Synchronizing Coupled Oscillators in Polygonal
Networks with Frustration
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

First, we investigate the synchronization phenomena in thesymmetrical coupled oscillatory systems. Two identical polygonal oscillatory networks are coupled by sharing a branch as
shown in Fig. 1. We call the first and the second oscillators
which are connected to both side of polygonal network “sharedoscillators.”
Nth osc.

4th osc.

1st osc.

(N+1)th osc.
(N+2)th osc.

6th osc.

1st osc.

5th osc.

(N+3)th osc.
2nd osc.

3rd osc.

4th osc.

2nd osc.

(N+4)th osc.
(2N-2)th osc.

3rd osc.

(a) 3-3 coupling network.
(b) N -N coupling (general) network.
Fig. 1. Two coupled polygonal oscillatory networks (N denotes the numberof coupled oscillators connected to the one side of the polygonal oscillatory
network).

ic1

3L

rm

3L

ia4
rm

ib2

ia2

3L

3L

iR4
C

v4

R

4th oscillator

ic2

rm

i R3

iR2
C

v3

3rd oscillator

Fig. 2.

rm

ib3

3L

ic3

R

3L

v1

3L

R

C
R

ia3

rm

3L

3L

i R1

ib4 ic4

3L
rm

ia1

rm

rm

1st oscillatorib1

rm

978-1-4244-9472-9/11/$26.00 ©2011 IEEE

II. S YMMETRICAL C OUPLED O SCILLATORY N ETWORKS

3L

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. have presented
the details of a theoretical analysis and corresponding circuit
experiments on electrical circuits oscillators arranged in aladder, a ring and in a two-dimensional array topology [1]-[3].
Moreover, coupled oscillatory systems can also produce interesting phase patterns, including wave propagation, clusteringand complex patterns [4].
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 studies oflarge-scale network using coupled oscillators, a ring, a ladderand a two dimensional array structure are often investigated.
However, there are not many discussions about coupled polygonal oscillatory networks by using electrical oscillators.
In this study, synchronization phenomena in coupled polygonal oscillatory networks with frustration is investigated. In
this system, van der Pol oscillators are connected to every
corner of polygonal network. First, we consider two identicaloscillatory networks are coupled by sharing a branch as asymmetrical system. In this oscillatory system, frustration is
occurred by the shared branch. We observe the different synchronization state than the coupled oscillatory system without
frustration. We confirm that the phase difference between theshared oscillators is shifted, then other oscillators synchronize
to compensate this phase shift. How do the shared oscillators
determine this phase shift? What effects will be occurred whensuch oscillatory system is applied for the large-scale network?
In order to make clear these questions, we focus on the power
consumption of the coupling resistors in the whole system andinvestigate the phase difference adjacent oscillators. By using
computer simulations and theoretical analysis, we confirm that
coupled oscillators tend to synchronize to minimize the powerrm

I. I NTRODUCTION

consumption of the whole system. Furthermore, the phasedifference of the shared oscillators is solved by finding theminimum value of the power consumption function. Finally,
we apply the analysis method of symmetrical system to theasymmetrical and general systems. We expect that the resultsof this study contribute to understanding of synchronization
phenomena observed in general complex networks.

3L

Abstract— In this study, synchronization phenomena observedin coupled polygonal oscillatory networks with frustration isinvestigated. We focus on the power consumption of coupling
resistors of the whole system. By using computer simulations
and theoretical analysis, we confirm that the phase differences of
the coupled oscillators are solved by finding the minimum valueof the power consumption function.

R

C

v2

2nd oscillator

Circuit model for 3 − 3 coupling oscillatory networks.

Next, we develop the expression for the circuit equations of
3 − 3 coupling oscillatory networks as shown in Fig. 2. The

745

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), (k = 1, 2, 3, 4).

In this equations, γ is the coupling strength, ε denotes thenonlinearity of the oscillators and yn denotes the current ofneighbor oscillator on coupling resistor.
A. Synchronization Phenomena

x1
x2

v2

x3

1.5
1
0.5
0
-0.5
-1
-1.5

v3

x4

Figure 3 shows the time wave form of the voltage charged at
the capacitance of each oscillator obtained from the computer
simulation and the circuit experiment. From this figure, we cansee that the first and the second oscillators are synchronized
at in-phase (phase difference: 0 degree). While, the othercombination oscillators synchronize with anti-phase (phasedifference: 180 degree).
We did not expect such synchronization state. Because, inthe case of three coupled van der Pol oscillators as ring topology, we can observe three-phase synchronization state (phasedifference: 120 degree) [5]. So we assumed that two pairof three-phase synchronization could be observed. By adding
one more oscillator, the shared oscillators have frustration tosynchronize for both side of polygonal networks. In such case,three-phase synchronization could not be observed.

