2023 International Symposium on Nonlinear Theory and Its Applications
NOLTA2023, September 26-29, 2023, Catania and Online

Frustrated Synchronization of Coupled Oscillators
Using Polygonal Structures
Yoko Uwate †, Martin Schule‡, Thomas Ott‡ 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‡Institute of Computational Life Sciences, Zurich University of Applied Sciences
Schloss 1, 8820 Wdenswil, Switzerland
Email: {scli, thomas.ott}@zhaw.ch
Abstract— In this study, we investigate an eﬀect of frustration to triangular oscillatory network with strong coupling.
Frustration as environmental factor is occurred by network
topology which is composed from polygonal structure. We
focus on the amplitude change of the proposed network using diﬀerent frustration levels. By using computer simulations, the eﬀect of frustration to triangular oscillatory networks with strong coupling is shown.

computer simulations, we confrim that frastration changes
depnding on the location of the oscillator.
2. Network Model using van der Pol Oscillators
The conceptual network models with diﬀerent frustration levels used in this study is shown in Fig. 1. In thisfigure, triangular oscillators are coupled with edges on 2dimensional space and a circle denotes a van der Pol oscillator.

1. Introduction
Synchronization phenomena observed in coupled oscillatory systems are excellent for modeling high-dimensional
nonlinear phenomena. In recent years, the synchronization phenomenon of oscillators and nonlinear circuits inrelated with complex networks has been the subject of significant research [1]-[10]. These research results are also
expected to be used for engineering applications, such as
the constructions of optimum communication transmission
networks [11].
Our research group has been studying synchronization
phenomena observed in polygonal networks using coupledoscillators [12]. The proposed model is that two odd polygons are shared by a single branch. Because two adjacentoscillators are coupled to be anti-phase states, the dilemma
frustrates the oscillators in odd numbered polygonal networks. In our previous works, we have investigated theamplitude and phase changes in polygonal oscillatory networks when the coupling strength is changed. These amplitude and phase have also been solved by using numericalanalysis of the averaging method, and the amplitude deathhas been observed.
In order to approach a more realistic network models,
we extend the two coupled polygonal network model totwo-dimensional space using 20 oscillators. The coupling
topology is to be a triangulra network, and we investigate
the ampliutde change with strong coupling. By using theORCID iDs First Author:
0000-0002-2992-8852, Second Author:
0000-0002-9379-4887, Third Author:
0000-0003-1635-7816, Fourth0000-0002-0247-0001
Author:

16591378n12161518411101417321920: van der Pol oscillator

Figure 1: Network model with triangular oscillators.
Figure 2 shows a van der Pol oscillator. This oscillator iscomposed by an inductor, a negative resistance and a condenser. The oscillator component is very simple, however
the oscillator could generate oscillation time wave. Whenthe parameter of the nonlinearity is set to small value (e.g.
ε=0.1), time wave form behaves similar to sin wave.
The circuit realization (target on 1st, 5th and 6th oscillators) of triangular oscillatory networks is shown in Fig. 3.
In this circuit model, we use unique coupling method for
two adjacent oscillators. Two adjacent oscillators are coupled by a resistor via a inductor which originally belongs
to each van der Pol oscillators. The inductor of one vander Oscillator is divided to six to connect the next oscilla-

This work is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivatives 4.0 International.

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tors. If the oscillators are located at boundary position, theinductors connect to the earth resistance via resistor R.
By using this coupling scheme, two oscillators tend tosynchronize at anti-phase state. However, in the triangular oscillatory network, two oscillators can not synchronized with anti-phase state because of the network structure. Then, the coupled oscillators synchronize with phase
diﬀerence to minimize the enegy consumption. This is really original part compared with the other networks focusing on synchronization.

irv

CLR

6L 1stL
oscillator 66L
6LR

3. Simulation Results
6Lrm

R6Li R1

rm

rm
6L

R

6L

R

6L

6L

rm

6L

rm

6L

6L

R

6L

rm

6L

6Li R5iR6v5

R

For the computer simulations, 20 van der Pol oscillators are coupled in triangular oscillatory space like Fig. 1.
By changing the coupling strength (γ) the network, the obtained amplitude of coupled oscillators are investigated.
Figures 4 to 8 show the simulation results of the obtained
amplitude when the coupling strength is changed from 0.1
to 30.0. In the case of the weak coupling, the amplitudeof all oscillators are the same (see. Fig. 4). By increasing the coupling strength, first, the amplitude of 6th, 11thand 14th oscillators decreases (see. Fig. 5). Then, the amplitude death of 11th oscillator can be confirmed when thecoupling strength is set to γ = 2.6 (see. Fig. 6).

