2014 International Symposium on Nonlinear Theory and its Applications
NOLTA2014, Luzern, Switzerland, September 14-18, 2014

Frustrations in Strongly Coupled Polygonal Oscillatory Networks
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
Abstract—In this study, we investigate synchronization
phenomena in two coupled polygonal oscillatory networks
with strong frustrations. We focus on the amplitude of each
oscillator when the coupling strength is increased. By using the computer simulations, we confirm that the amplitude of the oscillators decreases by increasing the couplingstrength and oscillation death is occurred at un-frustrated
oscillators.
1. Introduction
The synchronization phenomena observed from coupledoscillators are suitable model to analyze the natural phenomena. Therefore, many researchers have proposed different coupled oscillatory networks and have discovered
many interesting synchronization phenomena [1]-[4].
In our research group, we have focused on synchronization phenomena of coupled oscillators under a diﬃcult situation for the circuit. Setou et al. have reported the synchronization phenomena in N oscillators coupled by resistors asa ring. The oscillation stop in some range of the coupling
resistors was confirmed [5].
We have investigated the synchronization phenomenain the coupled polygonal oscillatory networks sharing
branches [6]. In this system, van der Pol oscillators areconnected to every corner of polygonal network. By using
computer simulations and theoretical analysis, we confirm
that the coupled oscillators tend to synchronize to minimize
the power consumption of the whole system. The phase
diﬀerence of the shared oscillators is solved by finding theminimum value of the power consumption function. Furthermore, we have discussed synchronization state of the
proposed oscillatory networks by changing the coupling
parameter [7]. In this circuit system, we confirm that theamplitude of the shared and the other oscillators obtains
diﬀerent value. The torus attractors can be also observedin the certain parameter region. However, we could not observe oscillation death even if the coupling strength is set
to very large value. In the circuit model of Ref. [7], theearth resistances are missing in the 3rd and the 4th oscillators. Namely, the network topology changes to just ringoscillators coupled by the inductors if the coupling strength

becomes large.
In this study, we consider new circuit model which isincluding the earth resistance in all ground parts. The synchronization phenomena in coupled polygonal oscillatory
networks with strong frustration are investigated. We confirm that the amplitude of the oscillators decreases by increasing the value of the coupling strength and oscillation
death is occurred at un-frustrated oscillators. We expectthat the results of this study contribute to understanding ofsynchronization phenomena observed in general complex
networks.
2. Two Coupled Oscillatory Networks
2.1. Symmetric Model
Two identical polygonal oscillatory networks are coupled by sharing a branch as shown in Fig. 1. In this circuit model, we consider the coupling method which twoadjacent oscillators are tend to synchronize at anti-phasestate. We call the first and the second oscillators which areconnected to both side of polygonal network “shared oscillators.”
Figure 2 shows the circuit model of the 3 − 3 coupling
networks. The novel part of this study is that the earth resistances are inserted to the 3rd and the 4th oscillators.
1st osc.

5th osc.

4th osc.

6th osc.
1st osc.

4th osc.

7th osc.
2nd osc.

3rd osc.

2nd osc.

3rd osc.

8th osc.

(a) 3-3 coupling network. (b) 5-5 coupling network.
7th osc.

8th osc.

6th osc.

9th osc.

1st osc.

5th osc.

10th osc.
2nd osc.

4th osc.

3rd osc.

12th osc.

11th osc.

(c) 7-7 coupling network.

Figure 1: Two Coupled Oscillatory Networks (SymmetricModel).

- 377 -ic1

3L

NEW

rm

iR4

3L

3L

3L

rm

3L

rm

ib2

ia2

3L

3L

Cv4

RR

4th oscillator

ic2

rm

i R3
R

rm

ib3

3L

R

ia3

3L

ia4

rm

v1

ic3

RC
R

3L

rm

3L

i R1

ib4 ic4

3L
rm

ia1

rm

rm

1st oscillator
ib1

iR2
Cv3

RCv2NEW

2nd oscillator

3rd oscillator

Figure 2: Coupling Model (3 − 3 coupling networks).
Next, we develop the expression for the circuit equationsof 3 − 3 coupling oscillatory networks as shown in Fig. 2.
The vk − iRk characteristics of the nonlinear resistor are approximated by the following third order polynomial equation,iRk = −g1 vk + g3 vk 3 (g1 , g3 > 0), (k = 1, 2, 3, 4).

