2012 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'12, Honolulu, Hawaii, USA. March 4-6, 2012Frustrated Synchronization in Coupled Oscillator Chainswith Unbalanced Parametric Distribution
Kosuke Matsumura, Takahiro Nagai, Hironori Kumeno, Yoko Uwate and Yoshifumi Nishio
Tokushima University
Tokushima 770-8506, JAPAN
Phone/FAX:088-656-7470/088-656-7471
Email: {kosuke, nagataka, kumeno, uwate, nishio}@ee.tokushima-u.ac.jp

frustrations by unbalanced parametric distribution. By using
computer simulations, we observe diﬀerent types of frustrated
In this study, we investigate frustrated synchronization in cou- synchronization states.
pled oscillator chains when symmetry of parametric distribution is broken. When one of coupling parameters is set to
diﬀerent value to the others, parametric distribution of cou- 2. Coupled Oscillator Chains [13]pled oscillator chains becomes unbalance. By using computer
In short, we explain about the coupled oscillator chains
simulations, we observe diﬀerent types of frustrated synchroproposed
and investigated in [13]. In this study, we focusnization states.
on the case of three row coupled oscillator chains. Figure 1
shows the circuit model for the case of 3 oscillator chains.
1. Introduction
Each oscillator-chain consists of 3 step van der Pol oscillators weakly coupled by resistors r. In this figure, the bottom
Coupled oscillators are suitable model to analyze the natu- three oscillators are coupled by relatively strong resistors Ri1
ral phenomena. Because, it is easy to observe the synchro- and Ri2 , while the top three oscillators are coupled by also relnization phenomena of coupled oscillators by various ele- atively strong resistors Ra1 and Ra2 via inductors. The middle
ments and techniques. Also, synchronization phenomena are oscillators are not coupled with oscillators located in the horvery basic phenomena. In particular we can observe them izontal direction but weakly coupled vertically.
everywhere in nature. For example, vibration of a pendu2L2L
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lum, firefly luminescence, gate patterns of four-leg animals,iiiiii
two frogs using voice religiously, periodic swinging of candleRRflames, and so on. Many researchers have proposed diﬀerent
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coupled oscillatory networks and have discovered many interesting synchronization phenomena [1]-[8]. The researchCv
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group of the authors is also working on coupled oscillatory
networks. Especially, we have been interested in coupledoscillators whose connections cause some kinds of frustrar
rrtions [9]-[12].
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In the previous study, we proposed coupled oscillator
Cv
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Lichains whose coupling structures are constructed to causefrustrations between oscillators and investigated synchronization phenomena observed in the circuits [13]. InterestingRR
synchronization phenomena were observed and the observedsynchronization were theoretically analyzed.
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Cv
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In this study, we focus on synchronization state of coupled oscillator chains when symmetry of parametric distribution is broken. In order to investigate such situation, oneof the coupling parameters is set to diﬀerent value to theothers. Namely, coupled oscillatory chains obtain stronger Figure 1: Circuit model for the case of 3 oscillators-chains.
Abstract

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The coupling structure on the bottom or in vertical couplings (by Ri1 and Ri2 or r) tends to make the oscillators tosynchronize in in-phase. While the coupling structure on thetop (by Ra1 and Ra2 via inductor) tends to make the oscillators
to synchronize in anti-phase.
We define the bottom three oscillators as Osc11 , Osc12 andOsc13 from the left, those on the middle row as Osc21 , Osc22
and Osc23 , and the top three oscillators as Osc31 , Osc32 andOsc33 .
The v − i characteristics of the nonlinear resistor in each
oscillator is given by the following third order polynomialequation.
iRk j = −g1 vk j + g3 vk j 3(1)

where xk j corresponds to the voltage across the capacitor andyk j , y3 ja , y3 jb are the currents through the inductors of Osck j .
In this system, we observed the interesting synchronization phenomenon in the previous study. Figures 2 and 3
show time waveforms of the oscillators and phase shifts between the oscillators when the parameters are set as ε = 0.10,
αi1 = αi2 = αa1 = αa2 = 0.5, and β = 0.02. The oscillatorsin the middle row have some amounts of phase shits to thebottom row, and those in the top row have twice amounts ofthe phase shifts between the middle row and the bottom row.
The observed synchronization was explained with theoreticalanalysis by considering the total power consumption of thecoupling resistors.

where g1 , g3 > 0, k = 1, 2, 3, and j = 1, 2, 3.
By using the following variables and parameters:√√√
g1
g1Ct = LCτ, vk j =x k j , ik j =yk j ,g3
g3 L
1
ε=g1

√

L
1, αin =CRin√L, αan = RanC√

L
1,β=Cr√L,
C

where n = 1, 2 and the normalized circuit equations are given
as follows:(1) Top oscillators:
ẋ31 = ε (1 − x31 2 )x31 − (y31a + y31b ) − β (x31 − x21 )ẏ31a = 0.5 {x31 − αa1 (y31a + y32b )}ẏ31b = 0.5 x31





ẋ
= ε (1 − x32 2 )x32 − (y32a + y32b ) − β (x32 − x22 ) 32
ẏ32a = 0.5 {x32 − αa2 (y32a + y33b )}ẏ32b = 0.5 {x32 − αa1 (y31a + y32b )}ẋ33 = ε (1 − x33 2 )x33 − (y33a + y33b ) − β (x33 − x23 )




ẏ33a = 0.5 x33
 ẏ33b = 0.5 {x33 − αa2 (y32a + y33b )}(2) Middle oscillators:
ẋ21 = ε (1 − x21 2 )x21 − y21 + β (x31 − 2x21 + x11 )ẏ21 = x21





 ẋ22 = ε (1 − x22 2 )x22 − y22 + β (x32 − 2x22 + x12 )ẏ22 = x22





ẋ23 = ε (1 − x23 2 )x23 − y23 + β (x33 − 2x23 + x13 ) ẏ = x23
23x11x12x13x21x22x23x31x32x33
τFigure 2: Time waveforms of the oscillators.

