2021 IEEE International Symposium on Circuits and Systems (ISCAS) | 978-1-7281-9201-7/20/$31.00 ©2021 IEEE | DOI: 10.1109/ISCAS51556.2021.9401462

Effect of Stochastically Coupling on Frustrated
Triangular Oscillatory Network
Yoko Uwate† , 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‡
Zurich University of Applied Sciences
Einsiedlerstrasse 31a, 8820 Waedenswil, Switzerland
Email: thomas.ott@zhaw.ch

Abstract— In this study, we focus on an effect of frustration to
triangular 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 triangularoscillatory networks with stochastically coupling is shown.

I. I NTRODUCTION
Coupled oscillatory circuits are excellent models for describing high-dimensional nonlinear phenomena occurring in
our 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.
In our research group, we have been studying the synchronization phenomena observed from the nonlinear oscillatory
networks. We have investigated synchronization phenomena
in coupled polygonal oscillatory networks. Through computer
simulations and theoretical analysis, we confirmed that the
coupled oscillators tended to synchronize to minimize thepower consumption of the whole system. The phase differenceof the shared oscillators was determined by finding the minimum value of the power consumption function. We consider
that the synchronization in complex oscillator networks areuseful for a deeper understanding of control methods in smart
grid, communication systems and so on.
However, many of the networks that have been studied
so far are static models, and it is necessary to propose amodel that changes the network topology like a network
observed in the real world including environmental factors.
In this study, we focus on an effect of frustration to polygonal
oscillatory network with stochastically coupling. We propose acoupled nonliear circuit network with stochastically coupling.
Frustration as environmental factor is occurred by network
topology which is composed from polygonal structure. In thissystem, van der Pol oscillators are connected to every node ofa polygonal network. I n this oscillatory system, two adjacentoscillators tend to synchronize with anit-phase. Hence, if thenumber of nodes is odd, frustration occurs in the polygonal
network. Each node has a coupling probability which is key
factor to connect or not (on/off coupling). At every certain
time in a simulation, the network topology is changed by
the coupling probability. We investigate synchronization of the
proposed network using different frustration levels by changing the coupling strength. By using computer simulations, theeffect of frustration to polygonal oscillatory networks withstochastically coupling is shown.
II. N ETWORK M ODEL USING VAN DER P OL O SCILLATORSThe conceptual network models with different frustrationlevels used in this study is shown in Fig. 1. In this figure,triangular oscillators are coupled with edges on 2-dimensional
space and a circle denotes a van der Pol oscillator. 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 asimulation, the network topology is changed by the coupling
probability.

1

6

5

9

13

7

n

Fig. 1.

8

12

16

15

18

4

11

10

14

17

3

2

19

20

: van der Pol oscillator

Network model with triangular oscillators.

978-1-7281-9201-7/21/$31.00 ©2021 IEEE
Authorized licensed use limited to: Tokushima Univ. Downloaded on August 23,2021 at 01:48:04 UTC from IEEE Xplore. Restrictions apply.

Figure 2 shows a van der Pol oscillator. This oscillator
is composed by an inductor, a negative resistance and acondenser. The oscillator component is very simple, however
the oscillator could generate oscillation time wave. When theparameter 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 thiscircuit model, we use unique coupling method for two adjacentosicllators. Two adjacent oscillators are coupled by a resistor
via a inductor which originally belongs to each van der Pol
oscillators. The inductor of one van der Oscillator is devided
to six to connect the next oscillators. If the oscillators arelocated at boundary position, the inductors connect to the earth
resistance via resistor R.
By using this coupling scheme, two oscillators tend to synchronize at anti-phase state. However, in the triangular oscillatory network, two oscillators can not synchronized with antiphase state because of the network structure. Then, the coupledoscillators synchronize with phase difference to minimize theenegy consumption. This is really original part compared withthe other networks focusing on synchronization.

ir

i

v
L

C

NR

Fig. 2.

van der Pol oscillator.