1.5
1
0.5
0
-0.5
-1
-1.5

1.5
1
0.5
0
-0.5
-1
-1.5

v4

v1

0

1000

2000

τ

3000

4000

5000

TABLE IP HASE DIFFERENCE OF SYMMETRICAL OSCILLATORY NETWORKS( COMPUTER SIMULATION )(1)

The normalized circuit equations governing the circuit are
expressed as[kth oscillator]
⎧

dxk
1 2
⎪
⎪
=
ε
1
−
xk xk − (yak + ybk + yck )
⎪ dτ
⎪
3
⎪
⎪

⎪ dy ak1⎪⎪=
xk − ηyak − γ(yak + yn )
⎪
⎪
⎨ dτ
3

1
dy bk
(2)
=
xk − ηybk − γ(ybk + yn )
⎪
⎪
dτ
3
⎪
⎪
⎪
⎪
1
dy ck
⎪
⎪
=
xk − ηyck − γ(yck + yn )
⎪
⎪
3
⎪
⎩ dτ
(k = 1, 2, 3, 4).

1.5
1
0.5
0
-0.5
-1
-1.5

see that the coupled oscillators do not synchronize with N phase state in each case.

Symmetrical Coupling
N − N (ON =2N − 2)
3-3 (ON =4)
5-5 (ON =8)
7-7 (ON =12)
9-9 (ON =16)
11-11 (ON =20)

Phase difference
Shared (1st-2nd) osc.
Other comb.
0◦
180◦
106◦
153◦ -154◦
130◦
158◦ -159◦
142◦
162◦ -163◦
149◦
164◦ -165◦

B. Power Consumption
In order to make clear the reason of that N -phase synchronization could not be observed in N -N coupling system, we
focus on the power consumption of coupling resistors in thewhole system. We assume the current of the inductor (3L) as
following equation.
√(3)
ik (t) = 2Ie sin(ωt + ϕ) (k = 1, 2, 3...ON ).
When the coupling resistance is fixed with R = 1, the average
power consumption of the coupling resistor between kth and(k + 1)th oscillators are described as Eq. (4).
2π
1{ik (t) + ik+1 (t)}2 dt(4)
2π 0
We calculate the power consumption of the whole system
by using the phase difference obtained from the computer simulations. For comparison, the power consumption of N -phasestate is also calculated. Table II summarizes the calculation
results when N is changed form 3 to 11. We confirm thatthe total power consumption of N -phase state is larger thanthe computer simulation results. In the case of 3-3 coupling
network (see. Fig. 2), the total number of coupling resistors isfive. When the coupled oscillators are synchronized at threephase state, the power consumption of each coupling resistor isIe 2 . Then, the power consumption of the whole system is 5Ie 2(=5×Ie 2 ). While, when the shared oscillators synchronize within-phase state, the power consumption of shared oscillators is
4Ie 2 (=(Ie + Ie)2 ). And other power consumptions become
zero by anti-phase synchronization.

P =TABLE IIP OWER CONSUMPTION FOR SYMMETRICAL OSCILLATORY NETWORKS
Symmetrical Coupling
N − N (ON =2N − 2)
3-3 (ON =4)
5-5 (ON =8)
7-7 (ON =12)
9-9 (ON =16)
11-11 (ON =20)t(a) Computer simulation.
(b) Circuit experiment.
Fig. 3. Time wave form of the voltage charged at the capacitance of each
oscillator. (a): ε = 0.1, γ = 0.1, η = 0.0001, (b): L = 30mH, C = 33nF .

Next, we investigate phase difference of two coupled polygonal oscillators when the number of oscillators in one side ofpolygonal network (N ) are changed from 5 to 11 with odd
number of steps. The simulation results are summarized inTable I. ON denotes the total number of oscillators in two
coupled polygonal oscillatory system. From this table, we canPower Consumption
Simulation
N -phase presumption
4Ie 2
5Ie 2
2
3.132Ie
3.437Ie 2
2.413Ie 2
2.580Ie 2
1.949Ie 2
2.050Ie 2
1.639Ie 2
1.700Ie 2

From these results, we can see that the coupled oscillators
tend to synchronize to minimize the power consumption of
the whole system. Then, N -phase synchronization can not
be obtained in two coupled polygonal oscillatory networks
sharing the branch with frustration.

746

1.5

C. Theoretical Analysis

36

1

ϕN = π − θ/(N − 1).

