2020 International Symposium on Nonlinear Theory and Its Applications,
NOLTA2020, Virtual, November 16-19, 2020

Amplitude Death observed from Coupled Oscillators
in Several Types of Polyhedron
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 Amplitude Death (AD) and amplitude change are important for understanding the control
mechanisms and efficient regulators of a system’s dynamics. Previously, the mechanism of AD observed from twodimensional coupled polygonal oscillatory networks wasmade clear by using mathematical analysis. In this study,
we extend the polygonal oscillatory networks to threedimensional space. We investigate the occurrence of AD
by using several types of 3D oscillatory network models.
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 fixed
points 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 powerconsumption 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.

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. We
explained the mechanism by which AD occurs using a theoretical approach.
In this study, we investigate the occurrence of ADof coupled polygonal oscillatory networks in threedimensional space. For considering the three-dimensional
networks, we propose two types of network models. The
first model is composed of a regular tetrahedron. As thesecond model, we propose a 3D model by adding depth tothe N–N coupling network models that has been proposedin the two-dimensional space. By using computer simulations, we observe similar AD phenomena in the proposed
three-dimensional coupled polygonal network with previous studies. We also find unique AD phenomena in 3D
oscillatory network model.
2. Coupled Oscillatory Networks in Two-Dimensional
Space [13]In our previous study, we investigated AD in two coupled oscillatory networks in two-dimensional space. Twopolygonal oscillatory networks are coupled by sharing abranch. And we investigate AD when the number of triangular or pentagonal networks connected to shared branch
is increased. Examples of the network models used in theprevious study are shown in Figs. 1 and 2. Here, the network model is composed of triangular oscillatory networks
as shown in Fig. 1. While, Fig. 2 shows the circuit model
which is composed of pentagonal oscillatory networks.
In these circuit models, we consider the coupling method
in which two adjacent oscillators tend to synchronize in theanti-phase state. The number of oscillators coupled to left
and the right polygonal networks is set to odd numbers toproduce anti-phase synchronization between adjacent oscillators. The first and second oscillators, which are connected to both sides of the polygonal networks, are called“shared oscillators.”
Figure 3 shows the circuit model of the 3–3 coupling
network. 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.

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4

ic3

ia4
iR4
Cv4

R

R

4th oscillator

ic2

rmiR2
Cv3

v2

CR

2nd oscillator

3rd oscillator

96th osc.
N-1

ib2

ia2

3L

3Li R3Figure 1: Coupled oscillatory networks composed of triangular network.
N-2

ib3

rm(b) N–triangular coupling network.

1st osc.

Ria3

R

5th osc.

rm3L3LRRrm

2nd osc.

(a) 3–3 coupling network.

v1

2

rm

3L

3rd osc.

C

ib4 ic4 3L

3L
rm

rm

3

3L

ia1
3L

5ic1

rm

N

63L

1

rm

1st oscillator
ib1

7

N-1

3L

4th osc.

rm

1st osc.

5

6

Figure 3: Coupling model (3–3 coupling network).

10

1
4th osc.

47th osc.
2nd osc.

3rd osc.

11

7

2

3

8th osc.

(a) 5–5 coupling network.

N8(b) N–pentagonal coupling network.

Figure 2: Coupled oscillatory networks composed of pentagonal network.
Next, we develop an expression for the circuit equationsof the N–M coupling oscillatory network. The vk −iRk characteristics of the nonlinear resistor are approximated by thefollowing third-order polynomial equation:same time. In the case of 3–3 coupling network, the amplitude of shared oscillators is smaller than the others. While,by increasing the triangular network as shown in Fig. 4(b),the amplitude of shared oscillators is larger than the others.
We consider that the number of triangular networks affectto frustration for whole network.
Figure 5 shows the simulation results of amplitudechange from the network model composed of pentagonal
networks. In these models, AD first occurs in the oscillators located farthest from the shared oscillators, and thenthe other oscillators stop oscillating at the same time.
22

1.8(1)

Tthe normalized circuit equations governing the circuitare expressed as: 1 dxk
= ε 1 − xk 2 xk − (yak + ybk + yck )


dτ3





dyak 1xk − ηyak − γ(yak + yn )=dτ3 dy1(2)bk
=xk − ηybk − γ(ybk + yn )dτ3dyck 1 x−ηy−γ(y+y)=kckckndτ3
(k = 1, 2, 3, ..., N + M − 2).

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 thenetwork type. We confirm that global AD occurs at the

1.6

1.4

1.4

1.2

1.2

amplitude

amplitude(k = 1, 2, 3, ..., N + M − 2).

1.6

1
0.8

1
0.8

0.6

0.6

0.4

0.4

0.2
0x1 (1st osc.)x3 (3rd osc.)

1.8

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

0.2
00.5

1γ

1.52(a) 3–3 coupling network.

02.500.5

1γ

1.5

2

2.5(b) 7–triangular coupling network.

Figure 4: Amplitude change of coupled triangular oscillatory networks.

22

1.8

1.8

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

1.6
1.4

1.4

1.2

1.2

10.8

10.8

0.6

0.6

0.4

0.4

0.2
0

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

1.6

amplitude(g1 , g3 > 0),

amplitude

iRk = −g1 vk + g3 vk

30.2
00.51γ1.5(a) 5–5 coupling network.

