IEEE Workshop on Nonlinear Circuit Networks
December 12-13, 2014

Theoretical Approach of Oscillation Deathin 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
Nth osc.

Abstract— We have observed oscillation death in two coupled
polygonal oscillatory networks with strong frustration. In orderto understand the mechanism of these phenomena, in this study,
theoretical analysis using power consumption is applied to solvethe amplitude of the coupled oscillators.

1st osc.

5th osc.

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

4th osc.

I. I NTRODUCTION
In recent years, we have focused on synchronization phenomena of coupled oscillators under a difficult situation forthe 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 the synchronization phenomena in thecoupled polygonal oscillatory networks sharing branches [2].
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 the coupled oscillators tend to synchronize to minimize the power consumptionof the whole system. The phase difference of the sharedoscillators is solved by finding the minimum value of thepower 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 [3], [4]. In thiscircuit system, we confirm that the amplitude of the oscillators
decreases by increasing the value of the coupling strength andoscillation death of the oscillators located farthest place fromthe shared oscillators is occurred.
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.
II. C IRCUIT M ODEL
A. Two Coupled Oscillatory Networks (Symmetric Case)Two identical polygonal oscillatory networks are coupled bysharing a branch as shown in Fig. 1. In this circuit model, weconsider 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 sideof polygonal network “shared oscillators.” Figure 1 (b) shows
the circuit model of the 5-5 coupling network.

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

6th osc.

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

3rd osc.

(a) N -N coupling network.
3L

6th oscillator

3L

5th oscillator

3L

3L
R

R

1st oscillator

iR5v7 3L

v6C

R
CC

3L

3L

v13L

iR8

R
3L

C

3L

iR43L

R

iR1v53L

R

R

iR7R

3L

3L

R
C

3L

7th oscillator

3L

iR6v8

R

3L
C

v43L

R

iR2

3L

C

v2

8th oscillatorRR

4th oscillator

3L

iR3
Cv3

2nd oscillator

3rd oscillator(b) 5-5 coupling network.
Fig. 1.

Two Coupled Oscillatory Networks (Symmetric Model).

Next, we develop the expression for the circuit equations ofN -N coupling oscillatory networks as shown in Fig. 1. 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).

(1)

The normalized circuit equations governing the circuit are
expressed as[kth oscillator]


dxk1

= ε 1 − xk 2 xk − (yak + ybk + yck ) dτ3ody ak1nx−ηy−γ(y+y)=kakakn
 dτ
3o
dy bk
1n(2)
x−
ηy−γ(y+y)=kbkbkndτ
3o
dy ck1nxk − ηyck − γ(yck + yn )=3 dτ
(k = 1, 2, 3, 4).

In this equations, γ is the coupling strength, ε denotes the

2nonlinearity of the oscillators and yn denotes the current ofneighbor oscillator on coupling resistor.

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

1.8
1.6

amplitude

1.4

B. Oscillation Death

1.2
1
0.8
0.6Figure 2 shows the observed attractors of 5 − 5 coupling
network by changing the coupling strength. In this circuit
model, the amplitude of fourth oscillator (which is locatedfarthest place from the shared oscillators) decreases with thecoupling strength. Then, we observe oscillation death of thefourth oscillator when the coupling strength is set to γ = 1.0.

0.4
0.2
00

0.2

0.4

0.6

0.8

1γ

1.2

1.4

amplitudey5y4y3(A) γ=0.1

x1x2

x5x4

x3

22x1 (1st osc.)x3 (3rd osc.)x4 (4th osc.)x5 (5th osc.)

1.8

1.4y2

1.8(a) 5-5 coupling network.

1.6

y1

1.6

1.2
1
0.8
0.6
0.4

y3

y2

y1

y5

y4

0.2
0(B) γ=0.4

y1

x3

x2

x1
y2

y4

y3

0

x5

x4

0.2

0.4

0.6

0.8

1

γ

1.2

1.4

1.6

1.8

2(b) 7-7 coupling network.

y52(C) γ=1.0

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

1.8

x1

x2

x4

x3

x5

1.6

y3

y2

amplitude

1.4

y1

y5

y4

(D) γ=1.5x1x3x2

x5x4

1.2
10.8
0.6
0.4(a) 1stFig. 2.

