Frustrated Synchronization in Two Coupled
Polygonal Oscillatory Networks
Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— In this study, synchronization phenomena observedin coupled polygonal oscillatory networks with frustration isinvestigated. We focus on the amplitude of each oscillator when
the coupling strength is changed. By using computer simulations
and circuit experiments, we confirm that the amplitude ofthe shared and the other oscillators obtains different value
by increasing the value of the coupling strength. Furthermore,theoretical analysis is applied to solve the amplitude and thephase differences between the adjacent oscillators.

simulations and circuit experiments, we confirm that the amplitude of the shared and the other oscillators obtains different
value by increasing the value of the coupling strength. Furthermore, theoretical analysis is applied to solve the amplitude
and the phase differences between the adjacent oscillators. We
expect that the results of this study contribute to understandingof synchronization phenomena observed in general complex
networks.

I. I NTRODUCTION

II. T WO C OUPLED O SCILLATORY N ETWORKSFirst, we investigate the synchronization phenomena in two
coupled oscillatory systems. Two identical polygonal oscillatory networks are coupled by sharing a branch as shownin Fig. 1. In this circuit model, we consider the coupling
method which two adjacent oscillators are tend to synchronize
at anti-phase state. We call the first and the second oscillators
which are connected to both side of polygonal network “sharedoscillators.”
Nth osc.

4th osc.

(N+1)th osc.
(N+2)th osc.

6th osc.

1st osc.

5th osc.

(N+3)th osc.
2nd osc.

3rd osc.

4th osc.

2nd osc.

(b) N -N coupling (general) network.

(a) 3-3 coupling network.

Two coupled polygonal oscillatory networks.

ic1

3L

v1
rm

3L

ib3

rm

3L

ia4
rm

ib2

ia2

3L

3L

iR4
Cv4

R

4th oscillator

ic2

rm

i R3

iR2
Cv3

3rd oscillator

Fig. 2.

rm

ic3

R3L

3L

RCR

ia3

rm

3L

3L

i R1ib4 ic43L
rm

ia1

rm

rm

1st oscillator
ib1

3LFig. 1.

(N+4)th osc.
(2N-2)th osc.

3rd osc.

rm

978-1-4673-5762-3/13/$31.00 ©2013 IEEE

1st osc.

3L

Synchronization phenomena with coupled oscillators 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].
On the other hand, there are several types of polygonal
network structures (e.g. honeycomb, soup bubbles and cracks
of drying mud structure) in the natural science. Generally, forthe studies of large-scale network using coupled oscillators, aring, a ladder and a two dimensional array structure are often
investigated. However, there are not many discussions about
coupled polygonal oscillatory networks by using electricaloscillators.
In our research group, we focus 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 as a ring. The oscillation
stop in some range of the coupling resistors was confirmed [5].
We have investigated synchronization phenomena in coupled
polygonal oscillatory networks sharing branches [6]. In thissystem, van der Pol oscillators are connected to every cornerof polygonal network. By using computer simulations andtheoretical analysis, we confirm that coupled oscillators tend
to synchronize to minimize the power consumption of thewhole system. Furthermore, the phase difference of the sharedoscillators is solved by finding the minimum value of thepower consumption function.
However, we have not discussed synchronization state of
the proposed oscillatory networks by changing the couplingparameter. In this study, synchronization phenomena observedin coupled polygonal oscillatory networks with frustration isinvestigated. We focus on the amplitude of each oscillator
when the coupling strength is changed. By using computerRCv2

2nd oscillator

Circuit model for 3 − 3 coupling oscillatory networks.

Next, we develop the expression for the circuit equations of
3 − 3 coupling oscillatory networks as shown in Fig. 2. The1781

(g1 , g3 > 0), (k = 1, 2, 3, 4).

