Analysis of Synchronization Phenomenonin Coupled Oscillator Chains
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 analyze a synchronization phenomenon observed from a circuit network consisting of several number of oscillator chains which are one-dimensional
arrays of weakly coupled van der Pol oscillators. Computer
simulations and circuit experiments show interesting unexpectedsynchronization phenomenon and theoretical analysis explains
the synchronization state.

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.

I. Introduction
Synchronization phenomena are basic but important phenomena observed everywhere in natures, for example vibrationof a pendulum, firefly luminescence, gait patterns of four-leg
animals, two frogs using voice religiously, periodic swinging
of candle flames, and so on. Coupled oscillators are goodmodels to investigate such interesting synchronization phenomena. Many researchers have proposed diﬀerent coupledoscillators 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 have been interested in coupled oscillators
whose connections cause some kinds of frustrations [9]-[12].
In this study, we consider a circuit network consisting ofseveral number of oscillator chains which are one-dimensional
arrays of weakly coupled van der Pol oscillators. In thisnetwork, we couple the oscillators at one edge of the chains
to constrain them to produce in-phase synchronizations. While
we couple the oscillators at the other edge of the chains toproduce anti-phase synchronizations. Middle oscillators in thechains are not coupled with the other chains. From computer
simulations and circuit experiments, we observe relatively
interesting unexpected synchronization phenomenon. By considering the total power consumption of the coupling resistors,
we can explain the stable synchronization state theoretically.
The results in this study would be a good model of various
natural and artificial systems.
II. Circuit Model
Figure 1 shows the circuit model for the case of 3 oscillator
chains. Each oscillator-chain consists of n van der Pol oscillators weakly coupled by resistors r. In the figure, bottom threeoscillators are coupled by relatively strong resistors Ri , while
the top three oscillators are coupled by also relatively strong
resistors Ra via inductors. Middle oscillators are not coupledwith oscillators located in the horizontal direction but weakly
coupled vertically.

978-1-4673-0219-7/12/$31.00 ©2012 IEEE620

Fig. 1.

Circuit model for the case of n oscillator-chains.

We define the bottom three oscillators as Osc11 , Osc12 , andOsc13 from the left, those on the kth row from the bottom as
Osck1 , Osck2 , and Osck3 , and the top three oscillators as Oscn1 ,Oscn2 , and Oscn3 .
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.
By using the following variables and parameters:
√√√
g1
g1C
t = LCτ, vk j =xk j , ik j =yk j ,g3
g3 L

(1)

ε=1
g1

√

L
1, αi =CRi√L1, αa =CRa√

L
1,β=Cr

√

x31
L,
C

x32

x33
x11x11x11

the normalized circuit equations are given as follows:(1) Bottom oscillators:
ẋ11 = ε (1 − x11 2 )x11 − y11 + β (x21 − x11 ) − αi (x11 − x12 )




ẏ11 = x11





ẋ = ε (1 − x12 2 )x12 − y12 + β (x22 − x12 ) 12+ αi (x11 − 2x12 + x13 )




ẏ
=x

1212




ẋ13 = ε (1 − x13 2 )x13 − y13 + β (x23 − x13 ) − αi (x13 − x12 ) ẏ = x1313(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 ) ẏ = xk3
k3(3)(3) 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





ẋ = ε (1 − xn2 2 )xn2 − (yn2a + yn2b ) − β (xn2 − x(n−1)2 )
 n2
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 ẏn3b = 0.5 {xn3 − αa (yn2a + yn3b )}(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. The circuit equations (2)-(4) are calculated
by using the fourth-order Runge-Kutta method with the step
size h = 0.02. The circuit parameters are chosen as ε = 0.10,
αi = αa = 0.5, and β = 0.02.
As we expected, the bottom three oscillators (Osc11 , Osc12 ,and Osc13 ) are synchronized in in-phase and the top threeoscillators (Osc31 , Osc32 , and Osc33 ) are synchronized in antiphase.
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 three oscillators in the left chain (Osc11 ,Osc21 , and Osc31 ) and in the right chain (Osc13 , Osc23 , andOsc33 ) are synchronized in in-phase. However, we can observefrom Figs. 2 and 3 that the oscillators in the second row (Osc21and Osc23 ) have some amount of phase shift to the first row,and those in the top row (Osc31 and Osc33 ) show larger phaseshift.

621x23x22x21x11x13x12x11x11x11x11x11x11

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

Fig. 2.

x11x12x13x21x22x23x31x32x33
τFig. 3.

Computer simulation results (time waveform) for n = 3.

