2011 20th European Conference on Circuit Theory and Design (ECCTD)

Synchronization and Frustration in Coupled
Large-scale Polygonal Oscillatory Networks
Yoko Uwate

Yoshifumi Nishio

Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: uwate@ee.tokushima-u.ac.jp

Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: nishio@ee.tokushima-u.ac.jp

Abstract— In this study, synchronization phenomena observedin coupled polygonal oscillatory networks with frustration isinvestigated. In this oscillatory system, dilemma is created by
the shared branch. We investigate the phase difference between
adjacent oscillators when the number of oscillators connected to
one side of the polygonal network is increased for considering
the large-scale oscillatory networks.

I. I NTRODUCTION
Coupled oscillatory systems are good models to express
essential role of high-dimensional nonlinear phenomena occurring in the field of natural sciences. Synchronization phenomena have been extensively reported in physical [1]-[4],biological [5],[6] and electrical [7],[8] systems. Endo et al.
have presented the details of a theoretical analysis and corresponding circuit experiments on electrical circuits oscillators arranged in a ladder, a ring and in a two-dimensional
array topology [9]-[11]. Moreover, coupled oscillatory systems
can also produce interesting phase patterns, including wave
propagation, clustering and complex patterns [12]. Therefore,
we expect that synchronization phenomena can be applied formodeling of the high-functional information processing in thehuman brain.
On the other hand, there are several types of polygonal
network structures (e.g. Honeycomb structure and crystal
structure) in the natural science. Generally, for the studiesof large-scale network using coupled oscillators, a ring, aladder and a two dimensional array structure are often investigated. However, there are not many discussions about coupled
polygonal oscillatory networks by using electrical oscillators.
Furthermore, it is interesting to investigate the effect of some
sort of dilemmas or frustrations for the synchronization of
coupled oscillatory systems and important to understand themechanism of synchronization observed in the field of naturalscience.
In our previous studies, synchronization phenomena in coupled polygonal oscillatory networks shared with one branchhave investigated. Here, the number of oscillators coupled tothe one polygonal network is set to odd number. We considered
two identical oscillatory networks were coupled by sharing abranch as a symmetrical system. In this oscillatory system,
frustration was occurred by the shared branch. We confirmedthat the phase difference between the shared oscillators is978-1-4577-0616-5/11/$26.00 ©2011 IEEE

shifted, then other oscillators synchronize to compensate thisphase shift.
In this study, we consider the circuit model which is that
odd number of polygonal network and even number polygonal
network are coupled by sharing one branch. For considering
the large-scale networks, the number of oscillators conncectedto the one side of polygonal network is increased untill around100. First, the phase difference and frustrations are investigatedwhen triangular oscillatory network and the even number
polygonal network are coupled by sharing one branch. Next,
we consider the quadrangular oscillatory networks and odd
number polygonal network are coupled with one branch.
II. C OUPLED T RIANGULAR N ETWORK AND E VEN
N UMBER P OLYGONAL N ETWORK
Figure 1 shows the circuit model which is that triangular
network and even number polygonal network are coupledwith one branch. Where Ne denotes the number of coupledoscillators of even polygonal network. As an example, the coupling method between the adjacent oscillators when coupled
triangular network and quadrangular network are coupled isshown in Fig. 2.
4th osc.

5th osc.

1st osc.
3rd osc.

Shared branch

6th osc.

2nd osc.
Ne-th osc.
Triangular
network

7th osc.

Even number
polygonal network

Fig. 1. Conceptual Circuit Model of Coupled Triangular Network and Even
Number Polygonal Network (Ne = 4, 6, 8, 10, ...100).

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).

(1)

The normalized circuit equations governing the circuit are
expressed as766

4th oscillator

rm

ic2

3L

3L
rm

3L

v3

3rd oscillatorR

ic5 3L

rm

iR2
C

ib5

ia5

3L
rm

rm

i R3

3L

rm

R

ib2

ia2

3L

rm

rm

ib3

v4

R

3L

C

3L

ia3

iR4

rm

ic3

ia4

R

R

[Fifth oscillator]
⎧

dx5
1 2
⎪
⎪
=
ε
1
−
x5 x5 − (ya5 + yb5 + yc5)
⎪
⎪
dτ
3
⎪
⎪

⎪
⎪
1
dy a5
⎪
⎨
=
x5 − ηya5 − γ(ya5 + yc2 )
dτ
3

1
dy b5
⎪
⎪
=
x5 − ηyb5 − γ(ya4 + yb5 )
⎪
⎪
⎪
dτ
3
⎪

⎪
⎪
⎪
⎩ dy c5 = 1 x5 − ηyc5
dτ
3

3L

ic4

rm

C

v1

3L

rm
3L

i R1

ib4

3L

3L

rm

1st oscillator
ic1
ib1
ia1

iR5
C

v2

2nd oscillator

R

C

v5

5th oscillator

Fig. 2. Circuit Model for Coupling Method (ex. Triangular Network andQuadrangular Network).

