2011 20th European Conference on Circuit Theory and Design (ECCTD)

Rotation of Phase Difference in Four Coupled
Oscillators as a Regular Tetrahedron Form
Takahiro Nagai, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {nagataka, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Coupled oscillatory systems are suitable to express
synchronization phenomena, and it is important to unveil nonlinear phenomena. In this study, we investigate the synchronization
phenomena observed in four coupled van der Pol oscillators
as a regular tetrahedron form. We observe the rotation of thephase difference between anti-phase state and asynchonization
state. Furthermore, the sojourn time of the anti-phase states iscalculated.

I. I NTRODUCTION
Synchronization is one of the basic natural phenomena, and
it is regarded typical nonlinear phenomena. Coupled oscillatory systems are good model to express high dimensional
nonlinear phenomena in the natural science fields. Therefore
investigations of the coupled oscillatory systems are reported
in various research fields [1]-[7]. Besides, many kind ofcoupled systems have been investigated to analyze the mechanism of the nonlinear phenomena. We can see that coupledoscillatory systems produce interesting phase wave patterns,
including wave propagation, clustering and complex patterns.
However, synchronization phenomena of the oscillators havenot been analyzed enough yet. Therefore, we have to research
the case of more complicated synchronization phenomena toelucidate high dimensional nonlinear phenomena.
Here, van der Pol oscillators were coupled in various form
and were investigated about their synchronization phenomena
by using the theoretically and the experimentally methods [8][10]. In [11], it is studied synchronization phenomena about
three van der Pol oscillators coupled as a ring topology.
In this circuit system, each oscillator was coupled by anFig. 2.

Conceptual circuit model for tetrahedron form.

inductor and the number of coupled oscillators was an odd
number. And then the coupled oscillators could not synchronize with in/anti-phase states. In other words, three-phasesynchronization (phase shift: 120∘ ) is obtained for the case of
three coupled oscillators by the effect of frustration. However,
the three-phase synchronization was always observed stably
in that system. Therefore we consider that what kind ofsynchronization phenomena are observed by using oscillatorswith stronger frustration.
In this study, we investigate synchronization phenomena incoupled oscillatory system which has stronger frustrations. We
consider four coupled van der Pol oscillators as the regulartetrahedron form as shown in Fig. 2. By computer simulations,
we observe the rotation of phase difference between anti-phasesynchronization and asynchronization periodically. Namely,
the adjacent oscillators are trapped into the anti-phase state
at certain time. In addition, we calculate the sojourn time ofanti-phase states in one-period, when the parameters of thenonlinearity and the coupling strength are changed.
II. C IRCUIT M ODEL

Fig. 1.

Three coupled van der Pol oscillators.

978-1-4577-0616-5/11/$26.00 ©2011 IEEE

The circuit model of four coupled van der Pol oscillators
as the regular tetrahedron form is shown in Fig. 3. Eachvan der Pol oscillator is composed of an inductor, a negative
resistor and a capacitor. In this study, the adjacent oscillators
are coupled by resistor via inductors.
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 currents flowing
from the negative resistance are 𝑖𝑅𝑘 . When we couple fouroscillators, tiny resistor 𝑟𝑚 is introduced to consider internal762

[Third oscillator]⎧ 𝑑𝑥3 = 𝜀(1 − 𝑥3 2 )𝑥𝑘 − (𝑦𝑎3 + 𝑦𝑏3 + 𝑦𝑐3 )
𝑑𝜏
}
𝑑𝑦1{𝑎3=𝑥3 − 𝜂𝑦𝑎3 − 𝛾(𝑦𝑎3 + 𝑦𝑐4 )⎨𝑑𝜏3{} 𝑑𝑦𝑏31 𝑑𝜏 = 3 𝑥3 − 𝜂𝑦𝑏3 − 𝛾(𝑦𝑏3 + 𝑦𝑏1 )}𝑑𝑦𝑐31{⎩=𝑥3 − 𝜂𝑦𝑐3 − 𝛾(𝑦𝑐3 + 𝑦𝑎2 ) .
𝑑𝜏
3Fig. 3.

[Fourth oscillator]⎧𝑑𝑥4= 𝜀(1 − 𝑥4 2 )𝑥𝑘 − (𝑦𝑎4 + 𝑦𝑏4 + 𝑦𝑐4 )

𝑑𝜏




{}


 𝑑𝑦𝑎4 = 1 𝑥4 − 𝜂𝑦𝑎4 − 𝛾(𝑦𝑎4 + 𝑦𝑐1 )⎨𝑑𝜏3{}𝑑𝑦1𝑏4
=𝑥4 − 𝜂𝑦𝑏4 − 𝛾(𝑦𝑏4 + 𝑦𝑏2 )𝑑𝜏3} 𝑑𝑦𝑐41{⎩=𝑥4 − 𝜂𝑦𝑐4 − 𝛾(𝑦𝑐4 + 𝑦𝑎3 ) .
𝑑𝜏3Four coupled van der Pol oscillators.