(b) 2nd(c) 3rd(d) 4th(d) 5th

0.2
0Attractor (5-5 coupling network).

00.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

2(c) 9-9 coupling network.

- 82 -2x1 (1st osc.)x3 (3rd osc.)x4 (4th osc.)x5 (5th osc.)x6 (6th osc.)x7 (7th osc.)

1.8
1.6
1.4

amplitude

Figure 3 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 at same 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. Then
next oscillators from the shared oscillators try to oscillate
to synchronize with the shared oscillators. Furthermore, afteroscillation death is occurred, the network topolgy is changed
from ring to two ladders.
Why only one oscillator of one side polygonal network
stops their oscillation? Figure 4 summarizes possibility of
the network structures after oscillation death for 5-5 coupling
network. We compare the power consumption of each pattern
by using the amplitude which is obtained at oscillation death.
From this figure, we can see that the power consumption ofpattern A is smaller than pattern B. Namely, 5-5 coupling
network changes the network structure to pattern A, afteroscillation death. Tables I-IV show the power consumption
for each network pattern of other coupling networks. Fromthese tables, the power consumption shows lowest value when
only one oscillator of one side polygonal network stops theiroscillation for any kinds of the coupling networks.

1.2
1
0.8
0.6
0.4
0.2
00

0.2

0.4

0.6

0.8

1

γ

1.2

1.4

1.6

1.8

2(d) 11-11 coupling network.
Fig. 3.

Amplitude.

C. Power Consumption
How is the amplitude of the coupled oscillators determined?
In order to make clear the mechanism of the oscillation death,
we focus on the power consumption of the coupling resistors
in the whole system.
We assume the current of the inductor (3L) as the followingequation.
ik (t) = ρk sin(ωt + ϕk )(k = 1, 2, 3...ON ),(3)where ω is the natural frequency of van der Pol oscillator
1. When the coupling resistance is assumed asas ω = √LC
R = 1, the average power consumption of the coupling resistor

TABLE I
E NEGRY OF 5-5 COUPLING NETWORK .
PatternA
BOscillation Death
473, 4, 56, 7, 8Power Consumption
1.654
2.100

TABLE II
E NEGRY OF 7-7 COUPLING NETWORK .
PatternAB
COscillation Death
5104, 5, 69, 10, 11
3, 4, 5, 6, 7
8, 9, 10, 11, 12

Power Consumption
2.800
3.096
3.181

TABLE IIIE NEGRY OF 9-9 COUPLING NETWORK .
Pattern
AB
C
D

Oscillation Death613
5, 6, 712, 13, 14
4, 5, 6, 7, 8
11, 12, 13, 14, 15
3, 4, 5, 6, 7, 8, 9
10, 11, 12, 13, 14, 15, 16

Power Consumption
3.463
3.692
3.608
3.504

TABLE IV
E NEGRY OF 11-11 COUPLING NETWORK .
PatternABCDEOscillation Death716
6, 7, 815, 16, 17
5, 6, 7, 8, 9
14, 15, 16, 17, 18
4, 5, 6, 7, 8, 9, 10
13, 14, 15, 16, 17, 18, 19
3, 4, 5, 6, 7, 8, 9, 10, 11
12, 13, 14, 15, 16, 17, 18, 19, 20

Network structure

5-5 network model

5Oscillation death

between kth and (k + 1)st oscillators are described as Eq. (4).
Z 2π1
P =
{ik (t) + ik+1 (t)}2 dt(4)2π 0Power

61

4Here, we try to derive each amplitude theoretically with thefollowing asummptions. First, let the phase difference between
the oscillators be π. Second,the amplitude of each oscillator
has different value. So, we fix each amplitude depending on
the located place of the oscillators. Figure 5 (a) shows theamplitude setting for 5-5 coupling network as one example.
The amplitude of the shared oscillators is set to ρ0 and thenext oscillators of the shared oscillators are set to ρ1 and ρ2 .
By using Eq. (4), we can obtain the power consumption of
the whole systems for 5-5 coupling network as follows.

7238

Vm2
5Vm2
Vm1

6

Pattern A

14

Vm22Vm2

3

Vm1

871.654

P = 4(ρ0 − ρ1 )2 + 4(ρ1 − ρ2 )2 .

Vm1

Pattern B

12364

758

2.100

Vm1

Fig. 4.

