2011 International Symposium on Nonlinear Theory and its Applications
NOLTA2011, Kobe, Japan, September 4-7, 2011

Investigation of Frustration in Several Types ofCoupled Polygonal Oscillatory Networks
Yoko Uwate† 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
Abstract—In this study, we consider the several typesof circuit models which are that odd and even polygonal
oscillatory networks are coupled by sharing a branch. We
apply the theoretical analysis by using the whole power
consumption to solve the phase diﬀerence between the adjacent oscillators. Furthermore, we discuss the general network model which is a number of polygonal oscillatory
networks are coupled by multiple branches.
1. Introduction
Coupled oscillatory systems are good models to express
essential role of high-dimensional nonlinear phenomena
occurring in the field of natural sciences. Endo et al. havepresented the details of a theoretical analysis and corresponding circuit experiments on electrical circuits oscillators arranged in a ladder, a ring and in a two-dimensionalarray topology [1]-[3]. Moreover, coupled oscillatory systems can also produce interesting phase patterns, including
wave propagation, clustering and complex patterns [4].
On the other hand, there are several types of polygonal
network structures (e.g. Honeycomb structure and crystalstructure) 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 electricaloscillators.
In our previous study, synchronization phenomena in
two coupled polygonal oscillatory networks with frustration was investigated. In this system, odd number of vander Pol oscillators are connected to every corner of polygonal network and frustration is occurred by the sharedbranch. We have confirmed that the phase diﬀerence between the shared oscillators was shifted, then other oscillators synchronized to compensate this phase shift. In order to solve the phase diﬀerence in the circuit system, wefocused on the power consumption of the coupling resistors in the whole system and proposed the theoretical analysis method by finding the minimum value of the power
consumption function. By using computer simulations andtheoretical analysis, we confirmed that coupled oscillators
tended to synchronize to minimize the power consumptionof the whole system [5].

In this study, we consider the circuit models which arethat odd and even polygonal oscillatory networks are coupled by sharing a branch. We apply the proposed theoretical analysis by using the whole power consumption to solve
the phase diﬀerence between the adjacent oscillators. Furthermore, we discuss the general network model which isa number of polygonal oscillatory networks are coupled bymultiple branches. We expect that the results of this study
contribute to understanding of synchronization phenomena
observed in general complex networks.
2. Coupled Odd-Even Polygonal Oscillatory Networks
First, we investigate the synchronization phenomena incoupled odd-even polygonal oscillatory systems. We consider the four types of circuit systems as shown in Fig. 1.
Two polygonal oscillatory networks are coupled by sharinga branch. We call the first and the second oscillators which
are connected to both side of polygonal network “sharedoscillators.”
4th osc.
1st osc.

1st osc. 4th osc.

5th osc.

3rd osc.

3rd osc.

6th osc.

2nd osc.

2nd osc. 5th osc.

(a) 3-4 coupling network.

7th osc.

(b) 3-6 coupling network.
6th osc.

5th osc.

5th osc.
1st osc.

1st osc.

6th osc.

7th osc.

4th osc.

4th osc.
2nd osc. 7th osc.
3rd osc.

(c) 5-4 coupling network.

8th osc.

2nd osc.
3rd osc.

9th osc.

(d) 5-6 coupling network.

Figure 1: Two coupled odd-even number polygonal oscillatory networks.
Next, we develop the expression for the circuit equationsof 3 − 4 coupling oscillatory networks as shown in Fig. 2.
The vk − iRk characteristics of the nonlinear resistor are ap-- 552 -4th oscillator

rm

ic2

3L

3L
rm

3L

v3

3rd oscillatorR

ic5 3L

rm

iR2
C

ib5

ia53L
rm

rm

i R33L

rm

rm

ib2

ia2

3L

3L

R

3L

ib3

v4

R

ia3

Crm

R

ic3

iR4

rm

v1

3L

ia4

RCthe even polygonal network. By adding some sort of frustration (eﬀect of the sharing branch), the diﬀerent types ofsynchronization can be observed.

3L

ic4

rm

3L

i R1

ib4

3L
rm

3L

rm

1st oscillator
ic1
ib1
ia1

iR5
C

v2

2nd oscillator

RCv5

5th oscillator

Figure 2: Circuit model for 3 − 4 coupling oscillatory networks.
proximated by the following third order polynomial equation,iRk = −g1 vk + g3 vk 3 (g1 , g3 > 0), (k = 1, 2, 3, 4, 5).

