IEEE Workshop on Nonlinear Circuit Networks
December 9-10, 2011

Synchronization of Coupled Oscillators with
Double Tetrahedrons as a Shared Triangular Face
Takahiro Nagai

Hironori Kumeno

Yoko Uwate

Yoshifumi Nishio

Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {nagataka, kumeno, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— This paper acquaints the synchronization of coupledoscillatory system with double tetrahedral forms as a sharing
triangular face. We confirm the synchronization phenomena bymeasuring the phase differences between adjacent oscillators.
Also, we obtain the amplitude of each oscillator using thesimulation results.

I. I NTRODUCTION
Coupled oscillatory system is one of the most proper model
to indicate the high-dimensional phenomena in natural sciencefields [1]-[7]. there are a lot of synchronization phenomena
in natural environment. Therefore investigations of the coupled oscillatory systems about synchronization phenomena
are reported in biology, physics and mathematics. Coupledoscillatory systems have various wave patterns and including
wave propagations, clustering and complex patterns. Hence,various coupled oscillators were proposed, and the part of themechanism of the non-linear phenomenon has been elucidated
until now. However investigations on the complex coupledoscillatory networks have not been studied enough yet.
In our investigations, we use several van der Pol oscillators.
Van der Pol oscillators have been coupled in various form andinvestigated about their synchronization phenomena [8], [9].
In particular, three coupled oscillatory system with a ring
topology as shown Fig. 1 have made the very interesting
results [10]. In this circuit system, each oscillator was coupled
by an inductor and the number of coupled oscillators was anodd number. Then, we could not observe the synchronization
phenomena with in/anti-phase states. That is to say, threephase synchronization (phase shift: 120◦ ) is obtained for theFig. 2.

Conceptual circuit model for tetrahedron form.

case of three oscillators by the effect of frustration. However,
the three-phase synchronization was always observed stably inthat system.
In our previous study, we have investigated several kinds ofinteresting synchronization phenomena in coupled oscillatorysystem which has stronger frustrations. We have researchedfour coupled van der Pol oscillators in the regular tetrahedron
form as shown in Fig. 1(b). By computer simulation, weobserved that the phase difference between adjacent oscillatorschanged and the synchronization was destroyed after the adjacent oscillators synchronize with anti-phase. And the anotherstudy, synchronization phenomena in two coupled triangularoscillatory networks sharing a branch was investigated in [11].
In this case, we could observed synchronization that the phasedifference of sharing branch is in-phase (phase difference: 0◦ ),and the other one between adjacent oscillators are synchronized with anti-phase (phase difference: 180◦ ).
This paper presents synchronization of coupled oscillatorswith double tetrahedrons as a shared triangular face. We show
the our coupled oscillatory system in Fig. 2 which is used
tetrahedral forms sharing a triangular face. We confirm thesynchronization phenomena by measuring the phase differences between adjacent oscillators.
II. C IRCUIT M ODELFig. 1. Coupled van der Pol oscillators. (a) Three coupled oscillator. (b) Fourcoupled oscillator.

The circuit model is shown in Fig. 3(a). In this circuit model,
two tetrahedrons oscillators are coupled by the triangular face
and the fourth and the fifth oscillators have no connection.
In the computer simulations, we assume that the vk − iRkcharacteristics of nonlinear resistor in each oscillator is givenby the following third order polynomial equation.

- 58 -Fig. 4. Attractor between adjacent oscillators (horizontal axis:xk , verticalaxis:yk (k = 1, 2, 3, 4, 5).

Fig. 3. Circuit model with double tetrahedrons. (a) Conceptual circuit model.
(b) Coupled structure.

iRk = −g1 vk + g3 vk 3

(g1 , g3 > 0),(k = 1, 2, 3, 4, 5).
(1)

Fig. 5.

The time waveform of the each oscillator.

