2015 International Workshop on Vision, Communications and Circuits
Keio University, Yokohama, Japan. October 31 – November 3, 2015

Synchronization in Bubble Structureusing Coupled Oscillators
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, we investigate the characteristics ofthe bubble structure using coupled oscillators. By using computer
simulations, we confirm that the stable type of the bubblestructure has symmetrical characteristics for the amplitude, thephase difference and the energy.

I. I NTRODUCTION
The synchronization phenomena of coupled oscillators are
suitable model to express essential behavior observed from thenatural science. Therefore, many researchers have proposed
different coupled oscillatory networks and have discovered
many interesting synchronization phenomena.
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 ofteninvestigated. 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 [1].
We have investigated synchronization phenomena in coupled
polygonal oscillatory networks sharing branches [2]. 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.
In this study, we consider the bubble structure using oscillators as one example of general nature patterns. Figure 1 shows
the bubble structures with four bubbles. In the case of Fig. 1(a),the center (intersection point) of the four bubbles becomes
unstable, then the four bubbles change the form as shownin Fig. 1(b). We apply the previous study of the polygonal
oscillatory network for the two bubble structures. Figure 2
shows the conceptual circuit model of the unstable and thestable bubble structures using coupled oscillators. The circle
corresponds to the van der Pol oscillators. By using computer- 93 -

simulations, we confirm that the stable type of the bubble
structure has symmetrical characteristics for the amplitude, thephase difference and the energy.

(a) Unstable model.

Fig. 1.

(b) Stable model.

Bubble structure with four bubbles.

(a) N=5.

(b) N=6.

Fig. 2.

Conceptual circuit model.

II. B UBBLE S TRUCTURE USING C OUPLED O SCILLATORS
Figure 3 shows the circuit model of the unstable and stablebubble structures using the coupled oscillators. In this circuit
model, we consider the coupling method which two adjacentoscillators are tend to synchronize at anti-phase state.
Next, we develop the expression for the circuit equations ofthe unstable and the stable bubble models as shown in Fig. 3(a)and (b). 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, 5). (1)The normalized circuit equations governing the circuit areexpressed as

4L[kth oscillator of the stable bubble model]


1dxk


= ε 1 − xk 2 xk − (yak + ybk + yck )
dτ3

no
dy1
ak
xk − ηyak − γ(yak + yn )= dτ3ody bk1nx−
ηy−γ(y+y)=kbkbkndτ3o
1ndy ck
xk − ηyck − γ(yck + yn )=3 dτ
(k = 1, 2, 3, 4, 5, 6).

1
4L
4L
4L

4L

R

R

4L

4L

4L

4L

4L

4L

4L

24L

54

R

R
4L

4L
4L

whereR

R
4L

4L

4L

t=

4L

3(a) Unstable bubble model (N =5).
3L

3L

413L

3L
3L

R
3L

3L
3L

3L

65R

R

R

3L

3L
3L
3LRR3Lrr
g1
g1
Cxk , iak =yak ,3g3
3g3 Lrrr
g1
CL
ibk =ybk , ε = g1,3g3 LCrr
CC
γ=R, η = rm,LL

LCτ ,vk =r

3L
3L

III. S YNCHRONIZATION P HENOMENA

3L

32
R

A. Amplitude and Phase Difference
Figures 4, 5 show the observed attractors of the two typesof the bubble structure models. The value of the amplitude,
the phase difference obtained from each bubble model aresummarized in Tables I, II and III. From Table I, the amplitude
of 5th oscillator has smaller value the other oscillators in theunstable bubble model. While, all amplitude have same value
in the stable bubble model.
Next, we focus on the phase difference in two types of thebubble models. In the case of the unstable bubble model, there
are three phase state such as ≈ 99, ≈ 106 and ≈ 153. In thecase of the stable bubble model, there are two phase state suchas ≈ 180 (anti-phase state) and ≈ 120 (3-phase state). Fromthese results, we assume that the synchronization state of thestable bubble model is also stable.

(b) Stable bubble model (N =6).

iRi

iC

vi

i-th oscillator(c) van der Pol oscillator.
Fig. 3.

√In this equations, γ is the coupling strength, ε denotes thenonlinearity of the oscillators and yn denotes the current ofneighbor oscillator on coupling resistor. We obtain the similar
circuit equations for the stable bubble structure model.
For the computer simulations, we calculate Eqs. (2),(3)using a fourth-order Runge-Kutta method with the step size
h = 0.005. The parameters of this circuit model are fixed as
ε = 0.1, γ = 0.1, η = 0.0001.

R
3L

R

(3)

Bubble structure models using coupled oscillators.

