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IEICETRANS,,F[INDAMENTALS,VOL.E9e-A,NO,1OOCT{]BER2007
2162

RAPER

SpeciaJts$ueonNonlinearTheoryanditsApplications

Characteristics
of
AveragingMethod Analysisof Synchronizationa Large Number of Nonlinearly
Coupledvan der Pol Oscillators
Kuniyasu SHIMIZU'I'a),
Student
Member, [Ebtsuro
ENDOtb),Flellow,andHisa-Aki[e4NAKAtt, Mlember

SUMMARY

The averaged equatgoll foran arbitrary number of oscillaby nonlinear coupling scheme invented
by S.Nagano,isdeters coupled

rived.This system isinventedas a model of uni-cellul ar slinne amoeba. By
the synchrenization characterusing theaveraged
equation, we investigate

ef coupledvanderPol oscillators can be derived,by using this equation, the steady-state characteristics

nurnberandof the same-phasesynchronizedsolution'(mode)

can

beob-

isticsof fivecoupled oseillators and a large
numbeTof ceupled oscillators,
In particular,
we psesent
the statisticalpropertyof ceupled oscir]ators interrns of coupling factorv.We also investigate
the effect of linearand non-tained. The reason why we concentrate on the same-phasemode
isbecause itispracticallyimportant and indeed,theuni-celluar slime arnoeba shows the same-phasesynchro-linearcoupljng

nization

terrns for achiev'ing synchrepiztttion.

and confirm thatthe
term plays an impertantrole forstrong synchronizatienthan]inear
coupling term does,
nonlinear coupling

kqy words:
nonlinearsynchrvnizatien,
coupling,

slime

coupledescillatersystem,

averagingmethocLa]noeba

1. Iiltroduction
The study
constantofsystemsofcoupledinterestin various

areas

oscillatorsofhas attracted
physics,

In particular,
mutual synchroniza[1]-[4],significant
due to itspracticalisextremely
needs [5],CoupledoscMatorsystemare often used in[6],modeling
biologicalsystems' [7],Recently,S.Naganoinvented a new nonlinear coupling scheme
betweenlimitcycle oscillators
basedon a model of an aggregateduni
elluartion of rhythms£

called"Dictyosteljum

with eachotherduringaggregation

be applicable totheoretically,
itisconfirmed thatin
our case theresults of the averaging method(synchronization probability
versusthecoupling factor)
are applicable
forconsiderably strong nonlinear case,"weak"

nonlinear

can

case

2. DeriytttionoftheAyeragedEquationengineering,and mathematics

slime amoeba

to cornmunicate

Althoughtheaveraging method[11]*,

discoideum"
[8],
[9].

He employeda van
der Pol oscillator as a typical modelof the limitcycle oscillator. One of the characteristics ofthisnonlinear
coupling
schemeisitsstrong same-phase synchronization ability. Namely, different
kindof oscillators
withdifferent
natura]frequencies
can be synchronizedeasily.Althoughthese characteristics have been confirmed bydirectcomputer simulation, thereisno theoretical
approachto thissystem so fhr.
In our previouspaper,we analyzed two coupled van
derPol oscillatorswith thisscheme via averaging method,and clarified various interestihg
characteristics peculiar
tononiinear coupling [10],In thispaper,we are succeeded"arbitrary"
in analyzing
annumberof vander Pol escillatorscoupledby thisscheme
via averaging
the explicit forrnof averaged equation

method.

for anNamely,A van derPol oscillator adopted inthis
system iswritten as
fbllows(AppendixA) ,ktl-sw(i-x2)/1+tu2x=o(l>s denotesa (sma]1
positive)parametershowing thedegreeof nonlinearity and where to dcnotes
the natural angularfrequency*'.
Couplingthe N-yan der Pol escillators via Nagano's
method givesthe following
equation [9]:wherexk + W:

(Xk
#.1
+ 7k

Xl)

2

=stukI]-(xk+7t#.1xl)

1xk,kT(2),lv.

where 7k > O denotesa linear
coupling factor,
and where
7t > O denotesa nonlinear coupling factoz Equation(2)can be rewritten
inthe fbllowing vector forrn:"arbitrary"

