2012 International Symposium on Nonlinear Theory and its Applications
NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012

Amplitude Analysis of Frustrated Systems of Coupled Oscillators
Yoko Uwate† , Yoshifumi Nishio† and Ruedi Stoop‡†Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp‡Institute of Neuroinfomatics, University / ETH Zurich,
Winterthurerstrasse 190, CH-8057 Zurich, Switzerland,
Email: ruedi@ini.phys.ethz.ch

2. Three Coupled Oscillators as a Ring Topology
2.1. Circuit Model and Equations
The circuit model of three coupled van der Pol oscillator
as ring topology is shown in Fig. 1.
1st oscillator
ia1
i R1

2L

rm

ib1

2L

C
v1
RR2L

rm

ib3

2L

ia3rm

Modeling of natural phenomena becomes very importantresearch topic to predict extreme weather consistent with
global warming, recently. Coupled oscillatory systems aregood models to express essential role of high-dimensional
nonlinear phenomena occurring in the field of natural sciences. Therefore, studies of synchronization phenomena
have been extensively reported in various areas such as
physical [1],[2], biological [3],[4] and electrical [5] systems.
On the other hand, there are several types of polygonal
network structures (e.g. Honeycomb structure and crystalstructure) in the natural science. Generally, for the studiesof large-scale network using coupled oscillators, a ring, aladder and a two dimensional array structure are often investigated. However, there are not many discussions about
coupled polygonal oscillatory networks by using electricaloscillators.
Furthermore, we have been interested in coupled oscillatory networks which has some sort of frustrations of network structure [6],[7]. Because, we consider that frustration have possibility to produce the other types of synchronization such as more complex phenomena and different
synchronization from the original system.
In our previous study, synchronization phenomena incoupled polygonal oscillatory networks with frustration
was investigated. In this system, odd number of van der
Pol oscillators are connected to every corner of polygonal
network and frustration is occurred by the shared branch.
We have confirmed that the phase difference between theshared oscillators was shifted, then other oscillators synchronized to compensate this phase shift. In order to solve
the phase difference in the circuit system, we focused on
the power consumption of the coupling resistors in thewhole system and proposed the theoretical analysis methodrm

1. Introduction

by finding the minimum value of the power consumptionfunction. By using computer simulations and theoreticalanalysis, we confirmed that coupled oscillators tended tosynchronize to minimize the power consumption of thewhole system [8]. In the proposed theoretical analysis, weuse the equal amplitude for all coupled oscillators. However, the accurate amplitude of coupled oscillators has different values by the effect of frustrations.
In this study, we propose amplitude analysis by using thepower consumptions of the coupling resistors and the nonlinear resistors in coupled oscillatory networks with frustrations.

2L

Abstract—By adding some kinds of frustrations to coupled oscillatory networks, interesting synchronization phenomena can be obtained. In this study, we propose amplitude analysis by using whole power consumption in coupled oscillatory networks with frustrations.

2Lrm

ib2

ia2
rm

i R3

iR2v3
C

3rd oscillator

RCv2

2nd oscillator

Figure 1: Three coupled van der Pol oscillators as a ringtopology.
In the computer simulations, we assume that the vk − iRkcharacteristics of the nonlinear resistor in each oscillator isgiven by the following third order polynomial equation- 848 -iRk = −g1 vk + g3 vk 3 (g1 , g3 > 0),(k = 1, 2, 3).

(1)

2

The normalized circuit equations governing the circuit inFig. 1 are expressed as[First oscillator]

1 2
dx1

=ε1−x1 x1 − (ya1 + yb1 )dτ3
 dya1 1(2)=x−ηy−βγ(y+y)

1a1a1b2

dτ2






dyb1 1 =x1 − ηyb1 − γ(ya3 + yb1 )dτ
2[Second oscillator]
dx21 = ε 1 − x2 2 x2 − (ya2 + yb2 )


dτ3




 dya2 1 =x−ηy−γ(y+y)2a2a2b3

dτ2



dy1
b2


=x2 − ηyb2 − βγ(ya1 + yb2 )dτ2[Third oscillator]dx31 2=ε1−x3 x3 − (ya3 + yb3 )

dτ
3





 dya3 1 =x−ηy−γ(y+y)3
a3a3b1

dτ2







dy1
b3


=x3 − ηyb3 − γ(ya2 + yb3 )dτ2√Amplitude

1.4
1.2
10.8
0.6
0.4
0.2
00.050.1

0.15

0.2

0.25

γ

0.3

0.35

0.4

0.45

0.5

Figure 2: Amplitude of oscillator by changing γ.