(1)

The normalized circuit equations governing the circuit are
expressed as[kth oscillator]
⎧

dxk1
⎪
⎪
= ε 1 − xk 2 xk − (yak + ybk + yck )
⎪⎪ dτ
3
⎪
⎪
⎪
1
dy ak
⎪
⎪
=xk − ηyak − γ(yak + yn )
⎪⎪
⎨ dτ
3

1dy bk(2)=xk − ηybk − γ(ybk + yn )
⎪
⎪
3
⎪ dτ
⎪

⎪
⎪
⎪ dy ck = 1 x − ηy − γ(y + y )⎪kckckn
⎪
⎪
3
⎪ dτ
⎩
(k = 1, 2, 3, 4).
In this equations, γ is the coupling strength, ε denotes thenonlinearity of the oscillators and yn denotes the current ofneighbor oscillator on coupling resistor.

110.5

0.5

00

yiRk = −g1 vk + g3 vk

3yvk − iRk characteristics of the nonlinear resistor are approximated by the following third order polynomial equation,

-0.5

-0.5

-1

-1
-2

-1.5

-1

-0.5

00.5

1

1.5

2

2.5

-2

-1.5

-1

-0.5

0x

Fig. 4.

x1x2

x3

x4

0.5

1

1.5

22.5

x

(a) 1st oscillator (γ=80.0)(b) 3rd oscillator (γ=200.0)
Poincare maps.

210-1-2
2
10-1-22
10-1-22
10-1-2

50000

60000

70000

80000

90000

100000τTime wave forms of 3-3 coupling network (γ=80.0).

Fig. 5.

A. Synchronization Phenomena
Figure 3 shows the attractor of each oscillator in 3-3
coupling network when the coupling strength is changed.
First, the amplitudes of the shared oscillators are decreased(Fig. 3(B)). By increasing the coupling strength, torus attractors are observed (Figs. 3(C), (D)). Poincare maps of torus
attractors are shown in Fig. 4.
Figures 5 and 6 show the time wave form of the voltage
charged at the capacitance of each oscillator obtained from thecomputer simulation. From this figure, we can see that the firstand the second oscillators are synchronized at in-phase (phasedifference: 0 degree). While, the other combination oscillators
synchronize with anti-phase (phase difference: 180 degree).
y2

y1y3

2
1x1 0-1-22
1x2 0-1-2
21x3 0-1-2
21x4 0-1-2

50000

60000

70000

τ8000090000

100000Time wave forms of 3-3 coupling network (γ=200.0).

Fig. 6.

y4(A) γ=0.10
x1x2
y2

y1x4x3y3y4(a) One periodic attractor.

(B) γ=10.0
x1x2
y2

y1x4x3y3Fig. 7.

(b) Torus attractor.

Attractors from circuit experiments.

y4(C) γ=80.0
x1x2
y2

y1

x4x3
y3y4(D) γ=200.0x1(a) 1stFig. 3.

x2(b) 2nd(a) v1 (R=5.1kΩ).

x4x3(c) 3rd(d) 4th

Attractors observed in 3-3 coupling network.

We also obtain the same synchronization states from thecircuit experiment as shown in Figs. 7, 8. The circuit experiments parameters are set to L = 50mH and C = 33nF . The
nonlinear resistor is realized by op-amp (TL082). The power
supply of the op-amp is fixed as ±12V .

1782(b) v3 (R=30.0 kΩ).
Fig. 8.

Time wave forms of torus from circuit experiments.

B. One Parameter Bifurcation

2.5

2.5

22

1.5

1.5x3x1

The one parameter bifurcation diagram of x1 and x3 isshown in Fig. 9. From this figure, we can confirm that theamplitudes of x1 and x3 have similar value when the couplingstrength is smaller than γ=0.02. Around γ=0.1, the amplitudeof x1 decreases and x3 keeps the amplitude almost 2.0. By
increasing γ, the both of the amplitudes start to decrease andtake extreme value around γ=1.5. After that the both of theamplitudes increase until γ=50.0. The torus attractor can beobserved in 1st oscillator (x1 ) from γ=50.0 to 100.0. When γreaches 100.0, the 1st oscillator switches one periodic attractor
and the 3rd oscillator generates torus attractor.
Generally, oscillation death can be occurred in symmetricalcoupled oscillatory system by increasing the coupling strength.
However, the oscillation death can not be observed in this circuit system. We consider that the unbalanced circuit structure
affect to synchronization state materially.

