2015 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing (NCSP'15)
Kuala Lumpur, Malaysia, February 27 - March 2, 2015

Investigation of Oscillation Deathin Strongly Coupled Polygonal Oscillatory Networks
Yoko Uwate and Yoshifumi Nishio
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
Email: {uwate, nishio}@ee.tokushima-u.ac.jp
Abstract
We have confirmed amplitude change and oscillation deathin two coupled polygonal oscillatory networks with strong
frustration. In order to understand the mechanism of thesephenomena, in this study, theoretical analysis using power
consumption is applied to solve the amplitude of the coupledoscillators.
1. Introduction
The synchronization phenomena of coupled oscillators are
suitable model to express essensial behavior observed fromthe natural science. 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 as a 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 are connected to every corner of polygonal network. By using computer simulations and theoretical analysis, we confirm that thecoupled oscillators tend to synchronize to minimize the power
consumption of the whole system. The phase difference ofthe shared oscillators is solved by finding the minimum valueof the power consumption function.
Recently, in order to investigate amplitude change of theoscillators with strong coupling parameter, we have proposeda new circuit model of coupled polygonal network which isinserted the earth resistances in all ground parts [7], [8]. In
this circuit system, we confirm that the amplitude of the oscillators decreases by increasing the value of the couplingstrength and oscillation death of the oscillators located farthest place from the shared oscillators is occurred. These phe-nomena are very interesting, however, we have observed such
oscillation death from only simple coupled polygonal oscillatory networks as basic model.
In order to make clear the mechanism of these phenomena
especially oscillation death, in this study, theoretical analysis
using power consumption of the coupling resistance is applied to solve the amplitude of the coupled oscillators.
We expect that the results of this study contribute to understanding of synchronization phenomena observed in generalcomplex networks.
2. Two Coupled Oscillatory Networks
2.1 Symmetric Model
Two identical polygonal oscillatory networks are coupledby sharing a branch as shown in 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 thefirst and the second oscillators which are connected to bothside of polygonal network “shared oscillators.”
Figure 2 shows the circuit model of the 5-5 coupling networks.
5th osc.

Nth osc.

6th osc.

6th osc.

1st osc.

4th osc.

7th osc.

5th osc.

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

2nd osc.
3rd osc.

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

1st osc.

8th osc.

(a) 5-5 coupling network.

4th osc.

3rd osc.

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

(b) N -N coupling network.

Figure 1: Two Coupled Oscillatory Networks (SymmetricModel).
Next, we develop the expression for the circuit equations ofN -N coupling oscillatory networks as shown in Fig. 2. The- 413 -3L

6th oscillator

7th oscillator

3L

5th oscillator

3L

3L
RR

3L

R

v7 3L

v6

C

3L

CRC

R

iR7R

3L

iR53L

iR6

1st oscillator

3L

RiR1v53L

C3L

v1

3L

3L

iR8R
3L

RC

3L

iR4

R

v83L
Cv4

3L

R

iR23L

C

v2

8th oscillator

RR

4th oscillator

3L

iR3
C

2nd oscillatorv3

3rd oscillator

Figure 2: Coupling Model (5-5 coupling networks).
vk − iRk characteristics of the nonlinear resistor are approxiFigure 3 shows the observed attractors of 5 − 5 coupling
mated by the following third order polynomial equation,
network by changing the coupling strength. In this circuit
model, the amplitude of fourth oscillator (which is locatediRk = −g1 vk + g3 vk 3 (g1 , g3 > 0), (k = 1, 2, 3, 4). (1) farthest place from the shared oscillators) decreases with theThe normalized circuit equations governing the circuit are coupling strength. Then, we observe oscillation death of thefourth oscillator when the coupling strength is set to γ = 1.0.
expressed as
We also obtain the same synchronization states from the cir[kth oscillator]
cuit experiment as shown in Fig. 4. The circuit experiments
dxk
1 2parameters
are set to L = 50mH and C = 68nF . The non
x−(y+y+y)=ε1−xkakbkckk
dτ3