Manuscriptreceived November 29, 2006.
Manuscriptrevised March 20, 2007.
Finalmanuscript received Apri1 16,2007,"'The
authors are with the Department of Electronics
and Communication, MeljiUniversity,
Kawasaki-shi,214-g571 Japan.
t'The author iswith the Department of Electronic
EngineeringTheUniversityofElectro-Communications(UEC),Chofu-shi,
182-8585Japan.
a) E-rnai1:
shimizu@isc,meiji.ac.jpb)E-mail:endoh@isc.melji.ac.jp
DOI:10.10931ietfecfe90-a.IO.2162

X + Bx = sCS - eG(x, k)(3a)[xl,x2,...,xN]T

(3b)

where
x

'More

ii

accurately,each(=cyclic adenosinecAMP

the same-phase"Intime waveformof the intereellular
3',5'-monophospltate)densLtypresents

synchronization

Nagano's model,

duningaggregatjon.

x associates

and drwith the cAMP
product,withthe intercellularcAMP

receptor.

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SHIMIZU

et al,: AYERAGINGMETHOD

ANAtiYSISOF SYNCHRONIZAI'ION

CHARACTERTSTICS

2163(1+7i)w?7ituii

In the averaging

method,wefirstassume+

ek),
9k. pk9k cos(9kt

Eq.(6a),and obtain theunperturbed(1+v2)w3 72bl; ･･････

hw;
B=-

271th)l72to;･''''''''''''

culated

uk

as

O in

Itcan be cal-solution.

pk sin(9kt

=

=

=

+

ek)fOrk=1,2,...,N.
When s t e, we assume pk and ek as functien$of time,From the theoryof averaging, the dynamics of thesefunctions can be calculated
from the fo11owingaveraged
equa-

I

,

easilyg27,NcaNt-------lst----- 7NCL)ft(1+){N)c,)ft(3c)tionfork=1,2,.,.,N[12]:
E

to1

o

O

tuN

=

(3d)

C.

where

Each componentof G(x,S) canbe written

as

fo11ows,

9ft

the

and

(4)

xi)

We definethe eigenyaluesofB

associatedaseigenvectors

9Z

<

922 <

as

pj

Sepk
itny,.lb(k,(Ukkib(k,
)

Pk =

Ml

=

plying a non-singular lineartransformation x = Py to
Eq. (3a)and multiplying P-i from the left-handside give
thefo11owingtransformed equation.
- sQG(y,seyy)

to infinthe non-fbrk= 1,2,..,,Nas fOllows(Appendix

condition

(5a)

,

m]

,

klp:,

k,k,klpZ+

fOrj --1,2,.,.,N,respectivelyt, Ap[pij,p2j,''',pNj]T

=

zero

C):

#.i

gk(x,x)=(xk+7,zaverageresonant

2tokxky + By(7)=-g,p,

from<･>denotesthe time
(3e)[gi(x,l),g2(x,S),...,gN(x,S)l'.
ity.Equation(7)can be calculated by assuming

G(x,k)･･･ <ek)>
Pk fiE(A(y,y)cos(nkt+.
Eek<ft(Y,Y)Sin(9kt+ek)>

Ok=o.'(sa)b(k,mi,m2,

whereisgivenin Eq,(6c)and akk isgiven

m3)

as
N

(sb)

ffkk=Zbllqklptk

where

IT1

g . p-IBp

The non-resonant condition
1,2,･･,
Ml, m2, m3

.9?

=

o.,,o

e i p-icp
P

=

=

92.

[aif]

[PiJ-]Qi P-'=[gij], i,J'--1,2,,,.,N' (5b)

Equation(5a)can be written inthe fo1]owing
scalar form:N+ 9Z"kijk

Ektek- shk(y,y)

sZ=sft(y,

Y),k= 1,2,,..,N(6a)(1)9m, ± 9m, ± 9m, ± 9k=O

for nzt 4 m2 4 m34k(2)29kformilm24k±

hk(y,S)= £Sinceek can be determinedas an arbitrary constant, we only
investigate
the amplitude
equation.
Therefore,
by defining
Uk i pZ,Eq.(8a)can be simplified as

Crk sUk
=

S
(ffkk
-

.Zl.lb(kim1,m1,k)U.,+N

Z Zb(k,mi,m2,m3)y.,u.,e.,
(6b)
fbrk=

where

tb(k,

l,2,...,N. We

k,k,k)Uk)(10)

ieFk(Ui,U2,･･･,Uiv)

ml=lm2=lm3=1

andforklm3=O(9)

thefunctions
hk isexpressed in terms of y,Y throughthe transformation x = Py as follows(AppendixB) :N

9m, ± 9m,=O(3)39k-n.,,

where

N

be written as fo11owsfor

,IV.

l=1
i

can

regardEq,(1O) as the fundamental

averagedequation.