The normalized circuit equations are expressed as:dxk
= ε(1 − xk 2 )xk − (yak + ybk + yck + ydk )
dτ

}
dyak1{
=xk − ηyak − γ(yak + yn )dτ4}dybk1{=xk − ηybk − γ(ybk + yn )dτ4
}
dyck1{xk − ηyck − γ(yck + yn )= dτ4{}
 dydk = 1 xk − ηydk − γ(ydk + yn ) .
dτ4(2)We use the following normalizations:
√√
√
g1
g1 C
t = LCτ , vk =xk , iak =yak ,g3
g3 L
√
√
g1 C
g1 C
ibk =ybk , ick =yck ,g3 L
g3 L
√√√
LCC
ε = g1, γ=R, η = rm,CLL(k = 1, 2, 3, 4, 5),
where ε is the nonlinearity, γ is the coupling strength, ηindicates the resistive component and yn denotes the currentof neighbor oscillator on coupling resistor. In the computer
simulations, we calculate the phase differences between adjacent oscillators. Here rm denotes the internal resistance of aninductor.

we fix the frequency errors ζ2 = 1.001, ζ3 = 1.002, ζ4 =1.003, ζ5 = 1.004.
We show the simulation result of the synchronization phenomena in Fig. 4. In this figure, we show the attractor ofeach oscillator and the horizontal axis is the voltage of each
oscillator, and the vertical axis is the electric current of eachoscillator. These electric currents are summed the four currents
yk = yak + ybk + yck + ydk and we set the parameters ε =0.10, γ = 0.10 and η = 0.00010. Because we define η to evadeL-loop, we fix this value in these simulations. Next, the time
waveforms of the voltage of each capacitor after sufficient time
has elapsed are shown in Fig. 5. And the phase differences
between the adjacent oscillators of this case is equal to theresult. As a result, in the case of this circuit model, it wasobserved that the phase difference finally converged with theconstant value. Also, this phase difference hardly varies evenif we change the initial conditions. The case of parameterγ > 0.0660, the synchronization phenomena between each
oscillator can be seen asynchronous.
When the coupling strength is strong, the phase differences converge for in/anti-phase. Shared triangular oscillators
are synchronized for in-phase but the shared oscillators areIII. S YNCHRONIZATION P HENOMENA
We calculate Eq. (2)-(6) using the fourth-order Runge-Kutta
method with the step size h = 0.002. In this simulation, weconsider frequency errors ζ at the voltage equation. In otherword, the voltage equations of the second, the third, the forthand the fifth oscillators are multiplied by ζ (k = 2, 3, 4, 5),- 59 -Fig. 6.

Amplitude of voltage.

[9] Y. Uwate, Y. Nishio and R Stoop, “Synchronization in three coupled vander Pol oscillators with different coupling strength,” Proc. of NCSP’10,
pp.109-112, Mar. 2010.
[10] Y. Uwate, Y. Nishio and R Stoop, “Synchronization in two polygonal
oscillatory networks sharing a branch,” NDES 2010, pp.62-65, May. 2010.

Fig. 7.

Difference of amplitude.

synchronized with other oscillators for anti-phase. Also, We
describe changes of the amplitude ρk of voltage in Fig. 6.
In this result, the amplitudes are divided into two groups
that one of the groups is the shared oscillators and theother one is the remained oscillators. We can find that allof amplitudes decrease by varying the parameter γ. Figure 7shows the difference of amplitude ρ4 − ρ1 , if the differenceof the amplitude becomes big to some extent, each amplitude
influences on the phase difference and each oscillator tends toconverge for in/anti-phase.
IV. C ONCLUSIONSThis paper has acquainted the synchronization of coupledoscillatory system with double tetrahedral forms as a sharing
triangular face. We have confirmed the synchronization phenomena by measuring the phase differences between adjacentoscillators. Unlike the case only for the tetrahedron form, it
was observed that the phase difference converged with certainvalues. Also, the coupling strength grew big, we could findthat the amplitude of each oscillator decreased by using thesimulation results.
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