2018 International Symposium on Nonlinear Theory and Its Applications,NOLTA2018, Tarragona, Spain, September 2-6, 2018

Amplitude Death in Large-Scale 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—The study of AD is important for understanding the control mechanisms and efficient regulators ofa system’s dynamics. In this study, we investigate the occurrence of AD in two coupled large-scale oscillatory networks.
1. Introduction
Synchronization phenomena in coupled oscillators aresuitable models for analyzing a number of natural occurrences [1],[2]. Therefore, many researchers have proposed
different coupled oscillatory networks, and some interesting synchronization phenomena have been discovered [3][6].
Oscillation quenching (oscillation and amplitude death),
another fundamental emergent phenomenon in coupled
nonlinear systems, can be caused by several factors [7],[8].
Amplitude death (AD) occurs in strongly coupled nonlinear oscillators when their interaction causes a pair of fixedpoints to become stable and attracting. Setou et al. reported
AD in ring coupled oscillators when the frequencies of thecoupled units differ [9].
We have investigated synchronization phenomena
in coupled polygonal oscillatory networks that sharebranches [10], [11]. In this system, van der Pol oscillators are connected to every corner of each polygonal network. The first and the second oscillators, which are connected to both polygonal networks, are called “shared oscillators,” and each polygonal network has an odd numberof oscillators. We then observe N-phase synchronization.
Through computer simulations and theoretical analysis, weconfirmed that the coupled oscillators tended to synchronize to minimize the power consumption of the whole system. The phase difference of the shared oscillators wasdetermined by finding the minimum value of the power
consumption function. Additionally, we proposed a newpolygonal circuit system that includes actual inductor models (with loss) at all ground parts [12]. Synchronization
phenomena in coupled polygonal oscillatory networks withstrong frustration are investigated. Strong frustration is realized using conflicting coupling organization in the network and by increasing the coupling strength.
We confirmed that the amplitude of the oscillators decreases as the value of the coupling strength increases, andthat AD occurs in the polygonal oscillatory networks. If
one of the polygonal networks is triangular, we observe

global AD. However, for other types of networks, AD appears in a complicated way. First, AD occurs at the oscillators located farthest from the shared oscillators. Next, ADoccurs simultaneously in all other oscillators as the coupling strength increases. We explained the mechanism bywhich AD occurs using a theoretical approach.
In this study, we investigate the occurrence of AD in
more large-scale networks. For considering the large-scalenetworks, the number of oscillators connected to the one
side of polygonal network is increased until around M =101. By using computer simulations, we observe similar
AD phenomena in the large-scale polygonal network withprevious studies.
2. Two Coupled Small-Scale Oscillatory Networks [12]In our previous study, we investigated AD in two coupled small-scale oscillatory networks. Two polygonal oscillatory networks are coupled by sharing a branch, asshown in Fig. 1. In this circuit model, we consider thecoupling method in which two adjacent oscillators tend tosynchronize in the anti-phase state. The number of oscillators coupled to left and the right polygonal networks is denoted by N and M, respectively. The values of N and M areset to odd numbers to produce N or M phase synchronizations. The first and second oscillators, which are connected
to both sides of the polygonal networks, are called “sharedoscillators.”
Nth osc.
6th osc.

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

5th osc.

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

4th osc.

3rd osc.

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

Figure 1: Two coupled oscillatory networks (N–M coupling network).
Examples of the network models used in this study are
shown in Fig. 2. Here, the 3–3 and 5–5 coupling networks
are symmetric models and the 5–7 coupling network is anasymmetric model.

- 538 -1st osc.

5th osc.

4th osc.

rrr
g1
Cg1
C
ick =
yck , in =yn ,3g3
L
3g3
Lrrr
L
CC, γ=R, η = rm,
ε = g1CLL

6th osc.

r

1st osc.

4th osc.

7th osc.
2nd osc.

3rd osc.

2nd osc.

3rd osc.

