2010 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 3-4, 2010

Synchronization Phenomena of Four Oscillators
Coupled as a Regular Tetrahedron Form
Takahiro Nagai

Hironori Kumeno

Yoko Uwate

Yoshifumi Nishio

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: {nagataka, kumeno, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— In our study, we research the synchronization phenomena of four oscillators coupled as a regular tetrahedron form.
We connect four van der Pol oscillators in the tetrahedron formand survey the phase difference between each oscillator.

I. I NTRODUCTION
Coupled oscillatory systems are good model to express highdimensional nonlinear phenomena in the natural science fields.
There are many studies on synchronization phenomena of
coupled oscillator systems in many research fields [1]-[7].
So, many kind of coupled systems have been investigated toanalyze the mechanism of the nonlinear phenomena. We cansee that coupled oscillatory systems produce interesting phasewave patterns, including wave propagation, clustering andcomplex patterns. However, synchronization phenomena of theoscillators have not been analyzed enough yet. Therefore, wehave to investigate the case of more complicated synchronization phenomena to elucidate high dimensional nonlinear
phenomena.
In our previous works, we have investigated synchronization
phenomena in three coupled van der Pol oscillators as a ring
topology [8]. In this circuit system, the number of coupledoscillators is an odd number. Then coupled oscillators can notsynchronize with in/anti phase states. In other words, threephase synchronization (phase shift: 120◦ ) is obtained by effectof frustration. However, three-phase synchronization is always
observed stably in this system.
In our study, we suppose that we can survey several kinds ofinteresting synchronization phenomena in coupled oscillatory
system which has stronger frustrations. We research four
coupled van der Pol oscillators in the form of the regulartetrahedron as shown in Fig. 1. In addition, we calculate thephase differences of each oscillator in four coupled van der
Pol oscillators. By changing the parameters of the nonlinearity
and the coupling strength, we observe how sojourn time in inphase or anti-phase changes.
II. C IRCUIT M ODELThe circuit model of four coupled van der Pol oscillators in
the form of the regular tetrahedron is shown in Fig. 2. Van derPol oscillator is constituted an inductor, a negative resistor anda capacitor. The adjacent oscillators are coupled by resistor viainductors.

v3
C3

ia3
3L

rm

ib3

ic3

iR3

3L

ic4
3

ia 3

rm
3L

ic 3

iR4

ib 4

R

ic1

ib1

Fig. 1.

13L

rm

ib1

rm
3L

ic 2
i c1

rm

3L

ic2

3L

4th oscillator

ia 4

R

rm

24

rm

R

ia4

v4

ib 2

ib2 3L
rm

C4ia2

rm

R

ib4

3L

ia 2

ic 4rmR3L

ib 3

3rd oscillator

rm

3Lrm

R

ia1
v1

2nd oscillator

C1 1st oscillator
i a1Conceptual circuit model for tetrahedron form.

Fig. 2.

- 12 -v2
C2

3L

iR1

iR2

Four coupled van der Pol oscillators.

In this figure, we assume that the electric currents flowing
from each top are iak , ibk and ick , the voltages of the capacitorof each oscillator are vk and the electric current flowing fromthe negative resistance iRk . When we couple four oscillators,tiny resistor rm is introduced to consider internal resistance. Inaddition, we connect tiny resistor to avoid L-loop. If L-loop
occurs in four coupled oscillators, the computer simulations
can not carry out normally.
In the computer simulations, we assume that the vk − iRkcharacteristics of nonlinear resistor in each oscillator is givenby the following third order polynomial equation.
iRk = −g1 vk + g3 vk 3

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

(1)

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

We use the following normalizations:
√√√
g1
g1 C
t = LCτ , vk =xk , iak =yak ,g3
g3 L
√√
g1 C
g1 Cibk =ybk , ick =yck ,g3 L
g3 L√√√LCC
ε = g1,γ = R, η = rm,CLL(k=1,2,3,4),
where ε is the degree of nonlinearity, γ is the couplingstrength, and η indicates the resistive component. In thecomputer simulations, we investigate the phase differences
between adjacent oscillators.

x1x2

x3x4
τ

Fig. 3.

The time wave form of the voltage.

states stay at the anti-phase whose sojourn time is longer thanthat of the in-phase.
Figure 5 shows one example of the changing phase differences for the parameters ε = 0.50, η = 0.02 and γ =0.15 (Fig. 5(a)) and γ = 0.40 (Fig. 5(b)). This figure shows
the change of the phase difference for time. The horizontal
axis shows the computer simulation time and the vertical axis
shows the phase difference of each oscillator. In Fig. 5, “1-2”indicates a phase difference between the first oscillator andthe second oscillator.
In this figure, we confirm that the phase differences of all
oscillators change periodically. In addition, we can see thatsojourn time of the in-phase is instant whereas the anti-phase isrelatively long. To increase γ, periods of the shifting the phasex2x3x1x1x4x2x3x1x3x2

III. S YNCHRONIZATION P HENOMENAx4

We carry out computer simulations for the four coupled
van der Pol oscillators in the regular tetrahedron form circuit.
Figure 3 shows the time wave form of four coupled van der Poloscillators. The horizontal axis shows the computer simulation
time and the vertical axis shows the time wave form of thevoltage. In this case, we input the parameters ε = 0.50, η =0.02 and γ = 0.15. Figure 4 shows the phase plane between
the adjacent oscillators at same time as Fig. 3.
In addition, we observe interesting synchronization phenomena that the phase differences between oscillators change periodically from 0◦ to 180◦ or the reverse. The synchronization- 13 -x4x4

x4x2Fig. 4.

Lissajous figures.

x3

180

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

140

Phase difference

120

(a)

100(a)

80
60
40
200

20000

40000

60000
τp

80000

100000

120
100
80
60
400

120000020000

40000

60000

τp

80000

100000120000180

180
1-21-3
1-4
2-3
2-4
3-4

140

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

160

120
100(b)

8060
40

phase difference

160

Phase difference140

200(b)

1-2
1-3
1-4
2-3
2-4
3-4160phase difference

160

140
120
100
80
60
40
20

20

000

20000

40000

60000
τp80000

100000120000Fig. 5. The phase difference between oscillators for ε = 0.50 and η = 0.02.
(a)γ = 0.15. (b)γ = 0.40.

differences become longer. In the case of Fig. 5(b), the intervalof the in-phase becomes long in comparison with Fig. 5(a).
In next simulation, we change the parameters. We fix γ andcarry out simulation whereas we fixed ε in former computer
simulation. Figure 6 shows the simulation results for theparameters γ = 0.40, η = 0.02 and ε = 0.20 (Fig. 6(a)) and ε =0.40 (Fig. 6(b)). In comparison with Fig 6(a) and Fig 6(b), we
can understand that there is time lag which switch of in-phaseor anti-phase. In particular, the switching time of anti-phasebecomes long to increase ε.
IV. C ONCLUSIONSIn this study, we investigated synchronization phenomena
observed in the four coupled oscillators in form of a regulartetrahedron. We observed synchronisation phenomena that thephase differences changed periodically. Moreover, when thecoupled strength was small, changes of phase differences
became sharp. The anti-phase state continued for a long timewith increasing γ.

020000

40000

60000

80000

100000120000

τpFig. 6. The phase difference between oscillators for γ = 0.40 and η = 0.02.
(a)ε = 0.20. (b)ε = 0.40.

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ACKNOWLEDMENTThis work was partly supported by CASIO Science Promotion Foundation.

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