(3)

By using the following variable for the current of thenonlinear resistor,rr
g1
CiRk =yRk .
3g3 L
And we assume the voltage of the capacitor as the following equation:
x = Am sin τ.
(5)

(4)

The power of the nonlinear resistor in van der Pol oscillator is described as Eq. (6).
PNR =r

g1
xk ,
3g3r
r
r
r
g1
C
g1
C
iak =
yak , ibk =
ybk ,
3g3 L
3g3 LrrrLCC
ε = g1, γ=R, η = rm,CLL
t=1.612πZ 2π01{−εA2m sin2 τ + εA4m sin4 τ}2 dτ3(6)By solving Eq. (6),LCτ, vk =11
PNR = − εA2m + εA4m .
28(k = 1, 2, 3).
In this equations, γ is the coupling strength and ε denotes
the nonlinearity of the oscillators. For the computer simulations, we calculate Eqs. (2)-(4) using a fourth-order RungeKutta method with the step size h = 0.005. The parametersof this circuit model are fixed as ε = 0.1, η = 0.001. In
this circuit system, we observe three-phase synchronization(phase shift: 120◦ ).

(7)

Figure 3 shows the graph of Eq. (7) for one van der Pol
oscillator. From this figure, we confirm that the amplitude
shows ±2 when the power is zero. We can see that the amplitude is determined by the balance of the total power (thepower of the nonlinear resistors and the coupling resistors)
in the circuit system.
0.6
0.5
0.4
PNR

where1.8

0.3
0.2
0.1

2.2. Amplitude Analysis by using Power0We investigate the amplitude of three coupled oscillators
when the coupling strength γ is changed. The simulation
result is shown in Fig. 2.
We can see that the amplitude is decreased with the coupling strength. How do the amplitude of oscillators determine? In order to solve this question, first, we consider thepower of the nonlinear resistor of one van der Pol oscillator
without coupling.

-0.1
-3

-2

-1

0
Amplitude

12

3

Figure 3: Power of nonlinear resistor.
Next, we apply this approach to three coupled van der
Pol oscillators as ring topology. The power of the nonlinear
resistor is calculated by using above method (Eq. (7)). In- 849 -order to calculate the power of the coupling resistors, wedefine the current of the inductor is described as follows.
πAm



sin(τ + )y = a22(8)



A
5π


 yb = m sin(τ + )2
6

the coupling strength obtained from the theoretical analysis and the computer simulation. From this figure, we cansee that the both methods match well when the couplingstrength is small.
21.8Simulation
Theory1.6The power of the coupling resistor is expressed byEq. (8).
Z 2πγAmπ Am
5πPCR =sin2 (τ + +sin2 (τ + }dτ (9){2π 02226Amplitude

1.4
1.2
1
0.8
0.6
0.4

By solving Eq. (8),

0.2
00.05

1
PCR = − γA2m .
80.1

0.150.2(10)The total power is obtained by calculating the following
equation.
PALL = 3PNR + 3PCR .
(11)
Figure 4 shows the graph of Eq. (10) when the couplingstrength is fixed with γ=0.1. In this case, the amplitudeof the oscillators denotes 1.73 when the total power shows
zero. This result matches very well with the computer simulation result.

0.25

γ

0.3

0.35

0.4

0.45

0.5

Figure 6: Amplitude of oscillator by changing γ.

3. Two Coupled Polygonal Networks
In this section, we apply the theoretical analysis to findthe amplitude by using total power for two coupled polygonal oscillatory networks. The conceptual circuit model of
5-5 coupled networks is shown in Fig. 7.

2.5

PALL, PNR, PCR

5th osc.

PALLPNR
PCR26th osc.
1st osc.

1.5

1

4th osc.

0.5

7th osc.
2nd osc.

3rd osc.

0-0.5
-3-2

-1
-1.73

0

1

2

Figure 7: Two coupled odd-number-node polygonal oscillatory networks (5-5 coupled networks).

3

1.73

Amplitude

Figure 4: Total power of three coupled van der Pol oscillators as ring topology.
3.5
g=0.1
g=0.2
g=0.3
g=0.4

3
2.5

8th osc.

Table I summarizes the obtained amplitude by using
computer simulation when the coupling strength is fixedwith γ=0.1. We confirm that the amplitude of the sharedoscillators is smaller value than the other combination oscillators. Namely, we define two amplitudes Am and Bm forthe two coupled polygonal networks.