(1)

The normalized circuit equations governing the circuit are
expressed as[kth oscillator]
(dxk1 2)=ε1−xk xk − (yak + ybk + yck )


dτ3

{}

dyak 1=x−ηy−γ(y+y)
kak
akn

dτ
3{
}
 dy
1(2)
bk


=xk − ηybk − γ(ybk + yn )



dτ
3

}


dyck 1 {=x−ηy−γ(y+y)

kck
ckn

dτ3




(k = 1, 2, 3, 4, 5).
In this equations, γ is the coupling strength, ε denotes thenonlinearity of the oscillators and yn denotes the currentof neighbor oscillator on coupling resistor. For the computer simulations, we calculate Eq. (2) using a fourth-order
Runge-Kutta method with the step size h = 0.005. The parameters of this circuit model are fixed as ε = 0.1, γ = 0.1,
η = 0.001.
2.1. Synchronization Phenomena
We investigate phase diﬀerence between adjacent oscillators in two coupled polygonal oscillators for four typesof circuit model. The simulation results are summarized inTable 1. There are three phase diﬀerence types as follows;phase diﬀerence between shared oscillators (θ), phase difference of odd polygonal oscillatory network (ϕNo ), phase
diﬀerence of even polygonal oscillatory network (ϕNe ).
Where, No and Ne denote the number of coupled van der
Pol oscillators to odd and even polygonal network, respectively.
From this table, we can see that the coupled oscillators
do not synchronize with No -phase state or anti phase statein each case. In the case of that the two coupled polygonal oscillatory networks are not coupled, No -phase synchronization could be occurred in the odd polygonal network and anti-phase synchronization could be obtained inTable 1: Phase diﬀerence of oscillatory networks (computer simulation)
Circuit Model
Phase diﬀerence
Odd-Even
Shared osc. Odd osc. Even osc.
No -Neθ
ϕ No
ϕNe3-4138.60 ◦109.50◦166.70◦3-6
133.30◦112.23◦
171.11◦◦◦
5-4
152.70141.00171.00◦◦◦5-6150.31142.56174.15◦

As one example, Fig. 3 shows the time wave forms of thevoltage charged at the capacitance of each oscillator andLissajous figures obtained from 3 − 4 coupling network byusing the computer simulation.
2(a)

(d)

1.5

1

10.5

00-0.5

-1

-1

-1.5

-2
-2

-1.5

-1-0.5

00.5

1

1.5

2

2

1(e)

1.5

(b)

1

0.5

00-0.5

-1

-1

-1.5

-2
-2

-1.5

-1-0.5

00.5

1

1.5

22(c)

(f)

1.5

1

1

0.5

00-0.5

-1

-1

-1.5

-2
-2

0500

1000

1500

2000

2500

3000

3500

-1.5

-1-0.5

00.5

11.5

24000

tFigure 3: Time wave forms and Lissajous figures observedin 3 − 4 coupling network. (a) Time wave forms of 1st (red)and 2nd (black) oscillators. (b) Time wave forms of 1st (red) and3rd (black) oscillators. (c) Time wave forms of 1st (red) and 4th(black) oscillators. (d) Lissagous figure with 1st and 2nd oscillators. (e) Lissagous figure with 1st and 3rd oscillators. (f) Lissagous figure with 1st and 4th oscillators.

2.2. Theoretical Analysis
In this section, we apply the proposed theoretical analysis [5] to solve the phase diﬀerence of the shared oscillatorsand the other combination oscillators.
We assume three points as following. First, the phase
diﬀerence between shared oscillators is θ. Second, thephase diﬀerence of the coupled oscillators in odd polygonal network is described by Eq. (3).

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ϕNo = π −θ.
No − 1(3)2Third, the phase diﬀerence of the coupled oscillators ineven polygonal network is described by Eq. (4).

6150π−θ.
Ne − 1-1(4)++++12π∫ 2π03dP/dΘ-4-52-6-7-180{sin ωt + sin(ωt + θ)}2 dt∫ 2π {No − 1sin ωt2π0(θ )}2sin ωt + π −dt
No − 1∫Ne − 1 2π {sin ωt2π0(π − θ )}2sin ωt + π −dt,
Ne − 1-90+1
18090
131.693. Application for General Networks(5)

We apply the previous theoretical approach to solve
the phase diﬀerence of shared branch to the general networks. One example of general coupled polygonal network
is shown in Fig. 5.