where
t=
r

[First oscillator]⎧⎪⎪ dx1 = ε 1 − 1 x1 2 x1 − (ya1 + yb1 + yc1 )
⎪
⎪
dτ
3
⎪
⎪
⎪
⎪
1
dy a1
⎪
⎨
=
x1 − ηya1 − γ(ya1 + yb2 )
dτ
3

1
dy b1
⎪
⎪
=
x1 − ηyb1 − γ(ya3 + yb1 )
⎪
⎪
⎪
dτ
3
⎪

⎪
⎪
⎪
⎩ dy c1 = 1 x1 − ηyc1 − γ(yb3 + yc1 )
dτ
3

iak =

g1
3g3

r
ε = g1

(2)

√

r
LCτ ,

r

L
,
C

vk =

C
yak ,
L

r
γ=R

g1
xk ,
3g3
r

ibk =
C
,
L

(6)

g1
3g3

r

C
ybk ,
L

r
η = rmC.
L

In these equations, γ is the coupling strength, ε denotes thenonlinearity of the oscillators and yn denotes the current ofneighbor oscillator on coupling resistor. For the simulations,the parameters are fixed as ε = 0.1, γ = 0.1 and η = 0.001.
A. Phase Difference[Second oscillator]
⎧

dx2
1 2
⎪
⎪
=
ε
1
−
x2 x2 − (ya2 + yb2 + yc2 )
⎪
⎪
dτ
3
⎪
⎪

⎪
⎪ dy a21⎪⎨=
x2 − ηya2 − γ(ya2 + yb3 )
dτ
3

dy b2
1
⎪
⎪
⎪
⎪ dτ = 3 x2 − ηyb2 − γ(ya1 + yb2 )⎪⎪⎪⎪ dy c21
⎪⎩=x2 − ηyc2 − γ(ya5 + yc2 )dτ3[Third oscillator]⎧⎪⎪ dx3 = ε 1 − 1 x3 2 x3 − (ya3 + yb3 + yc3 )
⎪
⎪
dτ
3
⎪
⎪
⎪
⎪
1
dy a3
⎪
⎨
=
x3 − ηya3 − γ(ya3 + yb1 )
dτ
3

1
dy b3
⎪
⎪
=
x3 − ηyb3 − γ(ya2 + yb3 )
⎪
⎪
⎪
dτ
3
⎪

⎪
⎪
⎪
⎩ dy c3 = 1 x3 − ηyc3dτ3
[Fourth oscillator]
⎧


dx4
1
⎪
⎪
= ε 1 − x4 2 x4 − (ya4 + yb4 + yc4 )
⎪
⎪
dτ
3
⎪
⎪
⎪ dy
⎪
1
a4
⎪
⎨
=
x4 − ηya4 − γ(ya4 + yb5 )
dτ
3

1
dy b4
⎪
⎪
⎪
⎪ dτ = 3 x4 − ηyb4 − γ(yb4 + yc1 )
⎪
⎪

⎪
⎪
⎪
⎩ dy c4 = 1 x4 − ηyc4
dτ
3

(3)

(4)

We investigate the phase difference between the adjacentoscillators when the number of oscillators connected to theeven number polygonal network is increased. In the circuit
model of Fig. 1, there are three types of the phase difference
depending on the oscillator as follows.
• Shared oscillators (1st and 2nd oscillators).
• Oscillators which are connected to the triangular network.
• Oscillators which are connected to the even number
polygonal network.
Figure 3 shows the simulation results of the phase difference. From this figure, we can see that the phase difference ofthe shared oscillators shows around 140◦ when the number ofoscillators connected to the even number polygonal network is
Ne = 4. By increasing Ne , the phase difference converges toaround 122◦. In the case of the oscillators which are connectedto the triangular network, the phase difference shows around110◦ and converge to 120◦ when the number of oscillators(Ne ) is larger than 50. The phase difference of the even number
polygonal oscillators converges to 180◦ with Ne .
We confirm that the triangular network closes to the threephase state and the even number polygonal network closes tothe anti-phase states when the circuit system becomes large.
B. Frastration(5)

Next, the frustration of coupled oscillators is investigated.
In the case of that the coupling strength is γ = 0 in Eq. (2),the amplitude of each oscillator is 2.0. By adding some kindsof frustrations, the amplitude is changed from 2.0. Namely the
amplitude of coupled oscillators is affected by the frustrations.
We research the change of the amplitude as the frustrations.

767

2

Anti 180
phase

1.5

170

0.5
x1, x2, x3

Phase difference

1

Shared oscillators
Oscillators connected to triangular network
Other oscillators

160
150

0
-0.5

140

-1
-1.5

130

-2

Three 120
phase

0

500

1000

1500

2000

2500

3000

2000

2500

3000

time

(a) Ne = 4.