[kth oscillator of the unstable bubble model]
1 2dxk=ε1−xk xk − (yak + ybk + yck + ydk )
dτ
3

no
 dy ak
1


=
x−
ηy−
γ(y+y)
kakakn

dτ4
o

1n
 dy bkxk − ηybk − γ(ybk + yn )=dτ4ody ck
1n


x−
ηy−
γ(y+y)=
kckckndτ4ody dk1nx−ηy−γ(y+y)=kckdkn dτ4
(k = 1, 2, 3, 4, 5).

TABLE IA MPLITUDE OF THE UNSTABLE AND THE STABLE BUBBLE MODELS .

(2)- 94 -

Oscillator No.
1st oscillator
2nd oscillator
3rd oscillator
4th oscillator
5th oscillator
5th oscillator

Unstable model
1.94
1.93
1.94
1.93
1.80
-Stable model
1.89
1.89
1.89
1.89
1.89
1.89

y1y3

y2

y5

y4

Attractor
x1x2x3x4x5
1st Osc. 2nd Osc. 3rd Osc. 4th Osc. 5th Osc.
Phase
difference
Phase12 Phase23 Phase34 Phase45 Phase51

TABLE IV

Phase
difference

E NERGY ( UNSTABLE BUBBLE MODEL )Phase14 Phase25 Phase35
Fig. 4.

B. Energy
Finally, we investigate the energy of the coupling resistor inthe bubble circuit models. The simulation results of the energy
are summarized in Table IV and V. As the results of the phasedifference, in the case of the unstable bubble model, there arethree energy states. While, in the case of the stable bubble
model, there are two energy states.
However, the total energy of the unstable bubble model issmaller than the stable bubble model.

Attractor and phase diffrence for the unstable bubble model (N =5).

y1y2y3y5y4y6

Attractor
x1x2x3x4x5x6
1st Osc. 2nd Osc. 3rd Osc. 4th Osc. 5th Osc. 6th Osc.

Bridge No.
bridge: 1-2
bridge: 2-3
bridge: 3-4
bridge: 4-5
bridge: 5-1
bridge: 1-4
bridge: 2-5
bridge: 3-5
Total energy

Energy
0.025
0.025
0.025
0.179
0.157
0.025
0.179
0.157
0.773

Phase
difference
Phase12 Phase23 Phase34 Phase45 Phase56 Phase61TABLE V
E NERGY ( STABLE BUBBLE MODEL )Phase
difference
Phase14 Phase26 Phase36

Fig. 5.

Attractor and phase diffrence for the stable bubble model (N =6).

TABLE IIP HASE DIFFERENCE ( UNSTABLE BUBBLE MODEL
Bridge No.
bridge: 1-2
bridge: 2-3
bridge: 3-4
bridge: 4-5
bridge: 5-1
bridge: 1-4
bridge: 2-5
bridge: 3-5

Phase difference [◦ ]153.21153.05153.2199.40
104.94153.21
99.40
107.25Statestate 3state 3state 3state 1state 2state 3state 1state 2Phase difference [◦ ]120.69179.78119.45119.23179.81120.62179.91120.56120.52

Energy
0.197
0.000
0.197
0.197
0.000
0.197
0.000
0.197
0.197
1.182

IV. C ONCLUSIONSIn this study, we have investigated the characteristics of thetwo types of the bubble structures using coupled oscillators.
By using computer simulations, we confirmed that the stable
type of the bubble structure has symmetrical characteristics
for the amplitude, the phase difference and the energy.
For the future work, we would like to investigate thestability of the solution obtained the bubble networks.

TABLE IIIP HASE DIFFERENCE ( STABLE BUBBLE MODEL )Bridge No.
bridge: 1-2
bridge: 2-3
bridge: 3-4
bridge: 4-5
bridge: 5-6
bridge: 6-1
bridge: 1-4
bridge: 2-6
bridge: 3-5

Bridge No.
bridge: 1-2
bridge: 2-3
bridge: 3-4
bridge: 4-5
bridge: 5-6
bridge: 6-1
bridge: 1-4
bridge: 2-6
bridge: 3-5
Total energy

ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Challenging Exploratory Research 26540127.

Statestate 1 (3-phase)state 2 (anti-phase)state 1 (3-phase)state 1 (3-phase)state 2 (anti-phase)state 1 (3-phase)state 2 (anti-phase)state 1 (3-phase)state 1 (3-phase)

R EFERENCES[1] Y. Setou, Y. Nishio and A. Ushida, “Synchronization Phenomena in
Many Oscillators Coupled by Resistors as a Ring,” Proc. of APCCAS’94,
pp. 570-575, Dec. 1994.
[2] Y. Uwate and Y. Nishio, “Synchronization in Several Types of Coupled
Polygonal Oscillatory Networks,” IEEE Trans. Circuits Syst. I, vol. 59,
no. 5, pp. 1042-1050, May 2012.

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