Nb(k,ml,m2,m3)iZqk,cv,a,.lasm:psm,
S=1

N

(6c)

asm=psm+7:Zpim

l=1

tltis not veri fiedso far thatthe eigenvalues of B are all
positiye
However, as faras our computer calculation shows,they are positive
real numbers.
Itshould be clarified inthe future
real numbers.
that in what case theybecomepositive
real numbers,

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2164
tA

3. Analysisof Steady-State
Solutions
via Averaging

1O.9o.s

TO,8 ,,..,
Fl,1)t:t.4ti",,..,..

MethodT=ri.6 m--i.

Fl.9 -"...
-14i,,,,
v=3.e--j-dltlrll-tl--ny1-L-ilttllptplPl--[ihesteady-state selution corresponds to an equilibrium
for
point{n Eq.(10). There are many equilibrium poirrts
mutuailycoupledoscMators.
[[hestabila 1argenumberofityof each equilibrium point
can be obtained by using thefollowing
Jacobianmatrix .
OFIOUiiOFNOUI
1-- OFIeUNJ=

ti//tieiJ//iTh/"///-///t//tb/i/"idiT//-/m/7d/vi/utt//t,"//""v/'i"i"/-itt:irf'ILt-/-/-i-l'il:[:'
llLl::lil::1'il:.････--tt

pdd-s-p---t-d･:ll::i111L-ql:iTdtlllt--li-1-1-[1----Tt-T-blibtny-1nyttt---b-LEd-IT---dlilsLlillil-l

o.2

o

(11)

1･OFN6UN,,e,7O,6O.5O.4O.3

.m

o.4

･

o.6

o.e

1

co5
Fig.1
Steady-state
ampHtude
fromO,OOIto
ps in terms of ws (vatying1.0)forvarious values of coupling factor7 for tut
O.1,bl2
O.2,tu3O.3,to4 O.4,=

Narnely,ifal1eigenvalues of J have negative real parts,
thesolution
isasymptoticallystable, If,at least,
one of themhas positive
real part,
itisunstable. In thismanner, we can

the
judge

stahilityof eachpoint.

equilibrium

ifthe nonlinBy directcomputersimulatien of Eq. (2),isweak (fdrexample,
s = O,1),
we
can observe
multisteadystatesolutionsdeterminedfrominitialcoilditions.
ple
example,On theother hand,forstrong nonlinear case (forearitys=1.0),weobservecantimewavefbrm

sides, eachonly

the sarne-phasesolution.

of the intercellular
cAMP

Be-

amoebathe samepresents
duringaggregation [11],Therefbre,
solution inthispaper.
we wi]1concentrateen thesame-phase
From Frobeniustheory [13],the sign ef al1compo=
nents
of eigenvector
1,2,.,,,N)associated with(l]kN,kthelargest
eigenvalue (9k)of B ispositive.
Therefbre,
thesame-phasetakes
fo11owingform:so}utionthephase synchronization

UN t O, Uk=O,k=1,...,N-1
4b(N,N, N, Nl)

=>

UoN=

="

PON =

=>

uN = poN sin(9Nt + eN),uk

O

XI =
x2

,Uok=O=

I
xN

rpNfvpoNsin(9Nt+

aN)

(12)

The piN ispositivefori = 1, 2,.., N from Frobeniustheory.
In Eq. (12),the phaseeN isan arbitrary constant, Therefbre,
we take ei O forall ifOrsimplicity.
,=

4. SynchronizationCharacteristicsof the Same-Phase
SolutionIn thissection, we

will show

computer

calculations of thesynchronization
based on the ayeraged
equation.At first,
we wM
investigate
thesynchronization characteristics
ef fivecoupledoscillators,
Then, we will inyesof coupled oscillators, Althoughwetigatea lar:ger
numbersame-phasecillators