110
2

100

0

10

20

30

40

50

60

70

80

90

100

1.5

The number of oscillators in even number of polygonal network

1

Phase difference with Ne .

0.5x1, x2, x3

Fig. 3.

0-0.5

The equation of the frustration is defined by decrease ratioof the amplitude as follows.
2.0 − Ak
× 100[%],(7)2.0
where, Ak denotes the amplitude of kth oscillator when theoscillators are coupled. There are three types of frustrations
depending on the coupled oscillators as follows.
• Shared oscillators.
• Third oscillator.
• Oscillators connected to the even number polygonal network.
The simulation results of the frustration by increasing Ne isshown in Fig. 4. We can see that the third oscillator has moststrong frustration in this circuit model of Fig. 1. Furthermore,
the frustration of oscillators connected to the even number
polygonal network has lower value than the other oscillatorsand show the most change from 0.7 to 0.2.
From this simulation result, the frustration decreases by
increasing the number of coupled oscillators of even number
polygonal network.

-1
-1.5
-2

0

F =

Decrease ratio of amplitude of oscillator(F [%])

1000

1500
time

Fig. 5.

(b) Ne = 100.
Time wave forms of x1 , x2 and x3 .

III. C OUPLED Q UADRANGULAR O SCILLATORY N ETWORKAND O DD N UMBER P OLYGONAL N ETWORKIn this section, we consider the circuit model which is that
quadrangular network and odd number polygonal network arecoupled with one branch as shown in Fig. 1. Where No denotes
the number of coupled oscillators of odd polygonal network.
The normalized circuit equations are obtained by same wayof above section.
5th osc.

5th osc.

4th osc. 1st osc.
Shared branch

10

7th osc.

3rd osc. 2nd osc.
No-th osc.
Shared oscillators
Third oscillator
Other oscillators

8th osc.

Quadrangular Odd number
polygonal network
networkFig. 6. Conceptual Circuit Model of Coupled Quadrangular Network and
Odd Number Polygonal Network (No = 3, 5, 7, 9, ...99).

1

A. Phase Difference
0.1

Fig. 4.

500

10The number of oscillators in even number of polygonal network

100

Decrease ratio of amplitude of oscillators (frustration) with Ne .

Next, the time wave forms of the voltage of the capacitancein the triangular oscillatory network when the number of evennumber polygonal network is set to Ne = 4, 100 is shown in
Fig. 5. We can see that the triangular network tends to thethree-phase synchronization by increasing Ne .

Figure 7 shows the phase differences when No is changed
from 5 to 101. The three types of the phase differences havethe different value when the No is smaller than 50. We can seethat all types of the phase differences tend to the anti-phasestate (phase difference: 180◦ ) by increasing No .
B. Frustration
The simulation results of the frustration by increasing No
is shown in Fig. 8. The frustrations of the quadrangularoscillatory network is smaller than the others. Furthremore, we

768

2

180

1.5

175
1
x1, x2, x3, x4

Phase difference

170
Shared oscillators
Oscillators connected to quadrangular network
Other oscillators

165
160

0.5
0
-0.5
-1
-1.5

155
-2

0

500

150

1000

1500

2000

2500

3000

2000

2500

3000

time

(a) No = 5.

145
2

140

1.5

0

10

20

30

40

50

60

70

80

90

100

110
1

The number of oscillators in odd number polygonal network

Phase difference with No .

x1, x2, x3, x4

Fig. 7.

0.5
0
-0.5
-1

confirm that the frustrations decrease by increasing the numberof coupled oscillators of odd number polygonal network.

-1.5
-2

0

500

1000

1500
time

10
Decrease ratio of amplitude of oscillator(F [%])

Fig. 9.
Shared oscillators
3rd and 4th oscillators
Other oscillators

ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid forYoung Scientists 23700269.

1

R EFERENCES

0.1

10

100

The number of oscillators in even number of polygonal network

Fig. 8.

(b) No = 101.
Time wave forms of x1 , x2 , x3 and x4 .

Decrease ratio of amplitude of oscillators (frastration) with No .

Finally, the time wave forms of the voltage of the capacitance in the triangular oscillatory network when the numberof even number polygonal network is set to No = 5, 101 isshown in Fig. 9. We can see that the quadrangular network
closes to the anti-phase synchronization by increasing No .
IV. C ONCLUSIONSIn this study, we have investigate synchronization observedin the circuit model which is that odd number of polygonal
network and even number polygonal network are coupled by
one branch. First, the phase difference and frustrations were
investigated when triangular oscillatory networks is fixed as
odd number polygonal network and the number of even number polygonal network is changed from 4 to 100. By computer
simulations, we confirmed that the triangular network closes tothe three-phase state and the even number polygonal network
closes to the anti-phase states when the circuit system becomeslarge.
Next, we considered the quadrangular oscillatory networks
and odd number polygonal network are coupled with one
branch. In this case, we confirmed that all of coupled oscillators tend to the anti-phase state by increasing No .

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