18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)

Synchronization in Two Polygonal Oscillatory
Networks Sharing a Branch
Yoko Uwate

Yoshifumi Nishio

Ruedi Stoop

UZH/ETH Zurich,
Dept. of Electrical and Electronic Engineering,
UZH/ETH Zurich,
Winterthurerstrasse 190,
Tokushima University
Winterthurerstrasse 190,
CH-8057 Zurich, Switzerland,
2-1 Minami-Josanjima, Tokushima, Japan
CH-8057 Zurich, Switzerland,
Email: yu001@ini.phys.ethz.ch
Email: nishio@ee.tokushima-u.ac.jp
Email: ruedi@ini.phys.ethz.chosc.

Abstract—In this study, synchronization phenomena
observed in two coupled polygonal oscillatory networks
sharing a branch is investigated. We focus on the phasedifference between two oscillators which are coupled tothe shared branch of the two polygonal networks. By
computer simulations, we observe that synchronization
state of the shared oscillators depends on the number ofcoupled oscillators of the each polygonal network.

osc.

osc.

osc.
osc.

osc.

osc.

osc.

osc.

Fig. 1.

osc.

General two coupled polygonal network oscillators.

I. I NTRODUCTION
II. T WO C OUPLED T RIANGLE O SCILLATORYN ETWORKS

Synchronization, in particular, is one of the mostimportant features that can be described and exploredwith the help of oscillators, because, upon their coupling, strongly correlated rhythms among the oscillators
emerge, called synchronized states. Synchronization phenomena have been extensively reported in physical [1][4], biological [5],[6] and electrical [7],[8] systems.
Moreover, there are several types of polygonal network structures (e.g. Honeycomb structure and crystalstructure) exists in the natural science. For the studiesof large-scale 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 this study, synchronization phenomena in two coupled polygonal oscillatory networks is investigated. Two
polygonal oscillatory networks is shared a branch of
two networks as shown in Fig. 1. First, we consider
the two triangle oscillatory networks are shared by abranch. By using computer simulations and theoreticalanalysis, the phase difference and the amplitude of each
oscillator are discussed. Furthermore, we investigate thephase difference between the shared oscillators when
two polygonal (quadrilateral, pentagon, hexagon andheptagon) oscillatory networks are coupled.

26-28 May 2010, Dresden, Germany

We consider the two coupled triangle oscillatory networks sharing a branch as a first circuit model.
A. Circuit Model
The circuit model of two coupled triangle oscillatory
networks sharing the branch is shown in Fig. 2.
We assume that the vk − iRk characteristics of thenonlinear resistor in each oscillator is given by thefollowing third order polynomial equation,iRk = −g1 vk + g3 vk 3

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

The normalized circuit equations governing the circuitin Fig. 2 are expressed as[First oscillator]
 dx

1 2
1

=
ε1
−x1 x1 − (ya1 + yb1 + yc1 )


dτ3
o

dy a11n=x1 − ηya1 − γ(ya1 + yb2 )dτ3n(2)ody1
b1
=x1 − ηyb1 − γ(ya3 + yb1 )dτ3no dy c1 = 1 x − ηy − γ(y + y )1c1c1b4dτ362

18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)
ic1

3L

rm

ia2

3L

3L

ib23L

rm

3L

3L

ib3

iR4
Cv4

rm

R

ia3

g1
iak =
3g3
r
L,
ε = g1Cr

3L

ia4

3L

3L

3L

R

rm

4th oscillator

ic2

rm

i R3

R

r

C
yak ,
Lγ=R

ibk =r

C,
L

iR2
v2

2nd oscillator

3rd oscillator

B. Synchronization Phenomena(a) Two coupled triangle oscillatory networks.
1st osc.

3rd osc.

For the computer simulations, we calculates Eqs. (2)(5) using a fourth-order Runge-Kutta method with thestep size h = 0.005. The parameters of this circuit model
are fixed as ε = 0.1, γ = 0.1, η = 0.0001.
Figure 3 shows the time wave form of the voltage
charged at the capacitance of each oscillator. From thisfigure, we can see that the first and the second oscillators are synchronized at in-phase (phase difference: 0degree). While, the other combination oscillators synchronize with anti-phase (phase difference: 180 degree).
Furthermore, the amplitude of between the first/second
and the third/fourth oscillators has small difference. The
phase plane of each combination oscillator is shown in
Fig. 4.

4th osc.

2nd osc.

(b) Conceptual circuit model.
Fig. 2.

Circuit model.