10.5

00-0.5
0.01

0.1

1γ(B)

(C)(D)

10100-0.5
0.01

0.1

1γ(C)(D)

10100

1000(b) x3 .
One parameter bifurcation.

200

Anti-phase
150

phase differencexk (τ ) = ρk sin(τ + θk ).

Shared oscillators
Other oscillators
10050In-phase
(A)

(B)

(C)(D)

10

100

0.1

1Fig. 10.

Phase difference.

γ(4)We pay attention to treat the non-resonance system and apply
for the averaging method to Eq. (2). The average method isdefined as following equations.
T0T
0

Fk cos(τ + θk )dτ(5)

1
Fk sin(τ + θk )dτρkBy solving the above equations, the phase difference andthe amplitude are obtained as follows.
Phase differences:θ1 − θ2 = 0.
θ2 − θ3 = θ3 − θ1 = π.

(6)

Amplitudes:
ερk 2γρ˙1 =(ρ − 4) +− 4x1 + x3 .
8 k2(4γ 2 + 9)ερk 2γρ˙3 =(ρk − 4) +− 2x3 + x1 .
82(4γ 2 + 9)

(7)
(8)

In the steady state, ρ˙k =0 must be satisfied. By using Newton
method for Eqs. (7) and (8), the amplitudes of ρ1 and ρ3 aresolved. In order to confirm the credibility of theoretical analysis, we compare the amplitude of oscillator of the theoreticalresult with the computer simulation result (see. Fig. 11). Fromthese results, we confirm that they match very well when thecoupling strength γ is smaller than 1.5. We consider that theerror between the theoretical and simulation results is occurredby harmonic component when the coupling strength γ is larger
than 1.5.

1800.01

Let us assume the solutions of Eq. (2) isT →∞Figure 10 shows the one parameter diagram of the phasedifference. The shared oscillators are synchronized at in-phaseand the other combination oscillators are synchronized at antiphase when the coupling strength is smaller than γ=100.0. The
all coupled oscillators synchronize with in-phase if γ is larger
than 100.0.

0Y3 ≡ ya3 + yb3 + ya2 + yb1 .

θ˙k = lim(a) x1 .
Fig. 9.

Y1 ≡ ya1 + yb1 + yc1 + ya3 + yb2 + yb4 ,T →∞

(B)

(3)

whereρ˙k = lim

(A)
1000

1dxk
d2 xk+ γYk ≡ Fk+ xk = ε(1 − xk 2 )dτ 2dτ
3(k = 1, 2, 3, 4)

10.5

(A)

3rd and 4th oscillators are same characteristics. Equation (2)is combined to give the second order nonlinear differential
equation as follows,

1000

2.5

2.5TheorySimulation

TheorySimulation
2In this section, we analyze the synchronization phenomena
by using the averaging method which is one of the theoreticalanalysis.
First, consider the case of that the tiny resistors to avoid Lloop for computer simulation is set to zero (η=0.0) in Eq. (2).
We assume that 1st and 2nd oscillators are same amplitude andphase difference by symmetrical circuit structure. Similarly,

1783

Amplitude

C. Theoretical Analysis

Amplitude

21.5

11

0.5
0.01

1.5

0.1

γ

1100.50.01

0.1γ(a) ρ1 .

(b) ρ3 .

Fig. 11.

Results of theoretical analysis.