We

investigate
the synchronization propertyof thesolution forfive
coupled oscillators.
Figure1 presentsthesame-phase amplitude ps (intheforvarious yalues
y-domain) in terms of tus = e,OOI-1.0
of 7, where we fixedbli
O,1,to2 O,2,to3 O.3 and=w4O,4i'.
In thiscase, thesame-phase solution exists for
7 2 O.8only, Both ends of each curve presentthe synchronization limit.
In thissystem, thesynchronization range
increaseswith the increaseof collpling faetor7. In particular,
fOr7 2 7. t 1.9,synchronization ranga expands
drastically.
Namely, synchronizationcanbe achieved forall values of ws
O,OOI 1.0. Such a propertycould
will

same-phase

=

=

=

-

when

we

fix

O･Ol,to2 O.02,to3 O.03,to4= O.04 and varying
tos = O.OOI-J 1.0in the samemamier,synchronization
cana]so be achieved forall values ef ws for7 Z 7c s 2,5,
Figure 2 (a)presentsthe synchronized angular frethis result with thequency stsin terms ef tus. We compare
empirical formulaobtained in [9]:tus =tol + 7Zr.i tu7,
toi =

eN)
+ eN)

p2NpoN sin(9Nt

4.1 Synchronization
Characteristics
foTFiveCeupledOs-

be obseryed forother cases, Forexample,

O

=

+

IJtNPoNSin(9Ntfixthelinearand nonlinear coupling terms 7k and 7L
independently,.
we take these two factors
equal ( such as
=
i
for
all
k)
for
most
cases
in
the succeeding
7
7k 7ylsections, unIess so identified.

=,POk=O

=

can

den-

sity of actual uni-cellulear slime

=

=

where

=

o,

=

is the angularchronizationand toifrequency of the same-phaseisthe maximum

natural angular

syn-

fre-

quency,i.e.toi ) bli fbr2 s i <. N in hisnotationH',
Figure2(b)compares the ws versus w, characteristics from
thefbrmulawith thews versus ns characteristics
from ourcomputer
calculation
for7 = 2,3forexample.
They show
TIt isenoughto vury O < tos s 1,because ifbljfcai 2 1,then1.
s'[''tSince
al1componentsof B are pesitive,
itissaid from a theorem inp.171in [13]thatthe1argest
eigenvalue of B (9k)satisfies
the followingrelation : 92N <- wi .Thcrefdre,
the curve from theformula
isclose to but 1argerthanour simulation curve inFig.
2(b).
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etal./PCVERAGINGMETHODANALYSISOFSYNCHRONIZMIONCHARACTERISTICS
2165
3
O,6O.4O.2e-O.2O,4-O,6
2.5

k"""e8.e-mu<2

aVi 1.s
rt
D.5

o

o

02

O,4

1

sooo

soo2

saotL

soo6

ooos

selo

T;.me
(a)C05O,6O,8

Fig.3

3

Time waveforms
from directcomputer
simulation
forfivecoupledcase for s = O.1,7 ,=1.4,colO.L,w2 O,2,w] O.3,tu4 O.4

escillator=

and tos =

=

=

=

O.5.

2,5

icsof the slime, thismay
be reasonable as follow.
An oscillator with low natural frequencycorresponds
to inactive
slime cell and an oscillator with highnatural frequencycorresponds to active slime cell, When they are coupled, thecontributionof the active
cell becomes 1arger
thanthat oftheinactive
cell.

2

A.tsttv

1,5
wnq

1
O.5
e
o

O,2

O.4

O.6

D.8

1

tu5

(b)Comparisonof 9s with tus.
'frequency
SynchTonizedangularns interms of a)s (varying
from

Fig.2
O.OOIto 1.0>for wl

O,1,bl2

result for various

calculation

between our

computerresult for7 =2.3.

Thblel==valuescalculation

O.2,to3ofO.3,w4O.4 Ca)Cornputer
coupling factor7 (b)Compurison

result=and=:Nagano'sheuristic
formulaComparisonbetweencomputersimulationresultsandaveraging

forN=5,v 1.4,wl
O.1,w2 O,2,bl3 O.3,bl4 O.4
O.5.
AveragingmethodComputersimu]ation
ns
O.9772
O.9775methodresutts