Next, Fig. 4 shows the observed attractors of 5 − 5 coupling network by changing the coupling strength. In thiscircuit model, the amplitude of fourth oscillator (which islocated farthest place from the shared oscillators) decreases
with the coupling strength. We observe the oscillation
death of the fourth oscillator when the coupling strengthis set to γ = 1.0. In the case of 7 − 7 coupling network,
we observe similar results with 5 − 5 coupling network as
shown in Fig. 5.
Figure 6 shows the change of the amplitude observed
from each network. In the case of 3 − 3 coupling network,
oscillation death of all oscillators is occurred at same time.
While, in the cases of 5 − 5 and 7 − 7 coupling networks,
first, the oscillation death of the oscillators located farthestplace from the shared oscillators is occurred. After that, theother oscillators stop to oscillate at same time.

y1

y2
x1

√

g1
xk , iak =3g3

√

g1
3g3

√x3
y3

x4
y4(B) γ=0.4

y2

y1

x3

x2

x1

x4
y4

y3

(C) γ=0.6

x2

x1

x4

x3
y3

y2

y1

y4

(D) γ=0.7

x3

x2

x1
(a) 1stLCτ, vk =x2y2y1

where√y4(A) γ=0.1(1)

The normalized circuit equations governing the circuit are
expressed as[kth oscillator]
(dxk1 2)



=ε1−
xk xk − (yak + ybk + yck )



dτ3


{}


dyak 1=xk − ηyak − γ(yak + yn )



dτ
3{
}
 dy
1(2)
bk


=x−
ηy−γ(y+y)kbkbkn

dτ3

}


dyck 1 {=x−ηy−γ(y+y)kckckn

dτ3




(k = 1, 2, 3, 4).

t=y3(b) 2ndx4(c) 3rd(d) 4th

Figure 3: Attractor (3-3 coupling network).

Cyak ,L√√
g1CL
ibk =ybk , ε = g1,
3g3
LC
√√
CC
γ=R, η = rm,
LL
√

y1

y2

y3

y5

y4(A) γ=0.1

x1

In this equations, γ is the coupling strength, ε denotes thenonlinearity of the oscillators and yn denotes the current ofneighbor oscillator on coupling resistor.
Figure 3 shows the observed attractors of 3 − 3 coupling
network by changing the coupling strength. When the coupling strength is weak, the amplitude of all oscillators arealmost same as shown in Fig. 3(A). By incrasing the valueof the coupling strength, the amplitude of the shared oscillators becomes smaller than the others (Fig. 3(B), (C)).
When the coupling strength is larger than γ=0.8, oscillation
death of all oscillators is observed as shown in Fig. 3(D).

- 378 -x3
y3

y2

y1

(k = 1, 2, 3, 4).

x2

x5x4
y5

y4(B) γ=0.4

y2

y1x3

x2

x1

y4

y3

x5

x4
y5(C) γ=1.0x2

x1y3y2y1x4x3x5y5y4(D) γ=1.5x1(a) 1stx3x2(b) 2nd(c) 3rdx5x4(d) 4th(d) 5thFigure 4: Attractor (5-5 coupling network).

y1y2

y3

2.2. Asymmetric Modely5y4(A) γ=0.1

x1x2
y3y2

y1x3x5x4
y5y4

Next, we considr the asymmetric models such as 3 − 5coupling network, 3−7 coupling network and 5−7 couplingnetwork. The asymmetric models are shown in Fig. 7.

(B) γ=0.4

4th osc.

y2

y1x3x2x1

y4

y3x5x4

1st osc.

y5(C) γ=1.0

x2

x1

x4

x3
y3

y2

y1

5th osc.

1st osc.

4th osc.

6th osc.

3rd osc.
2nd osc.

2nd osc.

y5

y4

5th osc.

3rd osc.

x5

7th osc.

6th osc.

(D) γ=1.5

8th osc.

(a) 3-5 coupling network. (b) 3-7 coupling network.

x3

x2

x1
(a) 1st(b) 2ndx5

x4(c) 3rd

6th osc.

(d) 5th(d) 4th

5th osc.

7th osc.

1st osc.

Figure 5: Attractor (7-7 coupling network).
8th osc.

4th osc.
2

2nd osc.
3rd osc.

x1 (1st osc.)x3 (3rd osc.)

1.8
1.6

9th osc.

(c) 5-7 coupling network.

1.4

amplitude

10th osc.

1.2

Figure 7: Two Coupled Oscillatory Networks (AsymmetricModel).

10.8
0.6
0.4
0.2
00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1γ

(a) 3-3 coupling network.
2x1 (1st osc.)x3 (3rd osc.)x4 (4th osc.)

1.8
1.6

amplitude

1.4
1.2
10.8
0.6
0.4
0.2
00

0.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

2(b) 5-5 coupling network.
2x1 (1st osc.)x3 (3rd osc.)x4 (4th osc.)x5 (5th osc.)