linear resistor is realized by op-amp (TL082). The power supnody ak1
ply of the op-amp is fixed as ±12V .
xk − ηyak − γ(yak + yn )
=

 dτ
3
no
1
dy bk(2)xk − ηybk − γ(ybk + yn )
=
y5

y1
y2
y3y4dτ3

no
dy1
(A)
γ=0.1ckxk − ηyck − γ(yck + yn )=x5x1x2x3x43 dτ(k = 1, 2, 3, 4).
y3
y2y5
y1
y4
wherer√g1
xk ,t = LCτ , vk =3g3r
r
r
r
g1
Cg1
Ciak =yak , ibk =ybk ,3g3 L
3g3 Lrrr
LCCε = g1,γ = R, η = rm,
CLL
In this equations, γ is the coupling strength, ε denotes thenonlinearity of the oscillators and yn denotes the current ofneighbor oscillator on coupling resistor.

- 414 -(B) γ=0.4

x3x2

x1
y1y2

y3x5x4y4y5(C) γ=1.0x1
y1x2x3y3y2x4x5y5y4(D) γ=1.5x1(a) 1stx3x2(b) 2nd(c) 3rdx5x4(d) 4th(d) 5thFigure 3: Attractor (5-5 coupling network).

oscillation death is occurred, the network topolgy is changed
from ring to two ladders. How is the amplitude of the coupledoscillators determined? In order to make clear the mechanismof the oscillation death, we focus on the power consumptionof the coupling resistors in the whole system. We assume thecurrent of the inductor (3L) as the following equation.
(b) R = 200Ω.

(a) R = 150Ω.

ik (t) = ρk sin(ωt + ϕk ) (k = 1, 2, 3...ON ),(3)where ω is the natural frequency of van der Pol oscillator as1
ω = √LC. When the coupling resistance is assumed as R =1, the average power consumption of the coupling resistor
between kth and (k + 1)st oscillators are described as Eq. (5).
P =(c) R = 300Ω.

(d) R = 390Ω.

12π

Z 2π0

{ik (t) + ik+1 (t)}2 dt(4)Here, we try to derive each amplitude theoretically withFigure 4: Circuit experiments results for 5-5 coupling network. upper:v1 (1st oscillator), lower:v4 (4th oscillator), The the following asummptions. First, let the phase difference between the oscillators be π. Second,the amplitude of each oshorizontal axis: [0.1ns/div]. The vertical axis: [2V/div]
cillator has different value. So, we fix each amplitude depending on the located place of the oscillators. Figure 6 shows theamplitude setting for 5-5 coupling network as one example.
xxxxx
The amplitude of the shared oscillators is set to ρ0 and thexx
next oscillators of the shared oscillators is set to ρ1 and ρ2 .
2

2

1.8

1.8

1 (1st osc.)

3 (3rd osc.)

1.6

1 (1st osc.)

3 (3rd osc.)

1.6

4 (4th osc.)

4 (4th osc.)

1.4

amplitude

amplitude

1.4
1.2
1
0.8

1
0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0

0.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

2

(a) 5-5 coupling network.

50

0.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

2

1.4

10.8

73

ρ1

0.4
0.2
0.4

0.6

0.8

1γ

1.2

1.4

1.6(c) 9-9 coupling network.

1.8

28

ρ1

10.6

0.2

ρ00.8

0.4

0

ρ2

1.2

0.6

0.2

6

ρ2 4

x1 (1st osc.)x3 (3rd osc.)x4 (4th osc.)x5 (5th osc.)x6 (6th osc.)x7 (7th osc.)

1.6

1.2

ρ12

1.8

amplitude

1.4

ρ0
1

2

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

1.6

amplitude

ρ1(b) 7-7 coupling network.

2
1.80

5 (5th osc.)

1.2

00

0.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

2(d) 11-11 coupling network.

Figure 6: Amplitude setting for calculating the power consumption.
By using Eq. (4), we can obtain the power consumption of
the whole systems for 5-5 coupling network as follows.