and ws=Amptitudeofxl
Ampljtudeofx2
Arnplitudeofx3

Amplitudeofx4
Amplitudeofxs

=

=

=

=

=

O.O164
O.0679ODI75
O.e677

O,1615
O.31L24
O.5505

O.1615
O.3126
O.55004.2 Synchronization
Characteristics
fora Large Numberof Coupled Oscillatorswith Randomly Distributed
NaturalFrequencies
From theabove results and our previouswotk [101,it[14],
seems that the synchronization
range becomes drasticallylargebeyond a certain coupling
factorx･, In particular,
fbrtwo coupled van derPol oscillator ¢ ase, we have confirmed analyticallythat the synchronization
rangeof thesame-phasesolutien becomes infinitely
1argefor7 ) I + Viiof coupledoscillator[10].However, fbra largenumber
cases,

thereare many

combinations

frequencies,
Therefore,
wecies

according

ctil

=

will

thenatural angularof

set naturalto the fbllowing statisticalangularfrequen-

rule.

1.0

(13)

tok=r+(1-r)crk,(2gkgN-1)(oN=r

In Eq.(13),ak denotesthe unifOrm random number distributingfrom O to 1.0 and r (O < r < 1,O)definestheminimum value of natural angular frequency.
Namely,natural angular firequenciescL)iand oL)N are set to 1.0and r, respectively, and other wk isset uniformly random
between r
and1.0.
We will show the results based on averaging
methodfor the 10 coupledoscillator
Here,
case.
we take lOO samfrom Eq. (13)forfixed7
plesof natural angular frequencies
andjudge
thestability firomEq,(11)i'. Then,we obtain thesynchronjzation
probabilityptqwhere q isthe nialnumber
andof stable
samples,Figure 4 shows thep isthenumber
synchronization
factorprobabilityin terms ot' coupling
7 for,in good agreement.

We confirmed

the empirical fbrmulafor other values of 7. Synchronizedangular fre9N
is much largerthan natural angular frequency
quencyof eachoscillator.
This isdue to the mannerof coupling
adoptedin Eq. (2). '[[lab]e1 compares the result of averaging method withagreeswel]thatof directcomputer
simulation of Eq. (2)fors = O.1,
=
=
=
=
1,4,
bli
O,1,
w2
bl4
O.2,to3 O.3,
O,4and7
bls = O.5. They show
in good agreement.
Figure 3 demon=
strates theresult ofdirect computer
simulation inthiscase.
The oscillation with highernatural frequencysuppresses thearnplitude of oscillation with lower natural frequencyfbr
achieving synchronization.
From the view pointof dynam='Wecheckthenon-resonantcondition(9)topickupthesesam-

ples.
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IEICETTIANS.FUNDAMENT}KLS,VOL.E90-A,NO.100CTOBER2007
2166
1,2

1.2

1

1

be

bta.o,ses££

eacUftop
O.6

BacUZbeop
O,6

O.4

O,4

O,2

O,2

o

o
oo.2

o,4

o.6･

(a)

oO.2

1.4

e.s

O.4

v '11.2

O.6

D.8

11.2

1.4

tf,2

IA

Y
(a)e=1.0.
12

1.2

1

1h･=r-

gh

g- O.B

O.B

R
e O,6
ac

£

BacO,6oftUk

O.4me.2

o.4

m

,,,

o

o
O

02

O.4

O.6

e.8

1

1,2

1.4

oO,2

O,4

(b)
Mg. 4

O.6

O,8

7
(b)e=3,e.

v
SyllchN)niztttion
probabilityinterms of 7 among

Mg.6

10 coupied

Comparisenof synchronization probability
in terms ofv obtainedwith thatobtained by directcomputer
simulationforby arJeraging method

oscillator case fOrvarious values of r.

(a)s=1.0andfor(b)e=3.0.

1.2

1

･=
=
p

e.B

B

2 o.6

at

U
Z

l

o,4

o,2
o

Fig.5

forr = O,OOt,O.O1,O.1and e,3,probability
Itshould be noted that the synchronizationisachieved fbr
comparatively small coupling factor7 comparedwith the1O coupled oscillator case, [[hisisbecause the natural angularfrequencies
are moredenselydistributedin limited frea 1arger
nurnberof coupled oscilquency range, Therefore,
latorsystems may be more easily synchronized, theughthe
amplitudeof each oscilrator ismore suppressed.
Itseemsthat forr s O the synchronization
curve
conprobability
vergesto a limiting
curve,
Figures6(a)and (b)compare the synchronization probability
in terms of 7 obtained by averaging
methodwiththatobtained by directcomputer simulation
for s
1,O
and 3,O,respectivelyt. The computer
simulation results for
s = 1.0 agrees
wellwith the results of averaging method,
Thismeans that although averaging method results is guaranteed
forsmal1 s only, in practice,itisapplicable fornonweak nonlinear cases arounde =
1.0, However, when 6
becomes 3.0thesetwo results differ
considerably.
Figures7(a) and (b)presentthe coupling factorltver'TIn
synchronization