resistance. In addition, we connect tiny resistor to avoid Lloop.
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:[First oscillator]⎧𝑑𝑥1= 𝜀(1 − 𝑥1 2 )𝑥1 − (𝑦𝑎1 + 𝑦𝑏1 + 𝑦𝑐1 )𝑑𝜏}𝑑𝑦𝑎11{=𝑥1 − 𝜂𝑦𝑎1 − 𝛾(𝑦𝑎1 + 𝑦𝑐2 )⎨𝑑𝜏3{}
 𝑑𝑦𝑏11 𝑑𝜏 = 3 𝑥1 − 𝜂𝑦𝑏1 − 𝛾(𝑦𝑏1 + 𝑦𝑏3 )



}

𝑑𝑦𝑐11{
⎩=𝑥1 − 𝜂𝑦𝑐1 − 𝛾(𝑦𝑐1 + 𝑦𝑎4 ) .
𝑑𝜏
3[Second oscillator]
⎧
𝑑𝑥22


 𝑑𝜏 = 𝜀(1 − 𝑥2 )𝑥𝑘 − (𝑦𝑎2 + 𝑦𝑏2 + 𝑦𝑐2 )}𝑑𝑦𝑎21{=𝑥2 − 𝜂𝑦𝑎2 − 𝛾(𝑦𝑎2 + 𝑦𝑐3 )
𝑑𝜏
3
⎨}
𝑑𝑦𝑏21{
=𝑥2 − 𝜂𝑦𝑏2 − 𝛾(𝑦𝑏2 + 𝑦𝑏4 ) 𝑑𝜏3}
𝑑𝑦1{
𝑐2


=𝑥2 − 𝜂𝑦𝑐2 − 𝛾(𝑦𝑐2 + 𝑦𝑎1 ) .

3
⎩ 𝑑𝜏

(2)

(4)

(5)

We use the following normalizations:
√√
√
𝑔1
𝑔1 𝐶
𝑦𝑎𝑘 ,
𝑡 = 𝐿𝐶𝜏 , 𝑣𝑘 =𝑥𝑘 , 𝑖𝑎𝑘 =𝑔3
𝑔3 𝐿
√
√
𝑔1 𝐶
𝑔1 𝐶
𝑖𝑏𝑘 =𝑦𝑏𝑘 , 𝑖𝑐𝑘 =𝑦𝑐𝑘 ,𝑔3 𝐿
𝑔3 𝐿
√√
√
𝐿𝐶𝐶, 𝛾=𝑅, 𝜂 = 𝑟𝑚,
𝜀 = 𝑔1𝐶𝐿𝐿(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.
III. S YNCHRONIZATION P HENOMENA
We calculate Eqs. (2)-(5) using a fourth-order Runge-Kutta
method with the step size ℎ = 0.02 in this circuit. In this case,the parameters are set to 𝜀 = 0.50, 𝜂 = 0.02 and 𝛾 = 0.15.
We exhibit the attractor of each oscillator in this simulation inFig 4. In this figure, we show the horizontal axis is the voltage(3)

Fig. 4.

763

Attractor of each oscillator.

phase difference between the first oscillator and the secondoscillator. The horizontal axis is the computer simulation time
and the vertical axis is the phase difference. From this figure,
we can see that the phase difference changes from 0∘ to360∘ periodically. The coupled oscillators can not maintainthe synchronization state and the phase differences change
periodically after the adjacent oscillators are synchronizedwith the anti-phase states temporarily. The phase differenceof all adjacent oscillators has a similar characteristic. Namely
we find that the synchronization is gotten out from anti-phaseand it becomes asynchronous.
IV. S OJOURN T IME OF A NTI -P HASE S TATE
Fig. 5.

Lissajous figures.

of each condenser, and the vertical axis is the electric currentof each oscillator.
Next, the phase differences between the adjacent oscillatorsin all combinations at a certain time are shown in Fig. 5.
We confirm that the rotation of phase difference is observed.
Namely, the adjacent oscillators are trapped into the antiphase state at the certain time and are disconnected thesynchronization periodically.
In order to investigate the mechanism of such synchronization phenomena in detail, the phase differences are calculated
as shown in Fig. 6. The parameters are fixed as 𝜀 = 0.50,
𝜂 = 0.02 and 𝛾 = 0.15. In Fig. 6, “1st-2nd” indicates a360

360
1st-2nd

1st-3rd315

Phase difference

Phase difference

315

270

225

180

13590

45

270

225

180

13590

45000

20000

40000

60000

τp

80000

100000120000020000

40000

(a)

60000

τp

80000

120000360
1st-4th

2nd-3rd315

Phase difference

315

Phase difference

100000

(b)

360

270

225

180

13590

45

270

225

180

135

90

45000

20000

40000

60000

τp

80000

100000120000020000

40000

(c)

60000

τp

80000

100000120000

(d)

360

360
2nd-4th

3rd-4thFig. 7.
phase.