1.8
1.6

amplitude

1.4
1.2

Figure 8 shows the change of the amplitude observed
from the asymmetric networks. In the case of 3−5 coupling
network, oscillation death of all oscillators is occurred atsame time. While, in the cases of 3 − 7 and 5 − 7 coupling
networks, first the oscillation death of the oscillators located farthest place from the shared oscillators is occurred.
After that, the other all oscillators stop to oscillate at sametime. These results are similar with the results with thesymmetric network models. However, we observe the oscillation of the amplitude in 3 − 7 coupling network when
the coupling strength is set to around γ = 0.75.
Figure 9 shows one example or the change of the ampliutde when the coupling strength is set to γ = 0.75. In thiscase, we confirm that the amplitude change can be observed
constantly as shown in Fig. 10(a). There are two pairs (1st4th-5th and 2nd-7th-8th) of oscillation types of the amplitude and these two pairs oscillate with anti-phase state.
While the other two amplitudes (3rd and 6th) oscillate within-phase state. By increasing the coupling strength, the oscillation death of the 3rd and 6th oscillators is occurred.
Finally, we summarize the network types depending on theamplitude change as shown in Fig. 10(b).

10.8

3. Conclusions

0.6
0.4
0.2
000.5

1γ

1.5

2(c) 7-7 coupling network.
Figure 6: Amplitude (Symmetric Model).

In this study, we have investigated synchronization phenomena in coupled polygonal oscillatory networks withstrong frustrations. We focused on the amplitude of each
oscillator when the coupling strength is increased. By
using the computer simulations, we confirmed that theamplitude of the oscillators decreases by increasing the- 379 -

2

1.6
1.4

amplitudey1

x1 (1st osc.)x3 (3rd osc.)x4 (4th osc.)x5 (5th osc.)

1.8

y2

y3

x1

1.2

x2

x3

x4

y7

y6

y5

1

y4

y8

0.8
0.6

x7

x6

x5

0.4

x8

0.2
00

0.1

0.2

0.3

0.4

0.5

γ

0.6

0.7

0.8

0.9

Figure 9: Attractors (3-7 coupling network, γ=0.75).

1(a) 3-5 coupling network.
1st oscillator
2nd oscillator
3rd oscillator
4th oscillator
5th oscillator
6th oscillator
7th oscillator
8th oscillator

1.4

2

1.2

1.6

amplitude

1.4
1.2

amplitude

x1 (1st osc.)x3 (3rd osc.)x4 (4th osc.)x5 (5th osc.)x6 (6th osc.)

1.8

1

1
0.8
0.6
0.4

0.8
0.2

0.6
0800

0.4

850

900

0950

1000

Ta

0.2
0

0.1

0.2

0.3

0.4

0.5

γ

0.6

0.7

0.8

0.9

1(a) Amplitude(b) 3-7 coupling network.

445455111263x1 (1st osc.)x3 (3rd osc.)x4 (4th osc.)x6 (6th osc.)x7 (7th osc.)x8 (8th osc.)

1.8
1.6

amplitude

1.4
1.226322786

37878(b) Network Types.

Figure 10: Amplitude Change (3 − 7 coupling network,
γ=0.75).

10.8
0.6
0.4
0.2
00

0.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

tors: Synchronization and Clustering” Chaos, vol.10, no.1,
pp.248-256, Mar. 2000.

2(c) 5-7 coupling network.

[3] M. Yamauchi, Y. Nishio and A. Ushida, ”Phase-waves in aladder of oscillators” IEICE Trans. Fundamentals, vol.E86A, no.4, pp.891-899, Apr. 2003.

Figure 8: Amplitude (Asymmetric Model).
coupling strength and oscillation death is occurred at unfrustrated oscillators. Furthermore, we observed the amplitude change in the asymmetric network model.

[4] H.B. Fotsina and J. Daafouza, “Adaptive synchronization ofUncertain Chaotic Colpitts Oscillators based on Parameter
Identification” Physics Letters A, vol.339, pp.304-315, May2005.
[5] Y. Setou, Y. Nishio and A. Ushida, “Synchronization Phenomena in Many Oscillators Coupled by Resistors as aRing,” Proc. of APCCAS’94, pp. 570-575, Dec. 1994.

Acknowledgment
This work was partly supported by JSPS Grant-in-Aidfor Challenging Exploratory Research 26540127.
References[1] M.G. Rosenblum, A.S. Pikovsky, and J. Kurths, “Phase
Synchronization of Chaotic Oscillators” Physical Review
Letters, vol.76, no.11, pp.1804-1807, Mar. 1996.

[6] Y. Uwate and Y. Nishio, “Synchronization in Several Typesof Coupled Polygonal Oscillatory Networks,” IEEE Trans.
Circuits Syst. I, vol. 59, no. 5, pp. 1042-1050, May 2012.
[7] Y. Uwate and Y. Nishio, ”Frustrated Synchronization in
Two Coupled Polygonal Oscillatory Networks,” Proc. of ISCAS’13, pp. 1781-1784, May 2013.

[2] W. Wang, I.Z. Kiss, and J.L. Hudson, “Experiments on Arrays of Globally Coupled Chaotic Electrochemical Oscilla-- 380 -