ϕ65θ26(6)θ1

ϕ6The extreme value is solved by Eq. (8).
((dPθ )π−θ )= − sin θ + sin π −+ sin π −. (7)dθ
No − 1Ne − 1
When dPdθ = 0 is satisfied, θ is phase diﬀerence of theshared oscillators to minimize the power consumptionsof the whole system. Table 2 summarizes the result of
numerically-calculated phase diﬀerence by using Eq. (8).
We confirm that the value of phase diﬀerence obtained
from the theoretical analysis has similar value with thecomputer simulation results (see. Table 1).
Table 2: Phase diﬀerence of oscillatory networks (theoretical analysis)
Circuit Model
Phase diﬀerence
Odd-Even
Shared osc. Odd osc. Even osc.
3-4
137.42 ◦111.29◦165.81◦◦◦
3-6
131.69114.16170.34◦◦◦5-4
152.32141.92
170.77◦5-6
150.02◦142.50◦174.00◦As an example, Fig. 4 shows the graphs of Eqs. (6), (7)for 3-6 coupling network. From this figure, we can see that
extreme value of Eq. (6) corresponds to 0 (zero) of Eq. (7).

θ1

θ2ϕ5ϕ5ϕ5ϕ6

1 + cos θ + (No − 1){1 + cos(π − θ/(No − 1))}(Ne − 1){1 + cos(π − (π − θ)/(Ne − 1))}.

0θ

Figure 4: Function of power consumption for 3-6 couplingnetwork.

where the amplitue of current is set to 1 and the coupling
resistance is fixed with R = 1.
P =

4-3

In the case of No − Ne coupling system, the power consumption of the whole system is expressed by followingequation.
P =P-2PϕNe = π −5θ1

θ1ϕ6θ26ϕ6θ1
θ1

θ1
θ2

θ2ϕ5ϕ65ϕ56ϕ6ϕ5ϕ6ϕ6

Figure 5: One example of general coupled oscillatory networks.
Then we can obtain the phase diﬀerences ϕ5 and ϕ7 as
described by the following equation.
4π − (θ1 + 2θ2 ).
2(π − θ1 ) + 2(π − θ2 )ϕ6 = π −.
3
The power consumption equation is expressed as
ϕ5 =

(9)

6(1 + cos θ1 ) + 6(1 + cos θ2 )(4π − (θ1 + 2θ2 ) )+ 6 1 + cos2((π − θ1 ) + 2(π − θ2 ) )+ 12 1 + cos.
(10)
3Figure 6 shows the function of power consumption with
3D plot when θ1 and θ2 are changed from 0 to π. By calculating this function, we obtain θ1 and θ2 which show the- 554 -P =(8)

minimum value of this function. In the range of [0:π], theminimum value of θ1 and θ2 are 167.30◦ and 152.41◦ , respectively.
Table 3 summarizes the phase diﬀerences obtained fromthe computer simulations and the theoretical analysis using
Eqs. (8)-(10). We confirm that the phase diﬀerences of thetheoretical analysis match pretty well to the results of thecomputer simulations. We consider that even if calculation
becomes more complex for the asymmetrical system, thesame theoretical analysis can be applied.

(θ1,θ2)=(167.30, 152.41)P 50403020100050403020References[1] T. Endo and S. Mori, “Mode analysis of a multimodeladder oscillator,” IEEE Trans. Circuits Syst., vol.23,
pp.100-113, Feb. 1976.
[2] T. Endo and S. Mori, “Mode analysis of twodimensional low-pass multimode oscillator,” IEEETrans. Circuits Syst., vol.23, pp.517-530, Sep. 1976.
[3] T. Endo and S. Mori, “Mode analysis of a ring of alarge number of mutually coupled van der Pol oscillators,” IEEE Trans. Circuits Syst., vol.25, no.1, pp.718, Jan. 1978.
[4] M. Yamauchi, Y. Nishio and A. Ushida, ”Phasewaves in a ladder of oscillators” IEICE Trans. Fundamentals, vol.E86-A, no.4, pp.891-899, Apr. 2003.

1002040 60

1208010080 100120140
60
θ140160
θ2180 0 20180140 160[5] Y. Uwate and Y. Nishio “Synchronizing Coupled Oscillators in Polygonal Networks with Frustration,”
Proc. of ISCAS’11, pp. 745-748, May 2011.

Figure 6: One example of general coupled oscillatory networks.

Table 3: Phase diﬀerence for general oscillatory networks
Phase Type
Phase Diﬀerence
Simulation Theoryθ1167.68◦
167.30◦◦θ2
153.85152.41◦◦
ϕ5122.78
123.94◦
ϕ6158.40◦
157.37◦

4. Conclusion
In this study, we have investigated synchronization phenomena in coupled polygonal oscillatory networks withfrustration. By using computer simulations and theoretical analysis, we have confirmed that coupled oscillators
tend to synchronize to minimize the power consumption ofwhole system. Investigation of synchronization phenomena observed in coupled chaotic oscillator in the polygonal
networks with frustration is our future work.

Acknowledgment
This work was partly supported by JSPS Grant-in-Aidfor Young Scientists 23700269.

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