P

2π

P

0

=
+

2(N − 1)2π+

sin ωt + π −

-1.5
-180

-90

−π

0

5

1

6

Oscillator

259
3 8

180

110

π

5th osc.

{sin ωt + sin(ωt + θ)} dt

0

4 7

208
90

phase difference

2

2π

54

(a) Power consumption.
(b) Polar representation.
Fig. 4. Functions of power consumption and polar representation for 5 − 5coupling network.

7th osc.

1st osc.

sin ωt

2
θ
dt,
N −1

2nd osc.
3rd osc.

P = 1 + cos θ + 2(N − 1){1 + cos(π − θ/(N − 1))}. (7)The extreme value is solved by Eq. (8).
dP
= − sin θ + 2 sin(π − θ/(N − 1)).
dθ

8th osc.

4th osc.

(6)

where the amplitue of current is set to 1 and the coupling
resistance is fixed with R = 1. By integrating Eq. (6), we
obtain the equation of the power consumption (Eq. (7)).

(8)

When dPdθ = 0 is satisfied, θ is phase difference of theshared oscillators to minimize the power consumptions of thewhole system. Table III summarizes the result of numericallycalculated phase difference by using Eq. (8). We confirm thatthe value of phase difference obtained from the theoreticalanalysis has similar value with the computer simulation results(see. Table I).

Fig. 5.

9th osc.

Asymmetrical coupled polygonal networks for (N − M )=(5 − 7).

Next, we consider the asymmetrical system which is composed two different types of polygonal oscillators as shown in
Fig. 5.
The phase difference obtained from the computer simulations and the power consumption are summarized in Table IV.
By increasing the total coupled number of oscillators, thephase difference of shared oscillators increases and the power
consumption becomes small.
TABLE IVP HASE DIFFERENCE FOR ASYMMETRICAL OSCILLATORY NETWORKS( COMPUTER SIMULATION )Asymmetrical
Coupling
N −M
3-5
3-7
3-9
3-11
5-7
5-9
5-11
7-9
7-11

P HASE DIFFERENCE FOR SYMMETRICAL OSCILLATORY NETWORKS( THEORETICAL ANALYSIS )Phase difference
Shared (1st-2nd) osc. (θ) Other comb. (ϕN )
0◦
180◦
110◦
152.5◦
131◦
158.2◦
142◦
162.3◦
149◦
165◦

As an example, Fig. 4 shows the graphs of Eqs. (7), (8)and the polar representation of the phase differences in thewhole system for 5 − 5 coupling network (see. Fig. 1 (b)).
From Fig. 4 (a), we can see that extreme value of Eq. (7)corresponds to 0 (zero) of Eq. (8). We confirm the the phasespace (4π-θ) is divided equally by the all oscillators other than
the shared oscillators (Fig. 4 (b)).

10th osc.

III. A SYMMETRICAL C OUPLED O SCILLATORY N ETWORKSTABLE III

Symmetrical Coupling
N − N (ON =2N − 2)
3-3 (ON =4)
5-5 (ON =8)
7-7 (ON =12)
9-9 (ON =16)
11-11 (ON =20)

34

P

-1

6th osc.

1
2π

0

35

-0.5

(5)

In the case of N − N coupling system, the power consumption of the whole system is expressed by following equation.

2 106

dP/dΘ

0.5

dP/dΘ

In this section, the phase difference of shared oscillators issolved by using theoretical analysis. We assume two points asfollowing. First, the phase difference between shared oscillators is θ. Second, the phase difference of the other combination
oscillators is described by Eq. (5).

Sharedosc. (θ)
67◦
86◦
94◦
102◦
119◦
125◦
129◦
136◦
140◦

Phase Difference
N osc.
M osc.
(ϕN )(ϕM )145◦ -146◦
163◦ -164◦136◦ -137◦
166◦ -167◦
132◦ -133◦
168◦ -169◦
129◦ -130◦
171◦ -172◦
150◦ -151◦
160◦ -161◦
160◦ -161◦
164◦ -165◦
147◦ -148◦
165◦ -166◦
158◦ -159◦
163◦ -164◦
156◦ -157◦
165◦ -166◦

Power
3.787
3.593
3.474
3.329
2.800
2.608
2.484
2.191
2.046

Next, the phase difference of shared oscillators is solved byusing same as previous theoretical analysis. For the case of
the asymmetric system, we calculate the power consumptionof each polygonal network separately. The equations of thetotal power consumption is described as follows.

747

P

=

1 + cos θ + (N − 1){1 + cos(π − θ/(N − 1))}+(M − 1){1 + cos(π − θ/(M − 1))}.

