Nonlinear Dynamics of Electronic Systems 2012, Wolfenbüttel

Synchronization of Coupled Tetrahedral Oscillatory
Networks Shared by Triangular Face
Takahiro Nagai, Hironori Kumeno, Yoko Uwate and 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—In this study, we investigate the synchronization phenomena observed in two coupled tetrahedral
oscillators shared triangular face. We focus on all of
the coupling strength between adjacent oscillators andwe show the change of the phase differences when wechange the coupling strength. Next, we vary the couplingstrength partially and observe synchronization states ofshared triangular oscillators.
Fig. 2.

I. M ANUSCRIPT S UBMISSION
Coupled oscillatory system is one of the most proper
model to indicate the high-dimensional phenomena innatural science fields [1]-[7]. There are a lot of synchronization phenomena in natural environment. For
example, swing of the pendulum, firefly luminescence,
cardiac heartbeat etc. are well known as the synchronization phenomena. Therefore investigations of the coupledoscillatory systems about synchronization phenomena
are reported in biology, physics and mathematics. Coupled oscillatory systems have various wave patterns andincluding wave propagations, clustering and complex
patterns. Hence, various types of coupled oscillators
were proposed, and the part of the mechanism of thenon-linear phenomenon has been elucidated until now.
However investigations on the complex coupled oscillatory networks have not been studied enough yet.

Fig. 1. Coupled van der Pol oscillators. (a) Three coupled oscillator.
(b) Four coupled oscillator.

Conceptual circuit model for tetrahedron form.

In our investigations, we use several van der Poloscillators. Van der Pol oscillators have been coupledin various form and investigated about their synchronization phenomena [8], [9]. In particular, three coupledoscillatory 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 andthe number of coupled oscillators was an odd 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 forthe case of three oscillators by the effect of frustration.
However, the three-phase synchronization was always
observed stably in that system.
In our previous study, we have investigated several
kinds of interesting synchronization phenomena in coupled oscillatory system which has stronger frustrations.
We have researched four coupled van der Pol oscillatorsin the regular tetrahedron form as shown in Fig. 1(b).
By computer simulation, we observed that the phasedifference between adjacent oscillators changed and thesynchronization was destroyed after the adjacent oscillators synchronize with anti-phase. In other words, thiscircuit model has the feature such as repeated antiphase synchronous and asynchronous. The another study,
synchronization phenomena in two coupled triangularoscillatory networks sharing a branch was investigated108in [11]. In this case, we could observed synchronizationthat the phase difference of sharing branch is in-phase(phase difference: 0∘ ), and the other one between adjacent oscillators are synchronized with anti-phase (phasedifference: 180∘ ). In this study, we investigate synchronization phenomena in coupled oscillatory system which
has two tetrahedrons. We consider two tetrahedron forms
sharing a triangular face as shown in Fig. 2. We carry out
the computer simulation and analyze the synchronization
phenomena.
First we investigate phase difference between adjacentoscillators and compare shared triangular oscillators to
the other oscillators. Next, we vary the coupling strength
partially and observe synchronization states of shared
triangular oscillators.

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

𝜂 indicates the resistive component and 𝑦𝑛 denotes thecurrent of neighbor oscillator on coupling resistor. In thecomputer simulations, we calculate the phase differences
between adjacent oscillators. Here 𝑟𝑚 denotes the internal resistance of an inductor.

II. C IRCUIT M ODEL

III. S YNCHRONIZATION P HENOMENA

The circuit model is shown in Fig. 3(a). In this circuit
model, two tetrahedrons oscillators are coupled by thetriangular face and the fourth and the fifth oscillators
have no connection. In the computer simulations, weassume that the 𝑣𝑘 − 𝑖𝑅𝑘 characteristics of nonlinear
resistor in each oscillator is given by the following thirdorder polynomial equation.
𝑖𝑅𝑘 = −𝑔1 𝑣𝑘 + 𝑔3 𝑣𝑘 3

(𝑔1 , 𝑔3 > 0),(𝑘 = 1, 2, 3, 4, 5).

(1)

The normalized circuit equations are expressed as:⎧ 𝑑𝑥1𝑘= 𝜀(1 − 𝑥𝑘 2 )𝑥𝑘 − (𝑦𝑎𝑘 + 𝑦𝑏𝑘 + 𝑦𝑐𝑘 + 𝑦𝑑𝑘 )
𝑑𝜏3
}
1{ 𝑑𝑦𝑎𝑘𝑥𝑘 − 𝜂𝑦𝑎𝑘 − 𝛾(𝑦𝑎𝑘 + 𝑦𝑛 )=𝑑𝜏4⎨}𝑑𝑦𝑏𝑘1{𝑥𝑘 − 𝜂𝑦𝑏𝑘 − 𝛾(𝑦𝑏𝑘 + 𝑦𝑛 )
=
𝑑𝜏4
}
𝑑𝑦𝑐𝑘1{𝑥𝑘 − 𝜂𝑦𝑐𝑘 − 𝛾(𝑦𝑐𝑘 + 𝑦𝑛 )
=
𝑑𝜏4