22.5

000.5

1γ

1.5

2

2.5(b) 17–pentagonal coupling network.

Figure 5: Amplitude change of coupled pentagonal oscillatory networks.

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3. Coupled Oscillatory
Dimensional Space

NetworksinThree-2
1.8

x1 (1st osc.)x2 (2nd osc.)x3 (3rd osc.)x4 (4th osc.)

1.6

Here, we investigate the amplitude change of the oscillators when the system model is extended to threedimensional space.

amplitude

1.4
1.2
10.8
0.6

3.1. Coupled Oscillatory Networks using Tetrahedrons
Two types of network models are proposed as shown inFig. 6. These models are composed of a triangular network.
The first model is a regular tetrahedron with four congruent
equilateral triangles. The second model is a combination of
two regular tetrahedrons sharing one regular triangle.

0.4
0.2
00

0.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

2

Figure 7: Amplitude change in three-dimensional coupledoscillatory networks (tetrahedron network).

2
1.8

x1 (1st osc.)x4 (2nd osc.)

1.6

amplitude

1.4(a) Tetrahedron network.

1.2
10.8(b) Two tetrahedrons network.

0.6
0.40.2

Figure 6: Coupled oscillatory networks in threedimensional space.
Figure 7 shows the simulation results of amplitudechange of tetrahedron network with the coupling strength.
From this result, it was found that global AD was observed in the same way as in the case of a triangular oscillatory network connected two-dimensionally. The amplitude of the all oscillators has same value. The couplingstrength causing AD is the lowest of all previous models.
It is therefore considered the tetrahedron network has large
frustration. Figure 8 shows the simulation results of amplitude change of two tetrahedron networks with the couplingstrength. In this model, there are two groups depending on
the amplitude value. The first group is 1st, 2nd and 3rd oscillators. The second groups is 4th and 5th oscillators. We
confirm that global AD occurs at the same time. In this network, the amplitude change is very similar with 3–3 coupling network, because the amplitude of shared oscillators
is smaller than the others.
To summarize the obtained results, we proposed a regular tetrahedron network as a 3D model, however we foundthat the occurrence of AD phenomena is similar with the2D models.

000.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

2Figure 8: Amplitude change in three-dimensional coupledoscillatory networks (two tetrahedrons network).

(a) 3D triangular oscillatory network (N=8).

3.2. Coupled Oscillatory Networks adding Depth
Next, we propose a 3D model by adding depth to the N–
N coupling network models that has been proposed in thetwo-dimensional space. Two types of 3D models are shownin Fig. 9. In these models, the square face is the shared faceof the networks on both sides.

(b) 3D pentagonal oscillatory network (N=16).

Figure 9: Coupled oscillatory networks in threedimensional space.

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Figure 10 shows the simulation results of amplitudechange of 3D triangular oscillatory network with the coupling strength. From this result, it was found that global
AD was observed in the same way as in the case of 3–3
coupling network.
Figure 11 shows the simulation results of amplitudechange of 3D pentagonal oscillatory network with the coupling strength. In this model, there are three groups depending on the amplitude value. One group (4th, 7th, 12thand 15th oscillators) stop their oscillation around γ=1.4.
The amplitude of the other two groups decreases with thecoupling strength. However, we can not observe the global
AD in 3D pentagonal oscillatory network. This is a phenomenon unique to 3D models.
2x1 (1st osc.)x3 (3rd osc.)

1.8

1.4

amplitude

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1.6

1.2
1

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0.8
0.6
0.4
0.2
0of the 3D pentagonal network model, we observe only partial AD and could not observe the global AD.
For the future works, we would like to make clear themechanism of AD using theoretical analysis. Further investigation of AD for larger and more complex polygonal
networks is one of the future issues.

00.2

0.4

0.6

0.8

1γ

1.2

1.4

1.6

1.8

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2Figure 10: Amplitude change in three-dimensional coupledoscillatory networks (triangular oscillatory network).

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

1.6

amplitude

1.4
1.2
1

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Reports, 531, pp. 173-199, 2013.

0.8
0.6
0.4
0.2
00

0.5

1

1.5

2γ

2.5

33.5

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4Figure 11: Amplitude change in three-dimensional coupledoscillatory networks (pentagonal oscillatory network).

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4. Conclusions[11] Y. Uwate and Y. Nishio, “Frustrated Synchronization in
Two Coupled Polygonal Oscillatory Networks,” Proc. of ISCAS’13, pp. 1781-1784, May 2013.

In this study, we investigated the occurrence of AD
observed from coupled oscillatory networks in threedimensional space. Two types of network models are proposed. The first models is composed of a tetrahedron network and the second model is adding the depth to 2D network models. By using computer simulations, we observed
similar AD phenomena in the tetrahedron network in threedimensional space with two-dimensional space. In the case[12] Y. Uwate and Y. Nishio, “Amplitude Death in Strongly Coupled Polygonal Oscillatory Networks with Sharing Branch,”
IEEE TNSE, DOI: 10.1109/TNSE.2018.2801465, 2018.
[13] Y. Uwate and Y. Nishio, “Amplitude Change of Coupled vander Pol Oscillators in Three-Dimensional Space,” Proc. ofNOLTA’19, pp. 101-104, Dec. 2019.

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