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

Synchronization Phenomenon in Four Chains of
Coupled Oscillators
Kosuke Matsumura, Takahiro Nagai, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Eng, Tokushima University, Tokushima 770-8506, JAPAN
Email: {kosuke, nagataka, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— In this study, we investigate a synchronization phenomenon from a circuit network which is composed of two kinds
of oscillator chains. They are one-dimensional arrays of weakly
coupled van der Pol oscillators. From computer simulations,we observe relatively interesting unexpected synchronization
phenomenon.

I. Introduction
Synchronization phenomena are very basic phenomena.
Also, the important phenomena observed everywhere in natures. For example, vibration of a pendulum, firefly luminescence, gate patterns of four-leg animals, two frogs using
voice religiously, periodic swinging of candle flames, andso on. Therefore, coupled oscillators are good models toinvestigate such interesting synchronization phenomena. Many
researchers have proposed diﬀerent coupled oscillatory networks and have discovered many interesting synchronization
phenomena [1]-[8]. The research group of the authors is also
working on coupled oscillatory networks. Especially, we havebeen interested in coupled oscillators whose connections causesome kinds of frustrations [9]-[12].
In this study, we investigate a circuit network which iscomposed of two kinds of oscillator chains. They are onedimensional arrays of weakly coupled van der Pol oscillators.
In this network, we couple the oscillators at bottom four
oscillators in the chains to constrain them to produce in-phasesynchronizations. While we couple the oscillators at the top
four oscillators in the chains to produce anti-phase synchronizations. Middle oscillators in the chains are not coupled withthe other chains. We observe relatively interesting unexpectedsynchronization phenomenon from computer simulations.
II. Circuit Model
Figure 1 shows the our proposed circuit model. Eachoscillator-chain consists of n van der Pol oscillators weakly
coupled by resistors r. In the figure, bottom four oscillators
are coupled by relatively strong resistors Ri , while the top
four oscillators are coupled by also relatively strong resistors
Ra via inductors. Middle oscillators are not coupled with oscillators located in the horizontal direction but weakly coupledvertically.
The coupling structure on the bottom or in vertical couplings(by Ri or r) tends to make the oscillators to synchronize inin-phase. While the coupling structure on the top (by Ra viainductor) tends to make the oscillators to synchronize in antiphase.

- 78 -Fig. 1.

Circuit model for the case of n oscillator-chains.

We define the bottom four oscillators as Osc11 , Osc12 , Osc13and Osc14 from the left, those on the kth row from the bottom
as Osck1 , Osck2 , Osck3 and Osck4 , and the top four oscillators
as Oscn1 , Oscn2 , Oscn3 and Oscn4 .
First, we assume that the v−i characteristics of the nonlinear
resistor in each oscillator is given by the following third orderpolynomial equation.
iRk j = −g1 vk j + g3 vk j 3
where g1 , g3 > 0, k = 1, 2, ..., n, and j = 1, 2, 3, 4.
By using the following variables and parameters:
√√√g1
g1C
t = LCτ, vk j =xk j , ik j =yk j ,g3
g3 L(1)

ε=1g1√

L1, αi =CRi√

L, α a = RaC√L
1,β=Cr√x11x12x13x14x21x22x23x24x31x32
x33x34L,C

the normalized circuit equations are given as follows:(1) Top oscillators:
ẋn1 = ε (1 − xn1 2 )xn1 − (yn1a + yn1b ) + β (xn1 − x(n−1)1 )ẏn1a = 0.5 {xn1 − αa (yn1a + yn2b )}
ẏn1b = 0.5 xn1





ẋn2 = ε (1 − xn2 2 )xn2 − (yn2a + yn2b ) + β (xn2 − x(n−1)2 )ẏn2a = 0.5 {xn2 − αa (yn2a + yn3b )}





 ẏn2b = 0.5 {xn2 − αa (yn1a + yn2b )}ẋn3 = ε (1 − xn3 2 )xn3 − (yn3a + yn3b ) + β (xn3 − x(n−1)3 )ẏn3a = 0.5 {xn3 − αa (yn3a + yn4b )}ẏn3b = 0.5 {xn3 − αa (yn2a + yn3b )}ẋn4 = ε (1 − xn4 2 )xn4 − (yn4a + yn4b ) + β (xn4 − x(n−1)4 )ẏn4a = 0.5 xn4ẏn4b = 0.5 {xn4 − αa (yn3a + yn4b )}Computer simulation results (time waveform) for n = 3.

Fig. 2.

x31x32x11x21x34x33

x11x22x11x11x24

x23

(2)

(2) Middle oscillators (k = 2, 3, · · · , n − 1) :ẋk1 = ε (1 − xk1 2 )xk1 − yk1 + β (x(k+1)1 − 2xk1 + x(k−1)1 )ẏk1 = xk1





ẋk2 = ε (1 − xk2 2 )xk2 − yk2 + β (x(k+1)2 − 2xk2 + x(k−1)2 ) ẏk2 = xk2
ẋk3 = ε (1 − xk3 2 )xk3 − yk3 + β (x(k+1)3 − 2xk3 + x(k−1)3 )ẏk3 = xk3
ẋk4 = ε (1 − xk4 2 )xk4 − yk4 + β (x(k+1)4 − 2xk4 + x(k−1)4 ) ẏ = xk4k4x11x11x12x11

Fig. 3.