(2)x32x31x11x33x11x23x22x21x11(3)x11(3) Bottom oscillators:
ẋ11 = ε (1 − x11 2 )x11 − y11 + β (x21 − x11 ) − αi1 (x11 − x12 )ẏ11 = x11





ẋ = ε (1 − x12 2 )x12 − y12 + β (x22 − x12 ) 12+ αi1 (x11 − x12 ) + αi2 (x13 − x12 )




ẏ
=x
1212



ẋ13 = ε (1 − x13 2 )x13 − y13 + β (x23 − x13 ) − αi2 (x13 − x12 ) ẏ = x1313(4)- 680 -x11x11x12x11x11x13x11

Figure 3: Phase shifts between the oscillators.

x11

Figure 4: Phase diﬀerences between oscillators (Osc22 ,Osc31 , Osc32 and Osc33 ) and Osc11 by changing αa2 .

x31x32x22x21x11x11x31

x11x11x21x11x22x11x11

Figure 5: An example of phase shifts between oscillators(αa2 = 0.01).

x11x23x11x11x13x12x11x33

x11x11x13x11x32x11x23x12x11x33

x11x11Figure 6: Phase diﬀerences between oscillators (Osc13 ,Osc22 , Osc32 and Osc33 ) and Osc11 by changing αi2 .

x11x11Figure 7: An example of phase shifts between oscillators(αi2 = 0.0033).

3. Coupled Oscillator Chains with Unbalanced Paramet- of phase diﬀerences between oscillators (Osc22 , Osc31 , Osc32
ric Distributionand Osc33 ) and Osc11 by changing of αa2 . Figure 5 shows anexample of phase shifts between oscillators when αa2 is quite
We investigate synchronization in the coupled oscillator small.
chains when symmetry of parametric distribution is broken.
With decreasing αa2 , the coupling strength between Osc32
Namely, one of the coupling parameters is set to diﬀerent and Osc33 becomes weak, then, Osc33 tend toward in-phasevalue to the others. We consider two cases of unbalanced synchronization with Osc11 via Osc23 . Moreover, the phaseparametric distribution as follows:
diﬀerences between oscillators (Osc31 and Osc32 ) and Osc11
tend toward |90| [deg.], the phase diﬀerences between oscilla1. Changing the coupling parameter (αa2 ) between Osc32 and tors (Osc21 and Osc22 ) and Osc11 tend toward |45| [deg.]. In
Osc33 of the top oscillators.
the parameter region where αa2 is relatively large, αa2 is from
2. Changing the coupling parameter (αi2 ) between Osc12 and 0.1 to 0.5, changes of the phase diﬀerences are small, where
Osc13 of the top oscillators.
as, in the parameter region where αa2 is small, αa2 is from 0to 0.1, the phase diﬀerences rapidly change. The phase dif(A) Synchronization in dependence on αaference between Osc31 and Osc32 and the phase diﬀerences
We investigate synchronization when αa1 is set at 0.5 and between Osc21 and Osc22 respectively keep 180 [deg.] andαa2 varies, where the other parameters are the same as the val- 90 [deg.] for any αa2 .
ues presented in the previous section. Figure 4 shows change- 681 -

(B) Synchronization in dependence on αi
We investigate synchronization when αi1 is set at 0.5 andαi2 varies, where the other parameters are the same as the values presented in the previous section. Figure 6 shows changeof phase diﬀerences between oscillators (Osc13 , Osc22 , Osc32and Osc33 ) and Osc11 by changing of αi2 . Figure 7 shows anexample of phase shifts between oscillators when αi2 is quitesmall.
With decreasing αi2 , the coupling strength between Osc12and Osc13 becomes weak, then, (Osc13 and Osc23 ) are synchronized with Osc33 in the in-phase. Moreover, the phase
diﬀerences between oscillators (Osc31 and Osc32 ) and Osc11
tend toward |90| [deg.], the phase diﬀerences between oscillators (Osc21 and Osc22 ) and Osc11 tend toward |45| [deg.]. In
the parameter region where αi2 is relatively large, αi2 is from0.1 to 0.5, changes of the phase diﬀerences are small, whereas, in the parameter region where αi2 is small, αi2 is from 0to 0.1, the phase diﬀerences rapidly change. The phase difference between Osc31 and Osc32 and the phase diﬀerences
between Osc21 and Osc22 respectively keep 180 [deg.] and90 [deg.] for any αa2 .
We could confirm diﬀerent synchronization phenomena
which are caused by asymmetric couplings.

[2] T. Endo and S. Mori, “Mode analysis of two-dimensional
low-pass multimode oscillator,” IEEE Trans. Circuits
Syst., vol. 23, pp. 517-530, Sep. 1976.

4. Conclusions[8] J. Lu, G. Chen and M. Bernardo, “On some recent advances in synchronization and control of complex networks,” Proc. ISCAS’10, pp. 3773-3776, May 2010.

In this study, we investigated synchronization state of coupled oscillator chains with unbalanced parametric distribution. By using computer simulations, we observed diﬀerent
types of frustrated synchronization states.
The results in this study would be a good model of various natural and artificial systems. For example, stone-paved
square of an old town (e.g. we observed one in Evora, Portugal) starts from one edge of the square to line up stones ina regular way like in-phase. However, sometimes the otheredge of the square has a diﬀerent constraint like anti-phase.
In that case, frustrations occur somewhere in the square andthey are not 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.
Acknowledgment
This work was partly supported by JSPS Grant-in-Aid forYoung Scientists 23700269.

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