h

O

O.2

O.4

b.6

O,8

･Y

1

1.2

1.4

Synehronizationinterrns of v arnong 30 coupledprobabilityos-cillator case forvarious values of r.

=

of r. Forr s O.9,the-same phase synchroisachieved easily. On theother hand,forr fs O, it
isdiMcultto synchronize,
However, when coupling factorvvarious values
nization

is suthciently
1arge,the same-phasesynchrenization
seems
to be achieved forany r. There exist a critica] 7 foreach r =constant curve above

which

synchronizationisachieved fbr

frequencies.Tliiscritical va]ue
decreaseswith the increaseof r,
Next we will show the results forlargersize of ceuany combinationofnaturalpled oscitlator case, namely

ourangularN=

30, Figure5 shows thevaluessteady-states

cornputer

simulation,

are given to Eq.
solution) 1.0,itseems

many(2)and we computeconverging

thatother modes

randomly

to the same-phase
exceptinitialchosentheprobabiljty
forthesolution.

the same-phaseForbecomeunstable,'
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SHIMIZUetal.:AVERAGINGMETHODANALYSISOFSYNC}IRONIZrmONCHARACIIERISTICS
2167
n.2

O.17and 7'

O, and findthat these two cases
difFerent
synchronization
characteristics.
qualitatively
Namely, the 7' = O.17 case can achieve synchronization
forfairly
smal1 values of r. In contrast,
the 7' = O case
can achieye synchronization
forr z O.8only. Namely,synchronizationabi]ity becomes stronger by adding even small
nonlinear coup]ing term. As a futureproblem,we will investigate
real slime
dynamicsby using thisequation.
tor as 7' =

=

show

1
hli.RBacUzbeen
o.s
O.6

O,4
O,2

A,cknowledgments

o

oe.2

o.4

o.6

(a)7'

e.s

11.2

1.4

Y
O.lx

=

The authors
S. Nagano

1,2

liketo express their gratitud
e to Prof.
Ritsumeikan University,
Japan,

wouldofReferences

1ta･==[1] R,H, Rand and PJ. Holrnes,
ofperiodic motiens intwo
weaklyeoupled van derPol oscillators;' tnt.J. Nonlinear Mechanics,
yol, L5,pp,387-399, ]98e.
]adder oscil[2]T,Endo and S,Mori, "Modeanalysis of a multimodelater;'IEEE Trans. Circuits
Syst,,
voLCAS-23.
no.2,
pp.100-113,
1976,`Land S,Oishi, Self-synchronizationof
L3jH. TleLnak4A.J.Lichtenberg,
coupledoscillators
withhystereticresponses;' PhisycuD, vol, LOO,
pp.279L300, 1997.
[4]S. Moro, Y Nishio, and S.Mori,
phenomena in
RCoseillatorscoupledbyoneresistor;'IEICETTans.Fundamentals,
vol.E78-A,
no.10, pp.1435-1439,
Oct. 1995.
[5] I.I.Blekhman, Synchronizationin Seienceand Technelogy,
ASME
Press,New Ybrk,i988.
AUni[6]A. Ptkovsky,M. Rosenblum, andJ. Kurthg,Synchronizatioi:
versal ConceptimNonlinear Sciences,Cambridge UniversityPress,
CambTidge,UK,2001.
[7] S.H, Strogatz,Sync: The Emerging Scienceof Spontaneous
Order,
HypeTion, New Ybrk,2003.
[81 S. Nagane,
coupling
scherne forsynchronization,"Prog.Theor. Phys.,vol. 103,no.2, pp.229-244, 2ooO.
ieceptor
schemeforrobust synchroniztttion[9]S. Nagano,"Biimcation

O,BRENO.6oZ
o,4bemO.2"Synchronization

o

oO,2

O.4

O,6

O.8

1

12

1,4

Y
(b)Y
Fig.7

=

O.