We could observe the same interesting unexpected synchronization in real circuit experiments. Figure 4 shows the circuit
experimental results corresponding to Fig. 2.
IV. Theoretical Analysis
In order to explain the above-mentioned interesting synchronization phenomenon, we focus on the power consumption of
the coupling resistors r in each chain.
First of all, we assume that
1. Oscillators in the bottom and the top rows are synchronized
in in-phase and anti-phase, respectively.
2. Phase shift between the neighboring two rows are the same.
These assumptions are reasonable, because of the couplingstrength of the bottom and the top rows and the symmetry ofthe coupling structure.
Under the above assumption, the voltage of each oscillatorfind the minimum value of P3 , we diﬀerentiate P3 by θ.
dP3 2V 2=(2 sin θ − cos θ) = 0.
dθr(10)Calculating Eq. (10), we obtainθ = arctan

1= 26.56 [deg.]2(11)Table I summarizes the phase shifts of all the oscillators
obtained by computer simulations and the theoretical analysis.
Note that the single value of θ gives all the phase shifts
in theoretical analysis, namely we can calculate all thesedata from Eq. (5) by substituting the value of Eq. (11).
We can see that computer simulation results agree with thedata obtained from the theoretical analysis. Small amount oferrors are considered to come from the assumptions of thev − i characteristics of the nonlinear resistors Eq. (1) and theconstant voltage amplitude Eq. (5).
Fig. 4. Circuit experimental results (phase shift) for n = 3. C = 22nF, L =10mH, Ri = Ra = 1.0kΩ, and r = 4.3kΩ. Horizontal and vertical: [5.0 V/div].

TABLE I
Phase shifts of oscillators for n = 3.

can be written as
vk1 = vk3 = Vm cos{τ + (k − 1)θ},{}(k − 1){(n − 1)θ − π}vk2 = Vm cos τ +.
n−1

(5)

where θ is a phase shift between two neighboring rows. Also,
we define nonlinear oscillators as a first-order approximation.
Next, we consider the power consumption of the resistors in
the left and the right chains. Because the phase shift between
the two oscillators in these chains is θ, the average power overone period can be calculated as∫
1 2π
V2PI =(vk1 − v(k+1)1 ) dτ =(1 − cos θ).
(6)r 0r
Next, the power consumption of the resistors in the middle
chain can be calculated as follows, because the phase shift
between the two oscillators in this chains is θ − π/(n − 1).
∫(1 2πV2 {π )}PII =(vk2 −v(k+1)2 ) dτ =1 − cos θ −. (7)r 0rn−1Hence, the total power consumptions are given asPn==

2 (n − 1) PI + (n − 1) PII((n − 1) V 2 {π )}3 − 2 cos θ − cos θ −. (8)rn−1(A) Case of n=3:
We consider the case of n = 3. Eq. (8) can be written as(2V 2 {π )}P3 =3 − 2 cos θ − cos θ −.
(9)
r2
It is considered that the total power consumption takes theminimum value at the stable synchronization state. In order to622

Osc11
Osc12
Osc13
Osc21
Osc22
Osc23

Simulation[deg.]
00.94
0.15
23.56
64.07
23.56

Theory[deg.]
000
26.56
63.44
26.56

Osc31
Osc32
Osc33

Simulation[deg.]
47.31
129.13
47.31

Theory[deg.]
53.13126.87
53.13(B) Case of n=9:
Next, we consider the case of n = 9. Computer simulated
results are shown in Figs. 5 and 6. We can observe interesting
synchronization phenomenon similar to the case of n = 3,
namely the bottom oscillators are synchronized in in-phase andthe top oscillators are synchronized in anti-phase, while themiddle oscillators (only the oscillators in 3rd, 5th, 7th rows areshown in the figure) look like non-synchronization. However,these phase shifts can be also explained by theoretical analysis
by considering the total power consumption.
For the case of n = 9, Eq. (8) can be written as(8V 2 {π )}P9 =3 − 2 cos θ − cos θ −.
(12)r
8Hence, the stable state can be given by calculating the following equation.
(π )}dP9 8V 2 {=2 sin θ + sin θ −= 0.
(13)
dθ
r
8Calculating Eq. (13) by the Newton method, we obtainθ = 7.45 [deg.](14)

Table II summarizes the phase shifts of some oscillators forthe case of n = 9. The computer simulation results and thetheoretical analysis agree even better than the case of n = 3.

x91x92x93

x11x11x72x71

x11x73x11x52x11x32x11x11

τx11

Fig. 6.

x13x12x11x91x92x93

x33
x11

x11

x51
x52
x53x71
x72
x73

x53
x11

x31

x33

x11

x11
x51

x11
x12
x13
x31
x32

x11

Computer simulation results (time waveform) for n = 9.

frustrations occur somewhere in the square and they are notcompensated 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.

x11

Acknowledgment
Fig. 5.

Computer simulation results (phase shift) for n = 9.

This work was partly supported by JSPS Grant-in-Aid forYoung Scientists 23700269.

TABLE II
Phase shifts of oscillators for n = 9.

Osc11
Osc12
Osc13
Osc31
Osc32
Osc33
Osc51
Osc52
Osc53

Simulation[deg.]
00.32
0.022
14.62
30.67
15.90
29.90
59.56
32.47

Theory[deg.]
000
14.90
30.10
14.90
29.80
60.20
29.80

Osc71
Osc72
Osc73
Osc91
Osc92
Osc93

Simulation[deg.]
46.08
87.37
49.98
63.12
115.1963.85References
Theory[deg.]
44.70
90.30
44.70
59.60
120.4059.60

V. Conclusions
In this study, we have investigated a synchronization phenomenon observed from coupled oscillator chains whose edges
were coupled to constrain their phase states to generate afrustration. By computer simulations and circuit experiments,
we could observe interesting unexpected synchronization phenomenon. Theoretical analysis considering the total power
consumption of the coupling resistors could explain the stablesynchronization state.
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,623[1] T. Endo and S. Mori, “Mode analysis of a multimode ladder oscillator,”
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