Oscillation Death and Amplitude Changein Coupled van der Pol Oscillatorswith Strong Frustrations
Yoko Uwate

Yoshifumi Nishio

Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: uwate@ee.tokushima-u.ac.jp

Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: nishio@ee.tokushima-u.ac.jp

Abstract— In this study, we investigate synchronization phenomena in 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 coupling strength and oscillation
death frustrated oscillators is occurred. Furthermore, we observe
the amplitude change in the asymmetric network model.

I. I NTRODUCTION
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 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 synchronization
phenomena in 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
coupled 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. Furthermore, we havediscussed synchronization state of the proposed oscillatory
networks by changing the coupling parameter [7]. In thiscircuit system, we confirm that the amplitude of the sharedc
978-1-4799-5230-4/14/$31.00 ⃝2014
IEEEand the other oscillators obtains different value. The torus
attractors can be also observed in the certain parameter region.
However, we could not observe oscillation death even if thecoupling strength is set to very large value. In the circuit modelof Ref. [7], the earth resistance is missing in the 3rd and the 4thoscillators. Namely, the network topology changes to just ringoscillators coupled by the inductors if the coupling strength
becomes large.
In this study, we propose the new circuit model which isincluding the earth resistance in all ground parts. Synchronization phenomena in coupled polygonal oscillatory networks
with strong frustration are investigated. We confirm that theamplitude of the oscillators decreases by increasing the valueof the coupling strength and oscillation death frustrated oscillators is occurred. Furthermore, we observe the amplitudechange when the network topology is asymmetric.
II. T WO C OUPLED O SCILLATORY N ETWORKSIn this study, we investigate the synchronization phenomena
in two coupled oscillatory systems.
A. Symmetric Model
Two identical polygonal oscillatory networks are coupledby sharing a branch as shown in Fig. 1. The new part of thiscircuit system is that the earth resistances are connected toground parts in the 3rd and the 4th oscillators. We consider
the coupling method which two adjacent oscillators are tend tosynchronize at anti-phase state. We call the first and the secondoscillators which are connected to both side of polygonal
network “shared oscillators.”

233

5th osc.

1st osc.

6th osc.
1st osc.

4th osc.

4th osc.

7th osc.
2nd osc.

3rd osc.

2nd osc.

(a) 3-3 coupling network.
Fig. 1.

3rd osc.

8th osc.

(b) 5-5 coupling network.

Two Coupled Oscillatory Networks (Symmetric Model).

3L

rm

rm

Cv4

R2

R

3L

4th oscillator

1.4

iR2
Cv3

x1 (1st osc.)x3 (3rd osc.)1.6ic2rmi R3
R

ib2

ia2

3L

3L

3L

3L
3L

rm

R

rm

ib3

iR4

1.8

3L

ia3

NEWia4

rm

R

ic3

RCv1

3L

rm

3L

i R1

ib4 ic4 3L

3L
rm

ia1

amplitude

ic1

rm

rm

1st oscillator
ib1

RCv2

1.2
1
0.8
0.6

NEW

0.4

2nd oscillator

3rd oscillator

Fig. 2.

0.2

Coupling Model (3 − 3 coupling networks).

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.

iRk = −g1 vk + g3 vk

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

(1)

2

1.6
1.4
1.2
1
0.8
0.6

The normalized circuit equations governing the circuit are
expressed as[kth oscillator]
⎧

dxk
1
⎪
⎪
= ε 1 − xk 2 xk − (yak + ybk + yck )
⎪⎪dτ
⎪
⎪
 3
⎪
dy1
⎪
ak
⎪
=
xk − ηyak − γ(yak + yn )
⎪⎪
⎨ dτ
3

dy bk
1(2)
=
xk − ηybk − γ(ybk + yn )
⎪⎪dτ3
⎪
⎪
⎪
1
dy ck
⎪
⎪
⎪
=xk − ηyck − γ(yck + yn )⎪⎪3⎪⎩ dτ(k = 1, 2, 3, 4).

0.4
0.2
000.2

0.4

0.6

0.8

1

1.2

γ

1.4

LCτ ,

ibk =

vk =
g1
3g3

g1xk ,
3g3
C
ybk ,
L

g1
3g3

iak =

ε = g1

C
yak ,
L

γ=R

η = rm2

Amplitude (Symmetric Model).

180

L,
C

Shared oscillators
Other oscillators

140
120
100
80
60
40
20
0
0.1C,
L

1.8200phase difference√

1.6(b) 5-5 coupling network.
Fig. 3.

160

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

1.8

amplitude

Next, we develop the expression for the circuit equations of
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,

0.2C,
L

0.3

0.4

0.5

γ0.6

0.7

0.8

0.9

1(a) 3-3 coupling network.
200In 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 change of the amplitude observed fromeach network. In the case of 3−3 coupling network, oscillation
death of all oscillators is occurred at same time. While, in thecase of 5−5 coupling network, first the oscillation death of theoscillators located farthest place from the shared oscillators isoccurred. After that, the other oscillators stop to oscillate atsame time.
Figure 4 shows the phase difference with the couplingstrength.

234

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

150

100
Shared oscillators
Other oscillators (1st-4th)
Other oscillators (4th-5th)

50

00.1

0.2

0.3

0.4

0.5

γ0.60.7

0.8

0.9(b) 5-5 coupling network.
Fig. 4.

Phase Difference (Symmetric Model).

1

y1

B. Asymmetric Modely2

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

1st osc.

x2

x3

x4

y7

y6

y8

5th osc.

5th osc.

Fig. 7.

6th osc.