Synchronizution
interms of 7 among 10 coupled osprobability
forvarious values of r for(a)11 O.l1,;{b)T,' O (lineaT

casecil]atorcoupling==case).

for(a)7' = O,17 and (b)probability
=
O.
In
these
figures,
weinvestjgate
theeffect of nonlin71
ear coupling
factor7' ii or1 7S ･･･ = 1{k･When 7' = O,
synchronization
can not be achieved
below r = O.8, However, for7'
O,17,itisachieved, at least,
above r = O,6.
=
Furtherfor7' 7, itisachieved above r s O (inFig.4),
From these results, theorigin of strong synchronization ofthis coupling scheme exists in the nonlinear couplingtermforsuch a 1argenumberof coupled system.
sus synchronization

=

=

=

averagedequation"Biologicaloflimitcycleoscillators;'Prog.Theor.Phys.voLI07,no.5,pp,g61-

877,2002.
IIO] K, Simizu,T. Endo, and H. Tar!aka,`Averaging

of synchronization

charactertstics of two

methodanalysis

nenlinearly-coupledvander poloscillators;' IEICE Trans. Fundamentals
Edition),(Japanese
vol,J90-A, no,2, pp,1[I-[21,Feb, 2007,il1] M.T. Laub and W,F,Loemis, molecular network thatproduces
spontane,ous oscillations inexcitable cells of dictyostelium;'Mo],
BLol.Ce", vol,9, pp,3521-3532, 1998.
oseil]ator[12]L.O. Chua and T, Endo,analysis vja integral
manit'olds,
Part I:Non-resonant case;' int.
J,Circuit
Theory Appl.,
tA`tMultimode5. Conclusions
The"Reeeptor-productfor an arbitrary number

voLl6,

of oscil-latorscoupled by nonlinear coupling
schemeinvented by
S.Nagapo,jsderived.
By using theaveraged equation, weinvestigatethe synchronization characteristics
of fivecouoscillators
and
a
large
numberofcQupledoscillators.
pled
In particular,
wethestatisticalpresent
propertyef coupledoscillators
in terms of coupling factor
7. Asaresu]t,itis
confirmedthat there isa critical coupling factor7. above
which
synchronization
can be achieved
forany combinationof natural angular frequencies
fora fixedr which detemines
therange of natural angular frequencydistribution.
We alsoinvestigate
the origin of thisstrong synchrenization
scheme.
Tb do this,we reduce the effect of nonlinear coupling fac-

25-58,19S8.

Baif--kan, Tb[131S.Furuya,Gyouretuto Gyouretusiki,
pp.159-172,
kyo, 2002.

T,Endo, and H. Tunaka,
L14]K. Shirnizu,`VtsJeraging

rnethod

analysisfor three-couptedvari der pol oscillators with nonlineaTNO[;TA06, pp.183-186,
Bologna,Ita}y,Sept.2006,

Appendix A:Van

coupling;'

der Pol Equation

Assuming thecubic nonlinearity forNC, i.e.,
iNc -eiv +>O
inFig.A･
1,
Kirchhoff's
current
law gives
g3v3,ei,g3
fo11owingequation:=

ifvdt+C:1-giv+g3v3=O

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IEICETRANS.FUM)AMENIALS,VOL.E90TA,NO.1eOCTOBER2007
2168

v

N

Z eks(asly!

+ a,2u2 + ･･･+ a,NuN)2(p,1el

=

itic

iNCNC

s.1

p,2e2+ ･･･+ p,NeN)tti,

+

V

c

L

=IX.lqkscvs(.Zll .ltr.lmZI.lasmlasm2psm3umlgm2em3]

NNN=

fig.A･ 1

Z Z Z b(k,ml,m2,m3)um,ym,em,=

A van derPol electric circuit where L isthe inductance,
C is
and NC isthe nonlinear conductance.

the capacitance,

ml=tm2=lml=1(A･7)'whereTalcingthetime derivativeof Eq, (A･1),welowingequation,

obtainAppendixC:a:li;2Vf/
Zil
iliv=o
3g9i3

-

v2)

By changing

the variables

g,

van

(A･2)

+

DerivatienoftheAveragedEquation(8a)'=Defining utk 9kt + ek,theaveraged equation forPkcan be,wrlttenas:as:Vllllii,
vliit=v = vox, vo =