(a) 3–3 coupling network.

(k = 1, 2, 3, ..., N + M − 2),

8th osc.

(b) 5–5 coupling network.
6th osc.

7th osc.

5th osc.
1st osc.

8th osc.

4th osc.
2nd osc.
3rd osc.

9th osc.

10th osc.

(c) 5–7 coupling network (N=5, M=7).

Figure 2: Examples of two coupled small-scale oscillatory
networks.
Figure 3 shows the circuit model of the 3–3 couplingnetwork. The earth resistances are inserted to the 3rd and4th oscillators to model actual inductors and realize symmetry in the circuit network model. Tiny resistors (rm ) are
inserted to avoid an L-loop in the computer simulations.
ic1

3L

rm

rm

rm

3L

iR4
CRv4

R

4th oscillator

iR2
Cv3

RC

3. Two Coupled Large-Scale Oscillatory Networksv2

2nd oscillator

3rd oscillator

Figure 3: Coupling model (3–3 coupling network).
Next, we develop an expression for the circuit equationsof the N–M coupling oscillatory network in Fig. 3. 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, ..., N + M − 2).
Using the variables and parameters
√r

g1
xk ,
3g3
r
r
r
r
g1
C
g1
C
iak =
yak , ibk =ybk ,3g3
L
3g3
L
t=

LCτ, vk =In this equation, γ is the coupling strength, ε denotes thenonlinearity of the oscillators, and y denotes the current of
the inductor of the connected oscillator with the kth oscillator. For the computer simulations, we calculate Eq. (2)using the fourth-order Runge–Kutta method with step size
h = 0.005. We set the parameters of this circuit model toε = 0.1 and η = 0.0001. The coupling strength γ between
the oscillators changes from a small to a large value.
Figure 4 shows the change in amplitude according to thesize of the network. In the case of the 3–3 coupling network, global AD occurs at the same time, whereas in the5–5 and 5–7 coupling networks, AD first occurs in the oscillators located farthest from the shared oscillators, andthen the other oscillators stop oscillating at the same time.

ic2

rm

i R3
R

ia2

3L

3L

ib2

rm

3L
rm

3L

R

ia3
ib3

R

3L

ic3

R

ia4

rm

C

v1

3L

rm

3L

i R1

ib4 ic4 3L

3L
rm

ia1

3L

rm

1st oscillator
ib1

the normalized circuit equations governing the circuit are
expressedas:
1 2dxk=ε1−xk xk − (yak + ybk + yck )dτ3

dyak 1 x−ηy−γ(y+y)=kakakndτ3
 dy1
(2)bk=xk − ηybk − γ(ybk + yn )



dτ3



dyck 1 x−ηy−γ(y+y)
=
kck
ckn



dτ
3




(k = 1, 2, 3, ..., N + M − 2).

(1)

In this section, we investigate the occurrence of AD in
more large-scale networks. Figure 5(a) shows the circuit
model which is that triangular network and odd number
polygonal network are coupled with one branch. WhereM denotes the number of coupled oscillators of odd polygonal network. Similarly, Fig. 5(b) and (c) show the circuit
model which is that pentagonal and heptagonal network areused for left side of polygonal network.
Figure 6 shows the change in amplitude when the number of oscillators M is set to 101. In the case of the triangle
network, global AD occurs at the same time, whereas in
the pentagonal and heptagonal networks, AD first occurs in
the oscillators located farthest from the shared oscillators,
and then the other oscillators stop oscillating at the sametime. These results are similar with the small-scale networkmodel. Namely, we can say that these AD phenomena do
not depend on the number of oscillators in polygonal networks.
Finally, we investigate the coupling strength when thefirst and second AD occurs. The simulation results are- 539 -

2

1.6

7

4

1.4

amplitude

65

x1 (1st osc.)x3 (3rd osc.)

