Frustrated Complex Networks of Nonlinear Circuitswith Stochastically Coupling
Yoko Uwate and Yoshifumi Nishio

Thomas OttDept. of E. E. Eng., Tokushima University
2-1 Minami Josanjima
Tokushima, Japane-mail: {uwate, nishio}@ee.tokushima-u.ac.jp

Institute of Applied Simulation
Zurich University of Applied Sciences
Zurich, Switzerlande-mail: ottt@zhaw.ch

Abstract—In this study, we focus on an effect of frustration topolygonal oscillatory network with stochastically coupling. We
propose a coupled nonliear circuit network with stochasticallycoupling. Frustration as environmental factor is occurred by
network topology which is composed from polygonal structure.
We investigate synchronization of the proposed network using
different frustration levels by changing the coupling strength. By
using computer simulations, the effect of frustration to polygonal
oscillatory networks with stochastically coupling is shown.

every certain time in a simulation, the network topology ischanged by the coupling probability. We investigate
synchronization of the proposed network using different
frustration levels by changing the coupling strength. By using
computer simulations, the effect of frustration to polygonaloscillatory networks with stochastically coupling is shown.
II. NETWORK MODEL USING VAN DER POL OSCILLATO
The conceptual network models with different frustrationlevels used in this study are shown in Fig. 1. In this figure,triangular oscillators are coupled with edges on 2-dimensional
space and black circle denotes a van der Pol oscillators.

Keywords; frustrated synchronization, coupled oscillatorsI.
INTRODUCTION
Coupled oscillatory circuits are excellent models fordescribing high-dimensional nonlinear phenomena occurring inour living world. In particular, synchronization is one of themost important functions that can be explained and exploredwith the help of an oscillator. This is because, when oscillators
are coupled, a strong correlation rhythm between oscillators
called a synchronized state appears. Therefore, many different
types of coupled oscillatory networks were proposed and many
interesting synchronization phenomena have been discovered.

Each edge has a coupling probability (p) which is key
factor to connect or not (on/off coupling). At every certain time
(τ=500) in a simulation, the network topology is changed by
the coupling probability.

In our research group, we have been studied thesynchronization phenomena observed from the nonlinear
oscillatory networks [1]. We consider that the synchronizationin complex oscillator networks are useful a deeper
understanding of control methods in smart grid,communication systems and so on [2]. However, many of thenetworks that have been studied so far are static models, and itis necessary to propose a model that changes the network
topology like a network observed in the real world including
environmental factors.

(a) Frustrate network with strong coupling.

In this study, we focus on an effect of frustration topolygonal oscillatory network with stochastically coupling. We
propose a coupled nonliear circuit network with stochasticallycoupling. Frustration as environmental factor is occurred by
network topology which is composed from polygonal structure.
In this system, van der Pol oscillators are connected to every
node of a polygonal network. In this oscillatory system, two
adjacent oscillators tend to synchronize with anit-phase. Hence,
if the number of nodes is odd, frustration occurs in thepolygonal network. Each node has a coupling probability
which is key factor to connect or not (on/off coupling). At

978-1-7281-8331-2/20/$31.00 ©2020 IEEE(b) Frustrate network with strong coupling.
Fig. 1. Network models using triangular oscillators.

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Figure 4 shows the simulation results of the average
amplitude with coupling probability (p). In the case of static
network (p=1.0), the average amplitude of 4 networks has big
difference from 0.93 to 1.94. While if the coupling probability
is set to 0.2 (dynamical networks), the average amplitude of all
4 networks has similar value around 1.99. From this result, weconfirm that the frustrated network (g=2.0) is affected a lot bystochastically coupling. By decreasing the coupling probability,
the network topology is changed dynamically, then thefrustration effect is defused.

Figure 2 shows a van der Pol oscillator. This oscillator iscomposed by an inductor, a negative resistance and acondenser. Two triangular oscillatory networks are coupled bya resistor via inductors as shown in Fig. 3.
irniv

CNRLFig. 2. van der Pol oscillator.
ic13L

rmi R3RCv3

3rd oscillator

rm

ia2

3L

3L

ib2

iR4
C3L
rm

3L
rm

3L

ib3

R

rm

ic3

R3L

3L

R

ia4v4

R

rm

Cv1
ia3

rm

3L

i R1

ib4 ic4 3L

3L
rm

ia1

3L

rm

1st oscillator
ib1

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4th oscillator

ic2

rm

iR2
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2nd oscillator

Fig. 3. Coupling scheme of two triangular oscillatory
networks.

(a)

(b)

Fig. 5. Conceptual network models using pentagonal
oscillators. (a) Frustrate network with strong coupling.
(b) Non-frustrated network with weak coupling.

III. SIMULATION RESULTSFor the computer simulations, 20 van der Pol oscillators arecoupled in triangular oscillatory space like Fig. 1 as the firststep. By changing the coupling strength (g) and the coupling
probability (p) in the network, the average amplitude ofcoupled oscillators are investigated.

IV.

CONCLUSIONSIn this study, we proposed coupled frustrated polygonal
oscillators with a stochastically coupling. This coupling isswitched on/off state depending on the coupling probability. As
the first step, 20 van der Pol oscillators are coupled intriangular oscillatory network was investigated. We calculate
the amplitude of coupled oscillators in order to capture theeffect of frustrations.
From computer simulations, we confirmed that bydecreasing the coupling probability in the proposed networks,
the average amplitude of all networks reached to 2.0 whichmeans that network has no frustration.
REFERENCES
[1]

[2]

Fig. 4. Simulation results of average amplitude depending on
the coupling strength (N=20).

978-1-7281-8331-2/20/$31.00 ©2020 IEEE

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Y. Uwate and Y. Nishio, “Amplitude Death in Strongly Coupled
Polygonal Oscillatory Networks with Sharing Branch” IEEE
Transactions on Network Science and Engineering, vol. 6, no. 3, pp.
188-197, Jul.-Sep. 2019.
F. Dorfler, M. Chertkov and F. Bullo, “Synchronization in Complex
Oscillator Networks and Smart Grid” PNAS, vol.110, no. 6, pp.20052010, Feb. 2013.

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