2011 International Workshop on Nonlinear Circuits,Communication and Signal Processing
NCSP'11, Tianjin, China, March 1-3, 2011

Synchronization in Coupled Triangular Oscillatory Networks
Shared with Two Branches
Yoko Uwate† and Yosifhimi Nishio††Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7662
Email: uwate@ee.tokushima-u.ac.jp, nishio@ee.tokushima-u.ac.jp

2nd osc.

Abstract

1.5
x1

x2

x3

1500

2000
τ

1

0.5x1, x2, x3

In this study, we investigate phase difference in three coupled
triangular oscillatory networks shared with two branches. We
apply theoretical analysis considering the power consumptionof coupling resistors to solve the phase difference. This theoretical analysis method matches well with the computer simulation results.

0-0.5

-1

3rd osc.

1st osc.

-1.5

0500

1000

(a)

2500

3000

3500

4000

(b)

Figure 1:

3rd osc.

2nd osc.
(a)

x1
x2

4th osc.

1st osc.

1.5
10.5
0-0.5
-1-1.5
1.5
10.5
0-0.5
-1-1.5x3

Coupled oscillatory circuits provide simple models for describing high-dimensional nonlinear phenomena occurring inour everyday world. Synchronization phenomena have been
extensively reported in physical, biological and electrical systems. Therefore, investigation of synchronization phenomena
observed in the coupled oscillatory systems is important issue
currently.
On the other hand, there are several types of polygonal network structures (e.g. Honeycomb structure and crystal structure) exists in the natural science. For the studies of largescale network using coupled oscillators, a ring, a ladder and a
two dimensional array structure are often investigated. However, there are not many discussions about coupled polygonal
oscillatory networks by using electrical oscillators.
In general, when three van der Pol oscillators are coupled
by resistor via inductor as ring topology, three phase synchronization (phase shift: 2π/3) can be observed (see. Fig. 1). We
assumed that some kinds of frustrations have ability to produce interesting synchronization phenomena. In our previousworks, synchronization in two coupled triangular oscillatory
networks sharing a branch (see. Fig. 2) was investigated [2].
By using computer simulations and theoretical analysis, weconfirmed that the shared oscillators are synchronized at inphase state. And the other combination oscillators synchronize with anti-phase state.
In this study, for understanding the general large system,
synchronization phenomena in coupled triangular oscillatory

1.5
10.5
0-0.5
-1-1.5x4

Three coupled van der Pol oscillators. (a) Circuit model.
(b) Three phase synchronization.

1. Introduction

1.5
10.5
0-0.5
-1-1.501000

2000

τ

3000

4000

5000

(b)

Figure 2: Two coupled triangular oscillatory networks. (a) Circuit model.
(b) Synchronization state.

networks sharing by two branches is investigated. By computer simulations, we confirm interesting synchronization
which means that the adjacent oscillators are not synchronized with three-phase state. Furthermore, we apply theoretical analysis considering the power consumption for solving
the phase difference.
2. Circuit Model
The circuit model of this study is shown in Fig. 3. Threetriangular oscillatory networks are coupled by sharing twobranches. We use van der Pol oscillator as a coupled oscillator in this system. Each van der Pol oscillator is connected- 143 -to every corners of the triangular networks.
1st osc.

3rd osc.

[3rd oscillator] dx(1 2)3=ε1−x3 x3 − (ya3 + yb3 + yc3 + yd3 )
 dτ

3
{}
 dy a31
=x3 − ηya3 − γ(ya3 + yb1 )

4

 dτ}
dy b31{=x3 − ηyb3 − γ(yb3 + ya2 )
4
 dτ
}
 dy c31{=x−ηy
3c3
 dτ

4
}
 dy d31{
=x3 − ηyd3dτ
4

4th osc.

2nd osc.

5th osc.

(a) Conceptual circuit model.
id1 ic1i R3v3

3rd oscillator

R

rm
4L

ic2

Cv2

2nd oscillator

rm

ib5

ia54L

4L

id2

iR2
C4L

4L
4L

rm

rm

R

rm

rm

ic5
4L

ia2

4L

4L

ib2

R

4L
4L

rm

id5
iR5

RCv5

5th oscillator(b) Circuit model.