1.8

8

1

1.2

9

Shared branch31100.8

20.6

110.4

M

0.2
000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1214

113γ

Odd number
polygonal network (M)Triangular
network (N=3)(a) 3–3 coupling network.
2(a) 3–3 coupling network (N=3, M=3, 5, 7, ... ,101).

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

1.8
1.6

amplitude

1.4

18

1

0.6

4

0.2
0

0.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

9

Shared branch

0.4

102

2
3(b) 5–5 coupling network.

11M

1213142x1 (1st osc.)x3 (3rd osc.)x4 (4th osc.)x6 (6th osc.)x7 (7th osc.)x8 (8th osc.)

1.8
1.6
1.4

amplitude

74

5

0.8

065

1.2

1.2

Pentagonalnetwork (N=5)

Odd number
polygonal network (M)

1(b) 5–5 coupling network (N=5, M=5, 7, 9, ... ,101).

0.8
0.6
0.4

06

5

0.2
0

0.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

7

4

26(c) 5–7 coupling network.

78

19

Shared branch

5

Figure 4: Amplitude of two coupled small-scale oscillatory
networks.

1042
311M

1214

shown in Fig. 7. In the case of triangular network (Fig. 7(a)), the coupling strength of global AD increases with thenumber of oscillators in right side polygonal network. If
M is larger than 20 the coupling strength of AD converges
to 1.0. While, in the cases of pentagonal and heptagonal
networks, the coupling strength of first AD increases with
the number of oscillators in right side polygonal network.
However, the coupling strength of second AD is almost
same even if the number of oscillators in right side polygonal network is increased. The coupling strength of second
AD of heptagonal network has larger value than the pentagonal network.

Heptagonalnetwork (N=7)

13

Odd number
polygonal network (M)(c) 5–7 coupling network (N=5, M=7, 9, 11, ... ,101).

Figure 5: Examples of two coupled large-scale oscillatory
networks.
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4. Conclusions
In this study, we investigated the occurrence of AD in
more large-scale networks. For considering the large-scalenetworks, the number of oscillators connected to the one
side of polygonal network is increased until around M =101. By using computer simulations, we observed similar
AD phenomena in the large-scale polygonal network withprevious studies.

[3] M. G. Rosenblum, A.S. Pikovsky and J. Kurths, “PhaseSynchronization of Chaotic Oscillators” Physical Review Letters, vol.76, no.11, pp.1804-1807, Mar. 1996.
[4] W. Wang, I.Z. Kiss and J.L. Hudson, “Experiments onArrays of Globally Coupled Chaotic Electrochemical- 540 -2

2

1.8
1.6

amplitude

1.4

Coupling strength of AD

x1 (1st osc.)x3 (3rd osc.)x53 (53th osc.)

1.2
10.8
0.6
0.4

1.5

1

0.5

0.2
00

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

20γ

110No

100

1000

(a) 3–101 coupling network.
(a) Triangle network (global AD).
21.8

Coupling strength of AD

1.4

amplitude

2x1 (1st osc.)x4 (4th osc.)x55 (55th osc.)

1.6

1.2
10.8
0.6
0.4
0.2
000.2

0.4

0.6

0.8

11.2

1.4

1.6

1.8

Second AD
1.5First AD
10.5

2

γ0(b) 5–101 coupling network.

1x1 (1st osc.)x5 (5th osc.)x57 (57th osc.)

1.6
1.4

100

1000

2

1.2

Coupling strength of AD

amplitude

No(b) Pentagonal network (partial AD).

2
1.8

10.8
0.6
0.4
0.2
010

00.2

0.4

0.6

0.8

11.2

1.4

1.61.82γ(c) 5–101 coupling network.

Second AD
1.5First AD
10.50110No1001000(c) Heptagonal network (partial AD).

Figure 6: Amplitude of two coupled large-scale oscillatory
networks.

Figure 7: Coupling strength at AD.
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[9] Y. Setou, Y. Nishio and A. Ushida, “Synchronization
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10.1109/TNSE.2018.2801465, 2018.

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