(3)(3) Bottom oscillators:
ẋ11 = ε (1 − x11 2 )x11 − y11 + β (x21 − x11 ) − αi (x11 − x12 )





ẏ11 = x11





ẋ12 = ε (1 − x12 2 )x12 − y12 + β (x22 − x12 )+ αi (x11 − 2x12 + x13 ) ẏ12 = x12
ẋ13 = ε (1 − x13 2 )x13 − y13 + β (x23 − x13 )+ αi (x12 − 2x13 + x14 )
ẏ
=x

1313



ẋ14 = ε (1 − x14 2 )x14 − y14 + β (x24 − x14 ) − αi (x14 − x13 ) ẏ = x1414(4)
where xk j corresponds to the voltage across the capacitor andyk j , yn ja , yn jb are the currents through the inductors of Osck j .
III. Synchronization Phenomena
Figures 2 and 3 show the computer simulation results forthe case of n = 3. Also, figures 4 and 5 show results for thecase of n = 4. The circuit equations (2)-(4) are calculated byusing the fourth-order Runge-Kutta method with the step size
h = 0.005. The circuit parameters are chosen as ε = 0.10,αi = αa = 0.5, and β = 0.02.
As we expected, the bottom four oscillators (Osc11 , Osc12 ,Osc13 and Osc14 ) are synchronized in in-phase and the top fouroscillators (Osc31 , Osc32 , Osc33 and Osc34 ) are synchronizedin anti-phase.

- 79 -x11x11x14x13x11x11x11x11

Computer simulation results (phase shift) for n = 3.

However, we observe unexpected synchronization phenomenon between the oscillators in each chain. Because theoscillators in each chain are coupled by r vertically, weexpected that the middle four oscillators (Osc21 , Osc22 , Osc23 ,Osc24 ,) are quasi-synchronization. And, the top four oscillators(Osc31 , Osc32 , Osc33 and Osc34 ) are which phase diﬀerence isaround 90[deg.]. Also, we can observe from Figs. 2 and 3
that the oscillators in the middle row (Osc21 , Osc22 , Osc23and Osc24 ) have some amount of phase shift to the first row,and those in the top row (Osc31 , Osc32 , Osc33 and Osc34 ) show
larger phase shift.
IV. Conclusions
In this study, we have observed a synchronization phenomenon from a circuit network which is composed of two
kinds of oscillator chains. Their edges were coupled to constrain their phase states to generate a frustration. By computer
simulations, we could observe interesting unexpected synchronization phenomenon.
The results in this study would be a good model of various
natural and artificial systems. For example, stone-paved squareof an old town (e.g. we observed one in Evora, Portugal) starts
from one edge of the square to line up stones in a regular
way like in-phase. However, sometimes the other edge of thesquare has a diﬀerent constraint like anti-phase. In that case,frustrations occur somewhere in the square and they are not[6] W. Wei, W. Zhou and T. Chen, “Cluster synchronization of linearly
coupled complex networks under pinning control,” IEEE Trans. Circuits
Syst. I, vol. 56, no. 4, pp. 829-839, Apr. 2009.
[7] Y. Uwate, Y. Nishio and R. Stoop, “Synchronization in three coupled
van der Pol oscillators with diﬀerent coupling strength,” Proc. NCSP’10,pp. 109-112, Mar. 2010.
[8] J. Lu, G. Chen and M. Bernardo, “On some recent advances in synchronization and control of complex networks,” Proc. ISCAS’10, pp. 37733776, May 2010.
[9] Y. Setou, Y. Nishio and A. Ushida, “Synchronization phenomena in many
oscillators coupled by resistors as a ring,” Proc. APCCAS’94, pp. 570575, Dec. 1994.
[10] Y. Setou, Y. Nishio and A. Ushida, “Synchronization phenomena inresistively coupled oscillators with diﬀerent frequencies,” IEICE Trans.
Fundamentals, vol. E79-A, no. 10, pp. 1575-1580, Oct. 1996.
[11] Y. Uwate, Y. Nishio and R. Stoop, “Synchronization in two polygonal
oscillatory networks sharing a branch,” Proc. NDES’10, pp. 62-65, May
2010.
[12] Y. Uwate and Y. Nishio, “Synchronizing Coupled Oscillators in Polygonal Networks with Frustration,” Proc. ISCAS’11, pp. 745-748, May 2011.

x11x12x13x14x21x22x23x24x31x32x33x34x41x42x43x44
Fig. 4.

Computer simulation results (time waveform) for n = 4.
x41x11x11x31x32x21Fig. 5.

x23x11x24x11x13x12x11x11x11x11x11x34x33x22x11x11x11x11x44x43x42x11x11x14x11x11

Computer simulation results (phase shift) for n = 4.

compensated at one or some particular points but over a wide
area of the square. That is one example of synchronization
phenomena observed from oscillator networks with frustration.
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