Figure 3: Three coupled triangular oscillatory networks sharing with twobranches.

(4)[4th oscillator]

C
rm

rm

ib3

4L

ic3

ic4
iR4v4

R

ia3

ia4

rm

R4L
4L

RCv1
id3

rm

4Li R1ib4 id4 4L 4th oscillator

4L
rm

4L

rm

4L

ia1

ib1

rm

4L

1st oscillator dx(1 2)4
=
ε1−x4 x4 − (ya4 + yb4 + yc4 + yd4 )

 dτ

3

{}
 dy a4
1


=x−
ηy−
γ(y+y)
1a4a4
c2

 dτ4

}
dy b41{=x1 − ηyb4 − γ(yb4 + yc1 )4 dτ}
 dy c41{

=x−
ηy−
γ(y+y)
1c4c4
b5
 dτ

4}
 dy d41{
=x1 − ηyd4dτ4(5)The vk − iRk characteristics of the nonlinear resistor is ap[5th oscillator]
proximated by the following third order polynomial equation,
 dx
(1 2)
5iRk = −g1 vk + g3 vk 3 (g1 , g3 > 0), (k = 1, 2, 3, 4, 5). (1)=ε1−x5 x5 − (ya5 + yb5 + yc5 + yd5 )
 dτ

3

{}
 dy a51
The normalized circuit equations governing the circuit are=x−ηy−γ(y+y)5a5a5d2 dτ
expressed as4

}
1{
dy b5
[1st oscillator]=x5 − ηyb5 − γ(yb5 + yc4 )4 dx( dτ}
1 2)
1
 dy c51{
=
ε1−xx−(y+y+y+y)


1
1
a1
b1
c1
d1

=x5 − ηyc5


dτ
3


4}
 dτ
}
dy a11{ dy d51{=x−ηy−γ(y+y)
1
a1
a1b2

=x−
ηy
5d54

 dτdτ4}1{
dy b1(6)=x1 − ηyb1 − γ(yb1 + ya3 )wheredτ4√}√
dy c11{g1
t = LCτ , vk =xk ,=x1 − ηyc1 − γ(yc1 + yb4 )3g3
dτ4}
 dy
√
√
1{
√√

d1
=x1 − ηyd1
g1
C
g1Cdτ
4
yak , ibk =ybk ,iak =3g3 L
3g3 L
(2)
√√
√
[2nd oscillator]
LCC dx)(ε=g,γ=R,
η=r,
1m1 22CLL
x−(y+y+y+y)=ε1−x2a2b2c2d22

 dτ
3

{}
(k = 1, 2, 3, 4, 5).
 dy a2 = 1 x − ηy − γ(y + y )2a2a2b3

4
 dτ

In these equations, γ is the coupling strength, ε denotes the}1{dy b2
=x2 − ηyb2 − γ(yb2 + ya1 )nonlinearityof the oscillators and η denotes the tiny resistor

dτ4


{}
toavoid
L-loop.
For the computer simulations, these param dy c2 = 1 x2 − ηyc2 − γ(yc2 + ya4 )etersarefixedas
γ = 0.1, ε = 0.1 and η = 0.0001, respec
dτ4

{}
tively.
By
using
Runge-Kutta
method to solve the normalized

1dy

d2
=x2 − ηyd2 − γ(yd2 + ya5 )circuitequation,
wecansimulate
synchronization phenomenadτ4(3) of this circuit system.

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networks [3]. We apply this theoretical approach to the circuit model of this study which is triangle oscillatory networks
sharing with two branches.

3. Synchronization Phenomena
3.1. Computer SimulationsFirst, the time wave forms obtained from each oscillator isshown in Fig. 4. Next, we calculate the phase difference between the adjacent oscillators. The simulated results is summarized in Tab. I. We confirm that coupled oscillators split
up into four groups depended on phase difference (groupA: ∼=96◦ , group-B: ∼=133◦ , group-C: ∼=130◦ and group-D:◦∼=167 ). The coupled oscillators connected to the sharedbranches synchronize with ∼=96◦ phase shift. Because, theseoscillators have some kinds of dilemma.