[Second oscillator]
 dx

1 22

= ε 1 − x2 x2 − (ya2 + yb2 + yc2 )

dτ

n 3o
dy1a2
=x2 − ηya2 − γ(ya2 + yb3 )dτ3ody b21n=x−ηy−βγ(y+y)
2a1b2b2 dτ3no dy c2 = 1 x − ηy − βγ(y + y )2c2a4c2dτ
3[Third oscillator]
 dx

1 2
3

=
ε
1−x3 x3 − (ya3 + yb3 + yc3 )


dτ
3
no

dy a31
=x3 − ηya3 − γ(ya3 + yb1 )dτ3nody b31=x−ηy−γ(y+y)3a2b3b3dτ3no dy c3 = 1 x − ηy3c3dτ3[Fourth oscillator]
 dx

1 2
4

=
ε1−x3 x3 − (ya4 + yb4 + yc4 )

dτ
3
no

dy a41
=x4 − ηya4 − γ(ya4 + yc2 )
dτ
3no
dy
1
b4

=x−
ηy−γ(y+y)4c1b4b4


dτ
3no

 dy c41
=x4 − ηyc4dτ3
wherer
√g1t = LCτ , vk =xk ,3g3

26-28 May 2010, Dresden, Germany

g1
3g3

r

Cybk ,LrCη = rm,L(k = 1, 2, 3, 4).
r

In this equations, γ is the coupling strength and εdenotes the nonlinearity of the oscillators.

(3)

(4)

x1

CR

1.5
10.5
0-0.5
-1-1.5

x2

v3

1.5
1
0.5
0-0.5
-1
-1.5

x3

C

1.5
1
0.5
0-0.5
-1
-1.5

x4

ic3

R

C

v1

3L

rm

rm

ia1
i R1

ib4 ic43L
rm

rm

rm

1st oscillator
ib1

1.5
1
0.5
0-0.5
-1
-1.5
01000

2000

τ

3000

4000

5000

Fig. 3. Time wave form of the voltage charged at the capacitanceof each oscillator.

Anti-phase synchronization(2-3, 3-1, 1-4, 4-2 oscillators)

x4

1st osc.

(5)

x1x2

x3
3rd osc.

x1x3

63

x3
x2

2nd osc.

x2Fig. 4.

4th osc.

x1

In-phase synchronization(1-2 oscillators)

Phase plane of each oscillator.

18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)nεγ6γρ1 + 2γρ2 cos(θ1 − θ2 )3ρk (4γ 2 + 9)−3ρ2 sin(θ1 − θ2 ) + 2γρ3 cos(θ1 − θ3 )

θ˙1 = −C. Theoretical Analysis
In this section, we explain the theoretical analysis byusing the averaging method for the circuit equations (2)(5) obtained from Fig. 2. Here, we consider the case ofthat the tiny resistance rm does not exist (η = 0).
First, Eq (2) can be described by second order nonlinear differential equation as follows.
d2 x1
dx1 1+ x1 = ε(1 − x1 2 )+ γY1 ≡ F12dτdτ3dY1 2111+ γY1 = x1 + x2 + x3 + x4dτ3333−3ρ3 sin(θ1 − θ3 ) + 2γρ4 cos(θ1 − θ4 )o−3ρ4 sin(θ1 − θ4 )(12)
We also apply the averaging method to Eqs. (3)-(5) assimilarity.
In the steady state,(6)ρ˙k = 0θ˙k = 0and(13)(7)(k = 1, 2, 3, 4).

where,must be satisfied. We obtain the solutions as follows.
For the amplitude:

Y1 ≡ ya1 + ya3 + yb1 + yb2 + yb4 + yc1 .

Equation (7) is the first order linear differential equation.
The solution is given as following equation.
R 2 2Y1 = e− 3 γτ e 3 γτ x1 + 13 x2 + 13 x3 + 13 x42(8)+Ce− 3 γτρk =s4−8γ
3ε(4γ 2 + 9)(14)

(k = 1, 2).

(C : const.)

ρk = 2(15)In the steady state, the second term of Eq. (8) becomesto zero.
Next, let assume the solution of Eq. (6) is(k = 3, 4).

For the phase difference:xk (τ ) = ρk sin(τ + θk ).

(9)
θ1 − θ2 = 0.

We pay attention to treat the non-resonance system andapply for the averaging method to Eq. (6). We obtain ρ1and θ1 as follows.
Z Tρ˙1 = lim
εF1 cos(τ + θ1 )dτT →∞ 0(10)Z Tεθ˙1 = lim
F1 sin(τ + θk )dτT →∞ 0 ρ1

θ3 − θ2 = θ3 − θ4 = θ4 − θ1 = θ4 − θ2 = π.

ρ˙1 = −TABLE I
C OMPARISON BETWEEN THEORETICAL AND SIMULATIONRESULTS FOR THE AMPLITUDE .