3rd osc.

x8

Oscillation of Amplitude (3-7 coupling network, γ=0.75).

2nd osc.

200

7th osc.

8th osc.

180160(b) 3-7 coupling network.

phase difference

6th osc.
(a) 3-5 coupling network.

x7

x6

x5

2nd osc.

Fig. 5.

y4

1st osc.

4th osc.

3rd osc.

x1
y5

4th osc.

y3

Two Coupled Oscillatory Networks (Asymmetric Model).

Figure 6 shows the change of the amplitude observed fromthe asymmetric networks. In the case of 3 − 5 coupling
network, oscillation death of all oscillators is occurred atsame time. While, in the case of 3 − 7 coupling network,
first the oscillation death of the oscillators located farthestplace from the shared oscillators is occurred. After that, theother all oscillators stop to oscillate at same time. These
results are similar with the results with the symmetric networkmodels. However, we observe the amplitude change in 3 − 7coupling network when the coupling strength is set to aroundγ = 0.75. Figure 7 shows one example or the oscillation of
the amplitude.
Figure 8 shows the phase difference with the couplingstrength.

Shared oscillators
Other oscillators (1st-3th)
Other oscillators (1th-4th)
Other oscillators (4th-5th)

140
120
10080
60
40
2000.1

0.2

0.3

0.4

0.5

γ0.6

0.7

0.8

0.910.8

0.91(a) 3-5 coupling network.
180160phase difference

14012010080
Shared oscillators
Other oscillators (1st-3th)
Other oscillators (1th-4th)
Other oscillators (4th-5th)
Other oscillators (5th-6th)

60
40
20
00.1

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

1.8
1.6

amplitude

1.4

0.3

0.4

0.5

γ

0.6

0.7(b) 3-7 coupling network.
Fig. 8.

Phase Difference (Asymmetric Model).

1.2
10.8

III. A MPLITUDE C HANGE

0.6
0.4
0.2
000.10.2

0.3

0.4

0.5

γ

0.6

0.7

0.8

0.9

1(a) 3-5 coupling network.
2x1 (1st osc.)x3 (3rd osc.)x4 (4th osc.)x5 (5th osc.)x6 (6th osc.)

1.8
1.6
1.4

amplitude

0.2

1.2
1
0.8
0.6
0.4
0.2
00

0.1

0.2

0.3

0.4

0.5

γ

0.6

0.7

0.8

0.9(b) 3-7 coupling network.
Fig. 6.

Amplitude (Asymmetric Model).

1In this section, we focus on the phenomena of the amplitudechange which are observed from 3−7 coupling network. First,
we confirm that the amplitude change can be observed aroundγ = 0.75. Figure 9 shows one example of the amplitudechange by changing the coupling strength. The horizontal axis
denotes the computer simulation time Ta = 10, 000τ .
Figure 9(a) shows the result of the amplitude when the coupling strength is fixed with γ = 0.72. We observe oscillationof the amplitude at the beginning of the simulation. When Ta
is larger than 300, the amplitude converges to constant. There
are three pairs (1st-2nd, 4th-8th and 5th-7th) of the amplitude
values and the 3rd and the 6th oscillators have own amplitude
value. In the case of γ = 0.75, we confirm that the amplitudechange can be observed constantly as shown in Fig. 9(b). There
are two pairs (1st-4th-5th and 2nd-7th-8th) of oscillation typesof the amplitude and these two pairs oscillate with anti-phasestate. While the other two amplitudes (3rd and 6th) oscillatewith in-phase state. The enlarged illustration of Fig. 9(b) is235

shown in Fig. 10. By increasing the coupling strength, theoscillation death of the 3rd and 6th oscillators is occurred.
Then, we confirm that the network type depending on theamplitude returns to three pairs again (see Fig. 9(c)). Finally,
we summarize the network types depending on the amplitudechange as shown in Fig. 11.

1.2

amplitude

amplitude

1.2

1

1
0.8
0.6
0.4
0.2

1st oscillator
2nd oscillator
3rd oscillator
4th oscillator
5th oscillator
6th oscillator
7th oscillator
8th oscillator

1.4

1st oscillator
2nd oscillator
3rd oscillator
4th oscillator
5th oscillator
6th oscillator
7th oscillator
8th oscillator

1.40800

850

900

950

1000

TaFig. 10.

Enlarged Illustration of Fig. 9(b).

0.80.6

4
50.4
10.2
06

30200

400

Ta

600

800

21000

7(a) γ = 0.72.

8(a) Type-A (3 pairs).
1st oscillator
2nd oscillator
3rd oscillator
4th oscillator
5th oscillator
6th oscillator
7th oscillator
8th oscillator

1.4

amplitude

1.2
14
5
10.8

6

30.6

2
70.4

8(a) Type-B (2 pairs).

0.2
00200

400

Ta

600

800

1000
4(b) γ = 0.75.

5
1

1st oscillator
2nd oscillator
3rd oscillator
4th oscillator
5th oscillator
6th oscillator
7th oscillator
8th oscillator

1.2

amplitude

631.4

10.8

2
78(c) Type-C (3 pairs).
Fig. 11.

Network Types Depending on Oscillation of Amplitude.

0.60.4

R EFERENCES

0.2
00200

400

Ta

600

800

1000(c) γ = 0.80.
Fig. 9.

Amplitude Change (3 − 7 coupling network).

IV. C ONCLUSIONSIn 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 frustrated oscillators is occurred. Furthermore, we observed the amplitude change in the asymmetricnetwork model.
ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Challenging Exploratory Research 26540127.

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