C
v1

6L

C

5th oscillator(2)

In these equations, γ is the coupling strength, ε denotes thenonlinearity of the oscillators. For the computer simulations, we calculate Eq. (3) using the fourth-order RungeKutta method with the step size h = 0.005. The parametersof this circuit model are fixed as ε = 0.1 and η = 0.0001.

Figure 2: van der Pol oscillator.

R


(1 2)
dxk


=
ε1−
xk xk − (yak + ybk + yck )

dτ3


{}


dyak 1=x−ηy−γ(y+y)kakakndτ6
}dybk 1 {=x−ηy−γ(y+y)kbkbkn

dτ6{
}


 dyck = 1 x − ηy − γ(y + y )kckckn


dτ6{
}


dy1
dk


=x−ηy−γ(y+y)
kdkdkndτ6{}
dy1
ek


=x−
ηy−γ(y+y)kekekn


dτ6
}


dy f k 1 {=x−ηy−γ(y+y)
kf
kf
kn
 dτ
6iNR

6L

The normalized circuit equations governing the circuit are
expressed asRCv6

6th oscillator

Figure 3: Circuit realization for 1st, 5th and 6th oscillators
in Fig. 1.
We develop the expression of the circuit equations of thismodel. The vk − iRk characteristics of the nonlinear resistor are assumed to be the following third order polynomialequation;iRk = −g1 vk + g3 vk 3 (g1 , g3 > 0), (k = 1, 2, 3, 4).
By using the variables and the parameters,√√
g1
t = LCτ, vk =xk ,3g3√√
g1
C
in =yn ,3g3 L√√√
LCC, γ=R, η = rm,
ε = g1CLL(1)Figure 4: Attractors of 20 oscillators coupling network(γ = 0.1).

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Figure 5: Attractors of 20 oscillators coupling network(γ = 1.0).

Figure 7: Attractors of 20 oscillators coupling network(γ = 10.0).

Figure 6: Attractors of 20 oscillators coupling network(γ = 2.6).

Figure 8: Attractors of 20 oscillators coupling network(γ = 30.0).

When the coupling strength is further increased, the amplitude of the oscillators located in the middle is reduced(see. Fig. 7). Finally, it is confirmed that the oscillators
located on the outer side of the system produce torus-likeoscillations (see. Fig. 8). In this network model, we confirm only the partial amplitude death and could not observe
the amplitude death of whole network.
Next, we investigate the time wave forms of 20 oscillators when the coupling strength is set to very large as shownin Figs. 9 and 10. In the case of γ = 10.0, it responds tofrustration with a decrease in the amplitude of the inner oscillators and the variation in the amplitude of the outer oscillators. In the case of γ = 30.0, it responds to frustrationby producing like a torus attractor in the outer oscillators.

investigated on the amplitude change of the proposed network using diﬀerent frustration levels. By using computer
simulations, we confirm that the eﬀect of frustration to triangular oscillatory networks with strong coupling.
For the future works, we need to investigate the phasestates at various frustration levels. Increasing the numberof oscillators and investigating the case of larger networks
is also one of a future task.

4. Conclusions[2] Y. Moreno and A. F. Pacheco, “Synchronization ofKuramoto Oscillators in Scale-Free Networks,” Europhys. Lett., 68, 603, 2004.

In this study, we have investigated the eﬀect of frustration to triangular oscillatory network with strong coupling.
Frustration as environmental factor is occurred by network
topology which is composed from polygonal structure. We

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x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20

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