110

In this section, we consider 7-7 coupling networks. The attractors are summarized in Fig. 13 when the coupling strengthis fixed with γ=10.0, 50.0 and 80.0. In the case of γ=10.0,
the all of oscillators generate one periodic attractors as shownin Fig. 13(a). When γ equals 50.0, the oscillation death canbe observed in 5th and 10th oscillators which are connectedmost far place from the shared oscillators (see Fig. 13(b)). By
increasing the value of γ, the 5th and the 10th oscillators start
to oscillate again (see Fig. 13(c)).

Figure 14(b) shows the amplitudes of each oscillator in thebubble network. We confirm that all amplitude have similarvalue when γ is smaller than 0.002. By increasing the couplingstrength, each amplitude has different value. One example ofattractors in the bubble network is shown in Fig. 15.
1.16
1.15

7

1.14

8

6
1
5

9AmplitudeIII. 7-7 C OUPLING O SCILLATORY N ETWORKS

8th osc.

6th osc.

1.12

1st oscillator
3rd oscillator
4th oscillator5th oscillator
8th oscillator
9th oscillator

1.11
1.1

2

7th osc.

1.13

1.09

4

9th osc.

1.08

103

1.07
0.001

1st osc.

0.01

0.1

γ(a) Bubble structure.

(b) Amplitude.

Fig. 14. Conceptual circuit model of bubble structure and result of amplitude.
5th osc.

10th osc.
2nd osc.

4th osc.

3rd osc.

Fig. 12.

y1

11th osc.

12th osc.

y2y2

x2x3(b) 2ndy7x9(h) 8th

y2x2x1(a) 1stx3(b) 2ndy7x9(h) 8th(f) 6th

y11x10(j) 10th(i) 9thx6(e) 5thy10x8x7(g) 7thx5(d) 4thy9y8y6x4(c) 3thy12x12(l) 12thx11(k) 11th(b) γ=50.0.
y2y1x2x1(a) 1stx3(b) 2ndy7y10x9x8x7(i) 9thx5(d) 4thy9y7y8y9y10x11(k) 11thx7(g) 7thx8(h) 8thx9(i) 9thx10(j) 10th

Attractors of bubble network (γ=400.0).

R EFERENCESx6(e) 5thy11x10(j) 10thy6ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid forYoung Scientists 23700269.

y6x4(c) 3th

y8y5y4y3(e) 5thV. C ONCLUSION
In this study, synchronization phenomena observed in coupled polygonal oscillatory networks with frustration is investigated. We focus on the amplitude of each oscillator when thecoupling strength is changed. By using computer simulations
and circuit experiments, we confirm that the amplitude of theshared and the other oscillators obtains different value byincreasing the value of the coupling strength. Furthermore,theoretical analysis is applied to solve the amplitude and thephase differences between the adjacent oscillators.

x12(l) 12thx11(k) 11th

y5y4y3(d) 4th

Fig. 15.

y12(a) γ=10.0.
y1x5(c) 3rd(f) 6th(f) 6th

y11x10(j) 10th(i) 9thx4x3(b) 2ndx6x6(e) 5th

y10x8x7x5(d) 4th

y9x2(a) 1sty6x4(c) 3th

y8(g) 7th

y5

y4

y3

x1(a) 1sty5

7-7 coupling network.
x1

y1

y4y3(f) 6th
y12x12(l) 12th(g) 7th(h) 8th

Fig. 13.

Attractors observed in 7-7 coupling network.

(c) γ=80.0.

IV. E XTENSION TO G ENERAL N ETWORKFinally, we consider the bubble structure as one exampleof general nature patterns. The bubble structure is shown inFig. 14(a). In this model, four different size of bubbles areconnected. Ten oscillators are coupled depending on the sizeof bubble.

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[2] W. Wang, I.Z. Kiss, and J.L. Hudson, “Experiments on Arrays of
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[3] M. Yamauchi, Y. Nishio and A. Ushida, ”Phase-waves in a ladder ofoscillators” IEICE Trans. Fundamentals, vol.E86-A, no.4, pp.891-899,
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[5] Y. Setou, Y. Nishio and A. Ushida, “Synchronization Phenomena in
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