01.50-1.5
1.5X30-1.5
1.5

X40-1.5
1.5

X5

θ13

θ224

θ1ϕΑ

θ25

Figure 5: Definition the phase difference.

-1.5

X2

ϕΑ

ϕΒ

First, we define the two phase differences θ1 and θ2 asshown in Fig. 5. Then we can obtain the phase differences
ϕA and ϕB as described by the following equations.

1.5

X1

10-1.5
01000

2000

3000

4000

5000

τ

Figure 4: Time wave forms obtained from each oscillator.

ϕA=

2π − (θ1 + θ2 ),ϕB=

2π − 2θ1 .

(7)

The power consumption of the whole system is expressedby the following equations.
∫ 2π2P ={sin ωt + sin(ωt + θ1 }2 dt
2π 0∫ 2π2+{sin ωt + sin(ωt + θ2 }2 dt
2π 0∫ 2π2+{sin ωt + sin(ωt + 2π − (θ1 + θ2 ))}2 dt
2π 0
∫ 2π1
{sin ωt + sin(ωt + 2π − 2θ1 )}2 dt, (8)+2π 0
where, the amplitude of the current is set to 1 and the coupling
resistance is fixed as R=1. By integrating Eq. (8), we obtain
the equation of the power consumption.

Table 1: Synchronization phenomena
Combination Phase difference
1st - 2nd95.22◦
2nd - 4th97.05◦
2nd - 3rd133.26◦
2nd - 5th133.20◦
1st - 3rd131.42◦
4th - 5th129.75◦
1st - 4th167.69◦

P=2(1 + cos θ1 ) + 2(1 + cos θ2 )

+2(1 + cos(2π − (θ1 + θ2 ))+1 + cos(2π − 2θ1 ).

(9)

Figure 6 shows the function of power consumption with
3D plot when θ1 and θ2 are changed from −π to π. By calculating this function, we obtain θ1 and θ2 which show theminimum value of this function. In the range of [0:π], theminimum value of θ1 and θ2 are 95◦ and 135◦ , respectively.
Table 2 summarizes the phase differences obtained fromthe theoretical analysis and the computer simulations using
Eqs. (11), (12). We confirm that the phase differences of
the theoretical analysis match pretty well to the results of
3.2. Theoretical Analysis
the computer simulations. We consider that even if calcuIn our previous research, we have proposed the theoretical lation becomes more complex for the asymmetrical system,
analysis to solve the phase difference of polygonal oscillatory the same theoretical analysis can be applied.

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(θ1, θ2)=(95, 135)

1412108642014121086420ππ/2-π-π/20

θ10π/2θ2-π/2π-πFigure 6: Power function of Eq. (9).
Table 2: Phase difference for oscillatory networks
Phase Type
Phase Difference
TheorySimulation
θ195.00◦
95.22/97.05◦
θ2135.00◦ 133.26/133.20◦
ϕA130.00◦ 131.42/129.75◦
ϕB
170.00◦167.69◦

4. Conclusions
In this study, we have investigated phase difference inthree coupled triangular oscillatory networks shared with twobranches. The interesting synchronization phenomena wasobserved. We assume that the adjacent oscillators are synchronized with some sort of rules.
Furthermore, we applied the theoretical analysis to solve
the phase difference and confirm that this theoretical approach matched well with the computer simulation results.
Acknowledgment
This work was partly supported by CASIO Science Promotion Foundation.
References[1] Y. Uwate, Y. Nishio and R. Stoop, “Synchronization in Three
Coupled van der Pol Oscillators with Different Coupling
Strength,” Porc. of NCSP’10, pp. 109-112, Mar. 2010.
[2] Y. Uwate, Y. Nishio and R. Stoop, “Synchronization in Two
Polygonal Oscillatory Networks Sharing a Branch,” Porc. ofNDES’10, pp. 106-109, May 2010.
[3] Y. Uwate and Y. Nishio, “Synchronizing Coupled Oscillators
in Polygonal Networks with Frustration,” Porc. of ISCAS’11,
May 2011 (Accepted).

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