⎩ 𝑑𝑦𝑑𝑘 = 1 {𝑥 − 𝜂𝑦 − 𝛾(𝑦 + 𝑦 )}.
𝑑𝜏𝑘4𝑑𝑘𝑑𝑘(2)𝑛We use the following normalizations:√
√
√
𝑔1
𝑔1 𝐶
𝑦𝑎𝑘 ,𝑡 = 𝐿𝐶𝜏 , 𝑣𝑘 =𝑥𝑘 , 𝑖𝑎𝑘 =
3𝑔3
3𝑔3 𝐿
√

𝑖𝑏𝑘 =
√𝜀 = 𝑔1

𝑔1 𝐶
𝑦𝑏𝑘 , 𝑖𝑐𝑘 =3𝑔3 𝐿
𝐿,𝛾 = 𝑅𝐶√

We calculate Eq. (2) using the fourth-order RungeKutta method with the step size ℎ = 0.01. We show
the simulation result of the synchronization phenomena
in Fig. 4. In this figure, we show the attractor of each
oscillator and the horizontal axis is the voltage of each
oscillator, and the vertical axis is the electric current ofeach oscillator. These electric currents are summed thefour currents 𝑦𝑘 = 𝑦𝑎𝑘 + 𝑦𝑏𝑘 + 𝑦𝑐𝑘 + 𝑦𝑑𝑘 and we set theparameters 𝜀 = 0.10, 𝛾 = 0.10 and 𝜂 = 0.00010.
Next, the time waveforms of the voltage of eachcapacitor after sufficient time has elapsed are shown inFig. 5. And the phase differences between the adjacentoscillator of this case is equal to the result as shown in
Fig. 6. As a result, in the case of this circuit model, it wasobserved that the phase difference finally converged withthe constant value. Also, this phase difference hardly
varies even if we change the initial conditions.
But the effect of varying initial conditions are able to√Fig. 4. Attractor between adjacent oscillators (horizontal axis:𝑥𝑘 ,vertical axis:𝑦𝑘 (𝑘 = 1, 2, 3, 4, 5).

𝑔1 𝐶
𝑦𝑐𝑘 ,3𝑔3 𝐿𝐶, 𝜂 = 𝑟𝑚
𝐿

√𝐶,
𝐿

(k= 1, 2, 3, 4, 5),
where 𝜀 is the nonlinearity, 𝛾 is the coupling strength,109

Fig. 5.

The time waveform of the each oscillator.

Fig. 9.
Fig. 6.

Amplitude of voltage.

Lissajous figures.

Fig. 7. The relationship of phase differences for the shared triangle.

confirm other point of view. The pattern of phase differences between shared three oscillators are rotated by theinitial conditions. Figure 7 shows the phase differences
between shared triangular oscillators. In this figure, wecan find that the three patterns of synchronization states
exit.
Here, we change the coupling strength from 𝛾 = 0.10
to 1.00. In this case, phase differences divide into twogroups and synchronize with in/anti-phase states. Shared
triangular oscillators synchronize with in-phase but othersynchronization states between adjacent oscillators shiftswith anti-phase. We summarize the changes of the phasedifference for the coupling strength in Fig. 8. We show
the phase difference of shared oscillators (Fig. 8(a)),
and the other oscillators (Fig. 8(b)). When the couplingstrength is strong, the phase differences converge forin/anti-phase. Also, We describe changes of the amplitude 𝜌𝑘 of voltage in Fig. 9. In this result, the amplitudes
are divided into two groups that one of the groups isthe shared oscillators and the other one is the remainedoscillators.
IV. C HANGE OF COUPLING STRENGTH
In this circuit system, if we break the coupling about
forth and fifth oscillators, the three coupled oscillatorswith ring topology exists. Namely, we can observe thethree phase synchronization between adjacent oscillatorsand other synchronization states are asynchronous. Here
we fix the coupling strength 𝛾 = 0.100, we multiply 𝛽(0 < 𝛽 < 1.0) coupling strength about forth and fifthoscillator. We observe change of the synchronization
states between adjacent oscillators. Table 1 shows thesynchronization states by varying 𝛽 .

(a)

TABLE IT HE CHANGE OF SYNCHRONIZATION STATES BY VARYING 𝛽.

(b)Fig. 8.
The change of phase differences by coupling strength.
(a) shared triangular oscillators. (b) other oscillators.