(9)

dP
dθ

=

− sin θ + sin(π − θ/(N − 1))+sin(π − θ/(M − 1)).

(10)

Table V shows the numerically-calculated phase differenceby using Eq. (10). From this table, we can see that the resultsof the theoretical analysis match well with simulation results(see. Table IV).
TABLE VP HASE DIFFERENCE FOR ASYMMETRICAL OSCILLATORY NETWORKS( THEORETICAL ANALYSIS )Asymmetrical
N −M
3-5
3-7
3-9
3-11
5-7
5-9
5-11
7-9
7-11

Phase Difference
N osc. (ϕN )140.5◦
133.5◦
130.0◦
149.8◦
128.0◦
148.3◦
147.5◦
157.3◦
156.7◦

Shared osc. (θ)
79◦
93◦
100◦
104◦
121◦
127◦
130◦
136◦
140◦

value of this function. In the range of [0:π], the minimum
value of θ1 and θ2 are 130.5◦ and 158.0◦, respectively.
Table VI summarizes the phase differences obtained fromthe computer simulations and the theoretical analysis using
Eqs. (11), (12). 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, the same
theoretical analysis can be applied.

M osc. (ϕM )160.3◦
164.5◦
167.5◦
159.8◦
169.6◦
164.1◦
167.0◦
163.0◦
166.0◦

P(θ1, θ2)=(130.5, 158.0)

260
240
220
200
180
160
140
120
100

180
-180

−π

90
0

θ1
Fig. 7.

ϕ7

ϕ5

ϕ7

ϕ7

4

3

θ2

ϕ7

θ1

ϕ5

θ2

5

θ2

θ1
θ2

ϕ7

θ1
θ2
ϕ5
ϕ7

ϕ7
ϕ7

7

10

θ1

θ2
θ1

5

9

From these results, we can apply this theoretical analysis forthe large-scale networks close to the real networks existing in
the our living world.

ϕ5

ϕ5
ϕ7
ϕ7

One example of general coupled oscillatory networks.

(Np − 1)π − (2θ1 + θ2 )Np − 3(Np = 5, 7).

(11)

V. C ONCLUSION
In this study, we have investigated synchronization phenomena in coupled polygonal oscillatory networks with frustration.
By using computer simulations and theoretical analysis, wehave confirmed that coupled oscillators tend to synchronize
to minimize the power consumption of whole system. Investigation of synchronization phenomena observed in coupledchaotic oscillator in the polygonal networks with frustration is
our future work.
R EFERENCES

The power consumption equation is expressed as
P

Phase Difference
Simulation
Theory130◦130.5◦
158◦
158.0◦
150.7◦
150.5◦
164.5◦
165.3◦

ϕ7

For this system, first, we define the two phase differences
θ1 and θ2 as shown in Fig. 6. Then we can obtain the phasedifferences ϕ5 and ϕ7 as described by the following equation.
ϕNp =

θ2

8

7

ϕ7

6

-90

TABLE VI
P HASE DIFFERENCE FOR GENERAL OSCILLATORY NETWORKS

7

2

θ1

ϕ5

Fig. 6.

θ2

ϕ7
1

5

θ1

7

ϕ5

180

Power function of Eq. (12).

Phase Type590

π -180−π

IV. A PPLICATION FOR G ENERAL N ETWORKS
Finally, 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 shownin Fig. 6.

π

0

-90

= 6(2π + π cos θ1 ) + 6(2π + π cos θ2 )(4π − (2θ1 + θ2 ))2(6π − (2θ1 + θ2 )).
(12)+ 12 2π +4Figure 7 shows the function of power consumption with 3Dplot when θ1 and θ2 are changed from −π to π. By calculating
this function, we obtain θ1 and θ2 which show the minimum+ 6 2π +748[1] T. Endo and S. Mori, “Mode analysis of a multimode ladder oscillator,”
IEEE Trans. Circuits Syst., vol.23, pp.100-113, Feb. 1976.
[2] T. Endo and S. Mori, “Mode analysis of two-dimensional low-passmultimode oscillator,” IEEE Trans. Circuits Syst., vol.23, pp.517-530,
Sep. 1976.
[3] T. Endo and S. Mori, “Mode analysis of a ring of a large number ofmutually coupled van der Pol oscillators,” IEEE Trans. Circuits Syst.,
vol.25, no.1, pp.7-18, Jan. 1978.
[4] 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.
[5] Y. Uwate, Y. Nishio and R. Stoop, “Synchronization in Three Coupled
van der Pol Oscillators with Different Coupling Strength,” Proc. of
NCSP’10, pp. 109-112, Mar. 2010.