315

Phase difference

315

Phase difference

In this section, we measure the one period length of thechanging phase difference and the sojourn time of the antiphase. We define the anti-phase state when the phase difference value is from 175∘ to 185∘ . Figure 7 shows the definitionof the one period and sojourn time of the anti-phase.
For this simulation, the iteration is set to 𝜏𝑝 = 1,000,000
for calculating the result more precisely. Figure 8 shows thesimulation result of one period length and the sojourn timeof the anti-phase state when the parameter of nonlinearity
is changed from 0.10 to .200. In this figure, we show thehorizontal axis is 𝜀, and the vertical axis is the length of one
period and the sojourn time of the anti-phase state. We confirm
that the length of one period and the sojourn time of the antiphase state increase with the nonlinearity 𝜀.
Besides, we show the rate of the anti-phase state in Fig 9.
In Fig 9, the ratio of the anti-phase state does not change somuch with the nonlinearity 𝜀.

27022518013590

45270180

V. P OINCARE M APS

13590

45000

20000

40000

60000

τp(e)

80000

100000120000Definition of the length of one-period and sojourn time of the anti-2250

20000

40000

60000

τp

80000

100000120000(f)

Fig. 6.
The phase differences between adjacent oscillators. (a) 1st-2nd.
(b) 1st-3rd. (c) 2nd-4th. (d) 2nd-3rd. (e) 2nd-4th. (f) 3rd-4th.

Poincare map is one of the important methods to understand the structure of obits in the topological space. The
one dimension of the dynamics system is reduced by using
Poincare map. Figure 10 shows the simulation results of thePoincare map, when the nonlinearity 𝜀 is fixed as 0.10, 0.70,

764

2

2

ε=0.10

1

1

0.5

0.5

0

-0.5

-1

-1

-1.5

-1.5

-2
-2

-1.5

-1

-0.5

(a)

00.5

11.5

-2

2

00.5

11.5

2

1.5

2

1.5

2x2
ε=0.701.5110.5

0.5

dx2/dτ

y2-0.52ε=0.70

0-0.5

0-0.5

-1-1-1.5-1.5-2-2-2-1.5-1-0.5(c)

00.5

11.5

2-2-1.5-1x2-0.5(d)

2

00.5

1x2

2

ε=1.601.5

ε=1.60

1.5

1

10.5

0.5

dx2/dτ

y2-1(b)20-0.50-0.5-1-1-1.5-1.5-2-2-2-1.5-1-0.5(e)Fig. 9.

-1.5

x2

1.5

Length of the one-period and sojourn time of the anti-phase.

0-0.5-2Fig. 8.

ε=0.101.5

dx2/dτ

y2

1.5

0x2

0.5

11.5

2-2-1.5-1-0.5

(f)

00.5

1x2

Fig. 10. Poincare map on the section 𝑦𝑎1 = 0.(a) Horizontal: 𝑥2 , vertical:
𝑦2 for the parameter 𝜀 = 0.10. (b) Horizontal: 𝑥2 , vertical: 𝑑𝑥2 /𝑑𝜏 forthe parameter 𝜀 = 0.10. (c) Horizontal: 𝑥2 , vertical: 𝑦2 for the parameter
𝜀 = 0.70. (d) Horizontal: 𝑥2 , vertical: 𝑑𝑥2 /𝑑𝜏 for the parameter 𝜀 = 0.70.
(e) Horizontal: 𝑥2 , vertical: 𝑦2 for the parameter 𝜀 = 1.60. (f) Horizontal:
𝑥2 , vertical: 𝑑𝑥2 /𝑑𝜏 for the parameter 𝜀 = 1.60.

Ratio of synchronization with anti-phase.

and 1.60. In the case of the left side figures (Fig. 10 (a), (c) and(e)), the horizontal axis denotes 𝑦2 . While, for the right sidefigures (Fig. 10 (b), (d) and (f)), the horizontal axis denotes
𝑑𝑥2 /𝑑𝜏 . From these figures, we can see that nonlinearity ofthe trajectory of Poincare map becomes strong with increasing
the value of 𝜀.
VI. C ONCLUSION
In this study, we have investigated synchronization phenomena observed in the four coupled oscillators as a regular tetrahedron form. In this circuit model, we observe the switching
synchronization state between the adjacent oscillators. Namely,
the coupled oscillators can not maintain the synchronization
state and the phase differences change periodically after theadjacent oscillators are synchronized with the anti-phase states
temporarily.
For the future work, we would like to investigate therelationship between the switching synchronization state observed in this study and the phase slip phenomena of noisily
oscillatory systems.
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