6L 1stL
oscillator 6
6L
6L

R

R

6L

rm

R
6L

i R1

rm

rm
6L

R

6L

R

6L

R

6L

6L

rm

6L

6L

rm

R
6L

6L

rm

6L

6L

i R5

iR6
v5

R

C

5th oscillator

Fig. 3.

RCv6

6th oscillator

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 polynomial equation;iRk = −g1 vk + g3 vk 3

Where, in denotes the current from the neighbor oscillator overthe corresponding coupling resistor. By using the variables andthe parameters,
r
√
g1
xk ,
t = LCτ , vk =
3g3
r
r
r
r
g1
C
g1
C
yak , ibk =
ybk ,
iak =
3g3 L
3g3 L
r
r
r
r
g1
C
g1
C
ick =
yck , idk =
ydk ,
3g3 L
3g3 L
r
r
r
r
g1
C
g1
C
yek , if k =yf k ,
iek =
3g3 L
3g3 L
r
r
g1
C
in =
yn ,
3g3 L
r
r
r
L
C
C
, γ=R
, η = rm,
ε = g1C
L
L
The normalized circuit equations governing the circuit are
expressed as

C
v1

6L

Then the circuit dynamics is described by the following
ordinary differential equation:
 dv
k


 C dt = −iRk − iak − ibk − ick




diak


6L
= vk − rm iak − R(iak + in )


dt



dibk



6L
= vk − rm ibk − R(ibk + in )


dt


dick
= vk − rm ick − R(ick + in )
6L
(2)

dt



didk


6L
= vk − rm idk − R(idk + in )


dt



diek



6L
= vk − rm iek − R(iek + in )


dt




 6L dif k = vk − rm if k − R(if k + in )dt

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



dxk
1 2


=
ε
1
−
xk xk − (yak + ybk + yck )


dτ
3


n
o

dy ak
1



=
xk − ηyak − γ(yak + yn )


dτ
6


o

dy bk
1n


=
xk − ηybk − γ(ybk + yn )



dτ
6

o
 dy
1n
ck
=
xk − ηyck − γ(yck + yn )
dτ
6


o

dy dk
1n



=
x
−
ηy
−
γ(y
+
y
)
k
dk
dk
n


dτ
6


o

1n
dy ek


=
x
−
ηy
−
γ(y
+
y
)

k
ek
ek
n


dτ
6


o
 dy f k
1n=
xk − ηyf k − γ(yf k + yn )

dτ
6

(3)

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 Runge-Kutta

Authorized licensed use limited to: Tokushima Univ. Downloaded on August 23,2021 at 01:48:04 UTC from IEEE Xplore. Restrictions apply.

For the computer simulations, 20 van der Pol oscillators arecoupled in triangular oscillatory space like Fig. 1. By changing
the coupling strength (γ) and the coupling probability (p) in
the network, the average amplitude of coupled oscillators areinvestigated.

1.5

1.5

1

0.5

1

0.5

0

0
0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

p

amplitude

0.6

0.8

1

0.8

1

p

(a) γ=0.1.

A. Amplitude Change
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,
we confirm that the frustrated network (γ=2.0) is affected
a lot by stochastically coupling. By decreasing the coupling
probability, the network topology is changed dynamically, thenthe frustration effect is defused.
The every amplitude are calclated as shown in Fig. 5.
When the coupling strength is weak (γ=0.1), all amplitude ofcoupled oscillaotr does not change by changing the coupling
probability (Fig. 5(a)). By increasing the coupling strength,
the amplitude of some oscillators decrease with the coupling
probability. The amplitude of the oscillators which are locatedin the middle of the network more decrease than the others(Fig. 5(b)-(d)).

2(b) γ=0.5.

2

2

1.5

1.5

amplitude

III. S IMULATION R ESULTS

2amplitude

amplitude

method with the step size h = 0.005. The parameters of thiscircuit model are fixed as ε = 0.1 and η = 0.0001.

1

0.5

1

0.5

0

0
0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

p

0.6
p

(c) γ=1.0.

(d) γ=2.0.

Fig. 5.

