C4rm
RR3L b43L

3rd oscillator

rm

rm

R

ic4

iR3

3L

ib3

3L

L/C , γ = R

where ε is the degree of nonlinearity, γ is the couplingstrength, and η indicates the resistive component.

(a)

C2

phase

rm

√

100

ia1

80
60
40
20

4th oscillator

iR1

C1

v1

0

2nd oscillator0

1st oscillator

20000

40000

60000

τp

80000

100000120000

180
1-2
1-3
1-4
2-3
2-4
3-4

160

Figure 1: Four coupled van der Pol oscillators.

140120(b)phase

The circuit model of four coupled van der Pol oscillators
in a regular tetrahedron form is shown in Fig. 1. Van der
Pol oscillator consists of an inductor, a negative resistor
and a capacitor. In addition, we assume that the vk − iRkcharacteristics of nonlinear resistor in each oscillator isgiven by the following third order polynomial equation.

10080
60
40
2000iRk = −g1 vk + g3 vk

3(g1 , g3 > 0),(k = 1, 2, 3, 4).

(1)

The normalized circuit equations are expressed as:

dτ3dτ

3

We used the following normalizations:√√√t = LCτ , vk = g1 /g3 xk , iak = g1 C/g3 L yak ,
ibk =

√

g1 C/g3 L ybk , ick =√g1 C/g3 L yck ,

40000

60000

τp

80000

100000120000Figure 2: Phase diﬀerence between oscillators for ε =1.0 and η = 0.02. (a)γ = 0.15. (b)γ = 0.40.

4. Conclusions

dxk


= ε(1 − xk 2 )xk − (yak + ybk + yck )
dτ

{}

 dyak = 1 xk − ηyak − γ(yak + yc(k+1) )}dybk1{=xk − ηybk − γ(ybk + yb(k+2) )
dτ
3
{}
 dyck = 1 xk − ηyck − γ(yck + ya(k+3) ) .

20000

(2)

In this study, we proposed the four coupled oscillators
in a regular tetrahedron form. We observed synchronisation phenomena that the phase diﬀerence changed periodically. Additionally, when the coupled strength was small,changes of phase diﬀerences became sharp.
References[1] Y. Uwate, Y. Nishio and R Stoop, “Synchronizationin Three Coupled van der Pol Oscillators with Diﬀerent Coupling Strength”,Proceedings of RISP International
Workshop on Nonlinear Circuits and Signal Processing(NCSP’10), pp. 109-112, Mar. 2010.