Power Consumption
1.399
1.411
1.468
1.554
1.804

Network structure after oscillation death (5-5 coupling network).

We calculate the power consumption of the whole system byusing the amplitude (ρ0 and ρ2 ) obtained from the computer
simulations. Figure 6 (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 networks isalso solved by using the following equations.
• 7-7 coupling network:
P = 4(ρ0 − ρ1 )2 + 4(ρ1 − ρ2 )2 + 4(ρ2 − ρ3 )2 .

- 83 -(5)

(6)

ρ1

ρ1

ρ05

ρ1

ρ26

ρ17

ρ2

8

ρ06

1

91

ρ2 4

2

ρ2

ρ1

ρ28

ρ1

ρ1
97

ρ0

ρ311

ρ39

ρ41212

ρ1

ρ1

ρ210

ρ1

ρ111

ρ0 12

81ρ4 613113

ρ2

1.4(b) 7-7 coupling network.

ρ210

ρ213

ρ3

1.2
10.8
0.6

1415 ρ41

ρ4 ρ5 70.4
0.2

ρ516025

ρ34

ρ23

ρ1

ρ01416

ρ1

ρ315

2

65

ρ3

ρ2(c) 9-9 coupling network.
Fig. 5.

ρ44

ρ2

Simulation
Theory1.6

ρ048(a) 5-5 coupling network.

ρ3

1.8

amplitude

ρ1

ρ3

102

ρ032

ρ2 ρ3 573

ρ0 20

ρ1

ρ1

1719

ρ2

18

00.2

0.4

0.6

0.8

ρ4

ρ31γ(a) 5-5 coupling network.

(d) 11-11 coupling network.

•=+24(ρ0 − ρ1 ) + 4(ρ1 − ρ2 ) + 4(ρ2 − ρ3 ) (7)4(ρ3 − ρ4 )2 .

amplitude

1.4

P=+1.2
10.8

0.4
0.2

4(ρ0 − ρ1 )2 + 4(ρ1 − ρ2 )2 + 4(ρ2 − ρ3 )2 (8)
2

2

0.6

11-11 coupling network:P

1.8

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

9-9 coupling network:
2

1.6

2

Amplitude setting for calculating the power consumption.

21.4(ρ1 by using ρ0 , ρ2 )

1.6

•

1.2

00

0.2

0.40.60.824(ρ3 − ρ4 ) + 4(ρ4 − ρ5 ) .

1γ

1.2

1.4

1.6

1.8

2(b) 7-7 coupling network. (ρ1,2 by using ρ0,3 )The results are shown in Fig. 6 (b), (c) and (d). From theseresults, we can see that the theoretical analysis can express
the characteristics of the amplitude change.

2
1.8

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

1.6

In this study, we have investigated synchronization phenomena in coupled polygonal oscillatory networks with strong
frustrations. We focused on the amplitude of each oscillator
when the coupling strength is increased. By using the computer simulations, we confirmed that the amplitude of theoscillators decreases by increasing the coupling strength andoscillation death is occurred at un-frustrated oscillators. Furthermore, we observed the amplitude change in the asymmetricnetwork model.

amplitude

1.4

III. C ONCLUSIONS

1.2
1
0.8
0.6
0.4
0.2
00

0.2

0.4

0.6

0.8

1γ

[1] Y. Setou, Y. Nishio and A. Ushida, “Synchronization Phenomena inMany 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.
[3] Y. Uwate, Y. Nishio and R. Stoop, “Frustrations in Strongly Coupled
Polygonal Oscillatory Networks,” Proc. of NOLTA’14, pp. 435-438, Sep.
2014.
[4] Y. Uwate and Y. Nishio, “Oscillation Death and Amplitude Changein Coupled van der Pol Oscillators with Strong Frustrations,” Proc. of
APCCAS’14, pp. 233-236, Sep. 2014.

- 84 -

1.6

1.8

22

ACKNOWLEDGMENT

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

1.6
1.4

amplitude

R EFERENCES

1.4(c) 9-9 coupling network. (ρ2,3 by using ρ0,1,4 )1.8This work was partly supported by JSPS Grant-in-Aid for
Challenging Exploratory Research 26540127.

1.2

1.2
1
0.8
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 )Fig. 6.

Theoretical and simulation results (Symmetric model).