Amplitude change.

synchronize when the coupling probability p is smaller than
0.4.
In the case of Fig. 7, the coupling strength is fixed with
γ=1.0. The coupled oscillatory network has frustration. Therefore, the phase difference can not converge one point as shownin Fig. 7(a),(b). By decreasing the value of the coupling
probability (p=0.6), the phase difference converges with small
vibrations at some points (Fig. 7(c),(d)). For the frustrated
network, the stocastically coupling is effective to obtained thesimilar phase difference with the non-frustrated network.

2

IV. C ONCLUSION

1.8

average amplitude

1.6
1.4
γ=0.1
γ=0.5
γ=1.0
γ=2.0

1.2
1
0.8
0.6
0.2

0.3

0.4

Fig. 4.

0.5

0.6
p

0.7

0.8

0.9

1

Average amplitude.

B. Phase Difference
Next, we investigate the phase difference between 1st oscillator and the others when the coupling probability is changed.
For the simulation, initial condition is set to that two adjacentoscilattors synchronize with anti-phase state. The simulation
results of the phase differences are shown in Figs. 6 and7. In the case of Fig. 6, the coupling strength is fixed with
γ=0.5. The phase difference converges several points when
the coupling probability is set to p=1.0. By decreasing thevalue of the coupling probability, the phase difference starts
to vibrate with small amplitude. And, some oscillators do not

In 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 the effect of frustrations. From computer simulations, we confirmedthat by decreasing the coupling probability in the proposed
networks, the average amplitude of all networks reached to
2.0 which means that network has no frustration.
Next, we also investigated the phase diffrence by changing
the coupling strength. When the network with weak coupling,
the oscillators does not synchronize with small value of thecoupling probability. While, the case of the network withstrong coupling, the original phase difference can be obtainedby using the appropriate coupling probability.
R EFERENCES[1] J. D. Watts and S. H. Strogatz, “Collective Dynamics of Small-World
Networks,” Nature, 393, 440, 1998.
[2] S. H. Strogatz, “Exploring Complex Networks,” Nature, 410, 268, 2001.
[3] E. Bullmore and O. Sporns, “The Economy of Brain Network Organization,” Nat. Rev. Neurosci., 10, 186, 2009.
[4] T. Nishikawa, A. E. Motter, Y. C. Lai and F. C. Hoppensteadt,“Heterogeneity in Oscillator Networks: Are smaller worlds easier tosynchronize?,” Phys. Rev. Lett., 91, 014101, 2003.

Authorized licensed use limited to: Tokushima Univ. Downloaded on August 23,2021 at 01:48:04 UTC from IEEE Xplore. Restrictions apply.

180

160

160

140

140

120

120
phase difference

phase difference

180

100

80

100

80

60

60

40

40

20

20

0

0
0

1000

2000

3000

4000

5000
t

6000

7000

8000

9000

10000

0

1000

2000

3000

180

160

160

140

140

120

120

100

80

60

40

20

20

8000

9000

10000

6000

7000

8000

9000

10000

6000

7000

8000

9000

10000

6000

7000

8000

9000

10000

6000

7000

8000

9000

10000

0
1000

2000

3000

4000

5000
t

6000

7000

8000

9000

10000

0

1000

2000

3000

(b) p=0.8.

4000

5000
t

(b) p=0.8.

180

180

160

160

140

140

120

120
phase difference

phase difference

7000

80

40

0

100

80

100

80

60

60

40

40

20

20

0

0
0

1000

2000

3000

4000

5000
t

6000

7000

8000

9000

10000

0

1000

2000

3000

(c) p=0.6.

4000

5000
t

(c) p=0.6.

180

180

160

160

140

140

120

120
phase difference

phase difference

6000

100

60

0

100

80

100

80

60

60

40

40

20

20

0

0
0

1000

2000

3000

4000

5000
t

6000

7000

8000

9000

10000

0

1000

2000

3000

(d) p=0.4.

4000

5000
t

(d) p=0.4.

180

180

160

160

140

140

120

120
phase difference

phase difference

5000t(a) p=1.0.