Figure 5: Amplitude (Symmetric Model).
Figure 5 shows the change of the amplitude observed from
5-5, 7-7, 9-9 and 11-11 coupling networks. From these figures, we can see that first, the oscillation death of the oscillators located farthest place from the shared oscillators is occurred. After that, the other oscillators stop to oscillate atsame time.
We explain the oscillation death as physical phenomena.
The earth resistance is not inserted in two shared oscillators,
namely the shared oscillators do not tend to oscillation deathby controling the phase difference to minimize enegy. Thennext oscillators from the shared oscillators try to oscillate tosynchronize with the shared oscillators. Furthermore, after

P = 4ρ20 + 8ρ21 + 4ρ22 − 8ρ0 ρ1 − 8ρ1 ρ2 .

(5)

We calculate the power consumption of the whole system byusing the amplitude (ρ0 and ρ2 ) obtained from the computer
simulations. Figure 7 (a) shows the theoretical results of theamplitude (ρ1 ) of 5-5 coupling networks. The results of thetheoretical analysis match well with simulation results.
The amplitude of other types of the coupling network isalso solved by usinv previous theoretical approch. The results
are shown in Fig. 7 (b), (c) and (d). From these results, we cansee that the theoretical analysis can express the characteristicsof the amplitude change.

- 415 -3. Conclusions
2
1.8

Simulation
Theory1.6
1.4

amplitude

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 thecomputer simulations, we confirmed that the amplitude of theoscillators decreases by increasing the coupling strength andoscillation death is occurred at un-frustrated oscillators.

1.2
1
0.8
0.6
0.4
0.2
0

Acknowledgment

00.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

2(a) 5-5 coupling network.

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

(ρ1 by using ρ0 , ρ2 )2

Simulation (x3)Theory (x3)
Simulation (x4)Theory (x4)1.8

References

1.6

amplitude

1.4[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.

1.2
10.8
0.6
0.4

[2] W. Wang, I.Z. Kiss, and J.L. Hudson, “Experiments onArrays of Globally Coupled Chaotic Electrochemical
Oscillators: Synchronization and Clustering” Chaos,vol.10, no.1, pp.248-256, Mar. 2000.

0.2
000.2

0.4

0.6

0.8

1

γ

1.2

1.4

1.6

1.8

2(b) 7-7 coupling network.
(ρ1,2 by using ρ0,3 )[3] M. Yamauchi, Y. Nishio and A. Ushida, ”Phase-waves
in a ladder of oscillators” IEICE Trans. Fundamentals,
vol.E86-A, no.4, pp.891-899, Apr. 2003.

Simulation (x4)Theory (x4)
Simulation (x5)Theory (x5)

1.6
1.4

amplitude[4] H.B. Fotsina and J. Daafouza, “Adaptive synchronization of Uncertain Chaotic Colpitts Oscillators based on
Parameter Identification” Physics Letters A, vol.339,
pp.304-315, May 2005.

21.8

1.2
10.8
0.6
0.4

[5] 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.

[7] Y. Uwate, Y. Nishio and R. Stoop, “Frustrations inStrongly Coupled Polygonal Oscillatory Networks,”
Proc. of NOLTA’14, pp. 435-438, Sep. 2014.
[8] Y. Uwate and Y. Nishio, “Oscillation Death and Amplitude Change in Coupled van der Pol Oscillators withStrong Frustrations,” Proc. of APCCAS’14, Nov. 2014.

000.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

2(c) 9-9 coupling network.
(ρ2,3 by using ρ0,1,4 )2Simulation (x3)Theory (x3)
Simulation (x4)Theory (x4)

1.8
1.6
1.4

amplitude[6] Y. Uwate and Y. Nishio, “Synchronization in Several
Types of Coupled Polygonal Oscillatory Networks,”
IEEE Trans. Circuits Syst. I, vol. 59, no. 5, pp. 10421050, May 2012.

0.2

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(d) 11-11 coupling network.
(ρ3,4 by using ρ0,1,2,5 )Figure 7: Theoretical and simulation results.

- 416 -