(11)ρ1 , ρ2
ρ3 , ρ4−3ρ3 cos(θ1 − θ3 ) + 2γρ4 sin(θ1 − θ4 )o−3ρ4 cos(θ1 − θ4 )26-28 May 2010, Dresden, Germany(17)

We confirm that the first and the second oscillators aresynchronized at in-phase. Other combination oscillators
synchronize with anti-phase. These theoretical resultscorrespond with the computer simulation results. Table I
summarizes the comparison between theoretical and simulation results when the parameters are set as ε = 0.1!$
γ = 0.1. From this table, we can see that they matchvery well.

By solving the above equations, Eqs. (11) and (12) areobtained.
ε2 ρk 2(ρk − 4)8nεγ−9ρ1 + 2γρ2 sin(θ1 − θ2 )3(4γ 2 + 9)−3ρ2 cos(θ1 − θ2 ) + 2γρ3 sin(θ1 − θ3 )(16)

64Theory1.765
2.000Simulation
1.790
1.979

18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)

TABLE IIS YNCHRONIZATION STATES FOR DIFFERENT TYPES OF

D. Case for polygonal network (N ≧ 6)

POLYGONAL OSCILLATORY NETWORKS

Finally, we investigate synchronization phenomena inseveral types of two coupled polygonal oscillatory networks as shown in Fig. 5. The computer simulation resultof the phase difference between the shared oscillatorsof each network structure are summarized in Tab. II.
From this table, we can see that the shared (the firstand the second) oscillators are synchronized at anti-phasestate when the number of the coupled oscillators of eachpolygonal network is even number. While in the caseof that the number of the coupled oscillators of eachpolygonal network is odd number, the shared oscillators
can not synchronized at anti-phase state.

Number of oscillators
N=6(quadrilateral)
N=8(pentagon)
N=10(hexagon)
N=12(heptagon)

Phase180◦ (anti-phase)180◦ (anti-phase)
107.2◦
152.8◦◦
180 (anti-phase)180◦ (anti-phase)
128.5◦
158.7◦

III. C ONCLUSIONIn this study, we investigated synchronization phenomena in two coupled polygonal oscillatory networks
sharing the branch. By computer simulations, we confirmed that synchronization state of the shared oscillators
depends on the number of coupled oscillators of the eachpolygonal network. We assume that the shared oscillators
synchronize to make balance for whole system. In orderto make clear the mechanism of such synchronization,
to discuss a power consumption of coupled resistor andto analyze the stability of solutions obtained from theaveraging method in detail are our future works.

4th osc. 1st osc. 5th osc.

3rd osc. 2nd osc. 6th osc.
(a) quadrilateral network (N=6).
5th osc.

Oscillator type
Shared osc. (1-2)
Independent osc.
Shared osc. (1-2)
Independent osc.
Shared osc. (1-2)
Independent osc.
Shared osc. (1-2)
Independent osc.

6th osc.
1st osc.

R EFERENCES
4th osc.

7th osc.

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phases in populations of nonlinearly coupled oscillators withbimodal frequency distributions,” Physica D: Nonlinear Phenomena, vol.113, no.1, pp.79-97, Feb. 1998.
[2] J.A. Sherratt, “Invading wave fronts and their oscillatory wakes
are linked by a modulated traveling phase resetting wave,”
Physica D: Nonlinear Phenomena, vol.117, no.1-4, pp.145-166,
June 1998.
[3] G. Abramson, V.M. Kenkre and A.R. Bishop, “Analytic solutions for nonlinear waves in coupled reacting systems,” Physica
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[4] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization and graph topology,” Int. J. Bifurcation and Chaos,vol.15, no.11, pp.3423-3433, Nov. 2005.
[5] C.M. Gray, “Synchronous oscillations in neural systems: mechanisms and functions,” J. Computational Neuroscience, vol.1,
pp.11-38, 1994.
[6] R. Stoop and C. Wagner, “Neocortex’s architecture optimizes
computation, information transfer and synchronizability, atgiven total connection length,” International Journal of Bifurcation and Chaos, vol.17, no.7, pp.2257-2279, 2007.
[7] T. Suezaki and S. Mori, “Mutual synchronization of two oscillators,” Trans. IECE, vol.48, no.9, pp.1551-1557, Sep. 1965.
[8] H.B. Fotsina and J. Daafouza, “Adaptive synchronization ofuncertain chaotic colpitts oscillators based on parameter identification” Physics Letters A, vol.339, pp.304-315, May. 2005.

2nd osc.
3rd osc.

8th osc.

(b) pentagon network (N=8).
7th osc.

6th osc.
5th osc.

1st osc.

8th osc.

4th osc.

2nd osc.

9th osc.
10th osc.

3rd osc.

(c) hexagon network (N=10).
7th osc.

8th osc.

6th osc.

9th osc.

1st osc.

5th osc.

10th osc.
2nd osc.

4th osc.

3rd osc.

12th osc.

11th osc.

(d) heptagon network (N=12).
Fig. 5. Several types of two coupled polygonal oscillatory networks.

26-28 May 2010, Dresden, Germany65