110

𝛽0.000 ∼ 0.4500.450 ∼ 0.500
0.500 ∼ 0.536
0.536 ∼ 1.000

Synchronization States
Asynchronous
Anti-phase Synchronization and Asynchronous
In-phase Synchronization and Asynchronous
Synchronization with a certain phaseregular intervals and they change the complex in antiphase area. Figure 11 shows the sojourn time of one
period, in-phase state and anti-phase. In this figure, wedefine the in-phase and anti-phase area from 0∘ to 10∘
and from 170∘ to 180∘ . Finally, 𝛽 is over 0.5360, eachphase difference converges a certain value. Namely, thephase differences change by 𝛾 , but they do not change
qualitatively.

Fig. 10.
Phase difference between adjacent shared triangularoscillators.

First, the case of 𝛽 is less than 0.450, we can findthat the phase differences between adjacent oscillatorschange periodically. Then, the bigger value of 𝛽 , thewidth of change of the phase difference becomes larger
and the case of 𝛽 is over 0.450, the synchronization
states shift the switching of anti-phase synchronization
and asynchronous. Figure 10 shows the phase differences
between shared three coupled oscillators about the caseof 𝛽 = 0.480. Each synchronization state is switchinganti-phase synchronization and asynchronous, and threechanges of phase differences remain the timing and shift.
Moreover, sojourn time of anti-phase synchronization
depend on the coupling strength and it tends to lengthen
the coupling strength bigger.
If 𝛽 reaches 0.500, we can find maintenance of inphase synchronization, and the case of 𝛽 is over 0.530,
the in-phase synchronization states are maintained thananti-phase states. Then, the phase differences temporarily
maintain about 160∘ in anti-phase area. The case of 𝛽
is over 0.5350, the phase differences stay at in-phase inFig. 11.

Change of sojourn time by changing 𝛽.

V. C ONCLUSIONSIn this study, we have investigated synchronization
phenomena observed in coupled two tetrahedrally form
sharing the triangular oscillatory system. Unlike the caseonly for the tetrahedron form, it was observed that thephase difference converged with certain values. Especially, the coupling strength grew big, we could find thesynchronization with in-phase between shared oscillatorsand synchronization with anti-phase between not sharedoscillators. Also, we could find that the switching ofasynchronous and anti-phase states when the couplingstrength differ between shared oscillators and other one.
R EFERENCES[1] L.L. Bonilla, C.J. Perez Vicente and R. Spigler, “Time-periodic
phases in populations of nonlinearly coupled oscillators withbimodal frequency distributions,” Physica D: Nonlinear Phenomena, vol.113, no.1, pp.79-97, Feb. 1998.
[2] J.A. Sherratt, “Invading wave fronts and their oscillatory wakes
are linked by a modulated traveling phase resetting wave,”
Physica D: Nonlinear Phenomena, vol.117, no.1-4, pp.145-166,
June 1998.
[3] G. Abramson, V.M. Kenkre and A.R. Bishop, “Analytic solutions for nonlinear waves in coupled reacting systems,” Physica
A: Statistical Mechanics and its Applications, vol.305, no.3-4,pp.427-436, Mar. 2002.
[4] 4. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization and graph topology,” International Journal of Bifurcation
and Chaos, vol.15, no.11, pp.3423-3433, Nov. 2005.
[5] C.M. Gray, “Synchronous oscillations in neural systems: mechanisms and functions,” J. Computational Neuroscience, vol.1,pp.11-38, 1994.
[6] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronization of nonlinear electronic oscillators for neural computation,” IEEE Trans. Neural Networks, vol.15, no.5, pp.13151327, Sep. 2004.
[7] R. Stoop and C. Wagner, “Neocortex’s architecture optimizes
computation, information transfer and synchronizability , at given
total connection length,” International Journal of Bifurcation andChaos, vol.17, no.7, pp.2257-2279, 2007.
[8] T. Suezaki and S. Mori, “Mutual synchronization of two oscillators,” Trans. IECE, vol.48, no.9, pp.1551-1557, Sep. 1965.
[9] T. Endo and S. Mori, “Mode analysis of a multimode ladderoscillator,” IEEE trans. Circuit Syst., vol23, pp.100-113, Feb.
1976.
[10] Y. Uwate, Y. Nishio and R Stoop, “Synchronization in three
coupled van der Pol oscillators with different coupling strength,”
Proc. of NCSP’10, pp.109-112, Mar. 2010.
[11] Y. Uwate, Y. Nishio and R Stoop, “Synchronization in two
polygonal oscillatory networks sharing a branch,” NDES 2010,
pp.62-65, May. 2010. T. Endo and S. Mori, “Mode analysis ofa multimode ladder oscillator,” IEEE trans. Circuit Syst., vol23,
pp.100-113, Feb. 1976.

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