180phase difference

phase difference(a) p=1.0.

4000

100

80

100

80

60

60

40

40

20

20

0

0
0

1000

2000

3000

4000

5000
t

6000

7000

8000

9000

10000

0

1000

2000

(e) p=0.2.
Fig. 6.

Phase difference (γ=0.5).

3000

4000

5000
t

(e) p=0.2.
Fig. 7.

Phase difference (γ=1.0).

Authorized licensed use limited to: Tokushima Univ. Downloaded on August 23,2021 at 01:48:04 UTC from IEEE Xplore. Restrictions apply.

[5] Y. Moreno and A. F. Pacheco, “Synchronization of Kuramoto Oscillatorsin Scale-Free Networks,” Europhys. Lett., 68, 603, 2004.
[6] M. Chavez, D. -U. Hwang, A. Amann, H. G. E. Hentschel and S. Boccaletti, “Synchronization is Enhanced in Weighted Complex Networks,”
Phys. Rev. Lett., 94, 218701, 2005.
[7] A. Arenas, A. Dı́az-Guilera and C. J. Perez-Vicente, “Synchronization
Reveals Topological Scales in Complex Networks” Phys. Rev. Lett., 96,114102, 2006.
[8] C. Zhou, A. E. Motter and J. Kurths, “Universality in the Synchronization of Weighted Random Networks,” Phys. Rev. Lett., 96, 034101,
2006.
[9] J. Gómez-Gardeñes, Y. Moreno and A. Arenas, “Paths to Synchronization in Complex Networks” Phys. Rev. Lett., 034101, , 2007.
[10] I. Lodato, S. Boccaletti and V. Latora, “Synchronization Properties of
Network Motifs” Europhys. Lett., 78, 28001, 2007.
[11] A. Arenas, A. Dı́az-Guilera, J. Kurths, Y. Moreno and c. Zhou, “Synchronization in Complex Networks” Phys. Rep., 469, 93, 2008.
[12] J. Gómez-Gardeñes, S. Gomez, A. Arenas and Y. Moreno, “Explosive
Synchronization Transitions in Scale-Free Networks” Phys. Rev. Lett.,
106, 128701, 2011.
[13] L. V. Gambuzza, A. Cardillo, A. Fiasconaro, L. Fortuna, J. GomezGardenes and M. Frasca, “Analysis of Remote Synchronization inComplex Networks,” Chaos, 23, 043103, 2013.
[14] S. Boccaletti, J. Kurths, G. Osipov, D. Valladares and C. Zhou, “The
Synchronization of Chaotic Systems” Physics Reports, 366, pp. 1-101,
2002.
[15] A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, “Phase
Synchronization of Chaotic Oscillators” Physics Reports, 469, pp. 93153, 2008.
[16] M.G. Rosenblum, A.S. Pikovsky and J. Kurths, “Phase Synchronizationof Chaotic Oscillators” Physical Review Letters, vol.76, no.11, pp.18041807, Mar. 1996.
[17] W. Wang, I.Z. Kiss and J.L. Hudson, “Experiments on Arrays of
Globally Coupled Chaotic Electrochemical Oscillators: Synchronization
and Clustering” Chaos, vol.10, no.1, pp.248-256, Mar. 2000.
[18] M. Yamauchi, Y. Nishio and A. Ushida, ”Phase-waves in a Ladder ofOscillators” IEICE Trans. Fundamentals, vol.E86-A, no.4, pp.891-899,
Apr. 2003.
[19] H.B. Fotsina and J. Daafouza, “Adaptive Synchronization of Uncertain
Chaotic Colpitts Oscillators based on Parameter Identification” Physics
Letters A, vol.339, pp.304-315, May 2005.
[20] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations in
a Modified van der Pol Oscillator Containing a Positive Nonlinear
Conductance,” Proc. of IEEE, vol. 69, pp. 394-395, 1981.

Authorized licensed use limited to: Tokushima Univ. Downloaded on August 23,2021 at 01:48:04 UTC from IEEE Xplore. Restrictions apply.