PALL

2
1.5

Table 1: Amplitude of 5-5 coupled networks (computersimulation)
Shared (1st-2nd) osc. Other combi.
1.933
1.986

10.5
0-0.5
-3-2-10
Amplitude

123Figure 5: Total power by changing γ.
Next, we show the graph of total power when the coupling strength γ is changed as shown in Fig. 5. By increasing the value of γ, the amplitude of the oscillators decreases. Figure 6 shows the change of the amplitude with

The power of the nonlinear resistor is obtained by
Eqs. (11), (12).
11



PNRa = − εA2m + εA4m



28

(12)



11


 PNRb = − εB2m + εB4m
28- 850 -The power of the coupling resistor is obtained by
Eqs. (13)-(15).
1PCRa = − γA2m {sin(4π + θ) − sin θ − π cos θ}

18



1

PCRab = − γAm Bm {sin(4π + θ) − sin θ − π cos θ}181 PCRb = − γB2m {sin(4π + θ) − sin θ − π cos θ}18(13)The total power is obtained by calculating the following
equation.
PALL = 2PNRa + 6PNRa + PCRa + 4PCRab + 4PCRb . (14)Figures 8 and 9 show the graph of the total power for
5-5 coupled networks. In Fig. 8, the amplitude Bm is set
to 1.98 which is obtained from the simulations. While, inFig. 9, the amplitude Am is set to 1.90 similary. From thesefigures, we obtain the amplitudes Am , Bm when the graphof total power shows zero.
Finally, the amplitude obtained from theoretical analysis
is summarized in Table II. We can see that these amplitudes
are very similar to the simulation results (see Table I).

Table 2: Amplitude of 5-5 coupled networks (theoreticalresult)
Shared (1st-2nd) osc. Other combi.
1.95
1.99

we have confirmed that the amplitude of coupled oscillators is decided by the balance of whole power consumption
in the circuit system. Furthermore, we have proposed thetheoretical analysis to solve the amplitude of oscillators byfinding zero value of the total power functions.
For the future work, we develop a efficiently method tofind the minimum value of the power function for the large
scale complex networks with many variables. Investigation of synchronization phenomena in coupled polygonal
oscillatory systems with strong nonlinearity or in coupled
polygonal chaotic oscillators is also our future work.
Acknowledgments
This work was partly supported by JSPS Grant-in-Aidfor Young Scientists 23700269.

1.2

PALL
PNR
PCR

PALL, PNR, PCR

1References

0.8
0.6
0.4
0.2
0-0.2
-3-2-1-1.95

0Amplitude

12

3

1.95

Figure 8: Total power of 5-5 coupled network (Am ).
3.5
PALLPNR
PCR

3PALL, PNR, PCR

2.5
2
1.5
1
0.5
0-0.5
-3-2-1-1.99

0Amplitude

12

3

1.99

Figure 9: Total power of 5-5 coupled network (Bm ).

4. Conclusions
In this study, we have proposed amplitude analysis byusing whole power consumption in coupled oscillatory networks with frustrations. By using computer simulations,[1] G. Abramson, V.M. Kenkre and A.R. Bishop, “Analyticsolutions for nonlinear waves in coupled reacting systems,” Physica A: Statistical Mechanics and its Applications, vol.305, nos.3-4, pp.427-436, Mar. 2002.
[2] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization and graph topology,” Int. J. Bifurcation andChaos, vol.15, no.11, pp.3423-3433, Nov. 2005.
[3] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronization of nonlinear electronic oscillators forneural computation,” IEEE Trans. Neural Networks, vol.15,no.5, pp.1315-1327, Sep. 2004.
[4] R. Stoop and C. Wagner, “Neocortex’s architecture optimizes computation, information transfer and synchronizability, at given total connection length,” International
Journal of Bifurcation and Chaos, vol.17, no.7, pp.22572279, 2007.
[5] H.B. Fotsina and J. Daafouza, “Adaptive synchronizationof uncertain chaotic colpitts oscillators based on parameter identification” Physics Letters A, vol.339, pp.304-315,May. 2005.
[6] Y. Uwate, Y. Nishio and R. Stoop, “Synchronization inthree coupled van der Pol oscillators with different couplingstrength,” Proc. of NCSP’10, pp. 109-112, Mar. 2010.
[7] Y. Uwate, Y. Nishio and R. Stoop, “Synchronization in Two
Polygonal Oscillatory Networks Sharing a Branch,” Proc.
of NDES’10, pp. 62-65, May 2010.
[8] Y. Uwate and Y. Nishio, “Synchronization in Several Typesof Coupled Polygonal Oscillatory Networks,” IEEE Transactions on Circuits and Systems I, vol. 59, no. 5, pp. 10421050, May 2012.

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