..

b (k,
mi, m2, m3) isdefinedinEq. (6c),the fol-sPk<fik(Y,Y)COSPk>
fill

=

(S.1

cot'

(A.3)

.VIIi

derPo]equationcan-be derivedas Eq.(1).

b(k,ml,m2, m3)p.,p.,p.,.ZN, =1.ZN,.1fl.,･Avl).ZN,.lDeriyationo £ theFunctionhk inEq. C6b)

AppefidixB:Nhk(y,Y)=

..ky

S=ny

S.i IX.i+72
qkstus(xs

(A･4)
Next,

#.i
li,III.i
ll=l
(p,.
il
#.1
V

itg

psmym +

vg

11heterm <cosuti-cos
ek>becornes112 fbrl = k,becauseutk> <1!2+ 112cos 2ek> = lf2,otherwise itiszero
forall t l k. Next,we will calculate IWi, Althoughthere
are many cases, the non-zero cases are limitedto the fo1lowing two cases, ifthe non-resonantcondition
Eq.(9)is'satisfied.
=

s=fy

=

case1

pim) um

.Zi.

8b)
<sin
gijm,･sinsbm,･costtirn,･cos-frk>, (A･

<cos2

xi)

..py

xi

E

irv1
2dr,

=

(A･8a)
where

Z gk,g,(x,k)1
s=1

xs +

ekw191 <cosut!･cosek>

Z{

=

: mi

AVi =

i(#.i

N

+ lt2

=

=

pl.)y.

(A･5)

m3

<sin2

:

tbm,･cos2 tuk>

(ii
-

t
(2tfrm, g (2bbm,

COS 21fr.,+-k cosN=Zasmy.

;k,k=

=m2

cos 2sbk-2bfrk)
-

cos

+

1

m=1

(A･9a)

=Z

where

a,.

2"bk))

isdefined inEq,(6c),Fumher,case2:

mi

N

Ayl .

(A･6)

drs=Zpsmem
mtl

Combining Eq.(A･4)-(A-6),the imction hk(y,Y)becornes
as fo11ows:

(.Zi
(2
ll.i
V

hk(y,s)

=

qk,
Z,l.]
.

ia,.y.)

2 N

p,. e.)ca,

=

m2

m3

=k:

ek .
wk>
<sin2

TherefOre,
theaveraged
Next, the averaged

c.s2

.

equation
equation(gg-

cos4utk)=e(A･9b)

forPk, becomes as Eq. (7).
fbrek can be written as :

-E

.

ek

=

=

g,p, <.ti(y,e)sinek>
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SHIMIZU

et al.,AVERAGING

METHOD

ANAI;YSiS OF SYNCHRONIZrtrlON

CHARACTERISTICS
2169

=

(#.1

g-k;k
N

-Z
rnl=1

N

N

rn2=1

ms=1

received
theB.E. degreeine]ecnical engineering and the M.E. and D.E.
degreesin eleetronics and coinmunication engineering fromWAseda University,
Tbkyo, Japan,

H;sa-Aki Thnaka

smefk>
eklplnl<costLrl

Z Z b(k,mi,m2,m3)pm,pm,pm,9ni,AV2)(A-1Oa)in 1990, l992, and 1995, respeetively.
From1997 to 2001 he was with theSony Computer,

ScienceLaboratory.
in2001,he joined
theDeFaculty ef
partment of ElectronicEngineerillg,
Electro-Communiealions,UniversityofElectroCornmunications,lbkyo, Japan, as an Associate
Professor,His current research interestsare inwhere'sin-ij.,･cos ger.,･sin Lfrk>
AV2!i <sin. (A･10b)utrni

synchronization

issuesincomputing

systems

as well as nQnlinear

physies,

The term <cosuti･sinek>
isequal to zero for al1k and l.
The term AY2 becomes zero fbral1possiblecombinationsof mi, m2, m3 andisk,ifthe non-resonantcondition Eq. (9)satisfied, Therefore,
the averaged equation forek becomesas Eq, (7).

Kuniyasu Shimizu
greein eleetronics andneeringandreceived

theB,E.de-communicationengi-

the M.E. degree inelectronicengi-

neeringfromMeijiUniversity,Kawasald,Japan,

in2DO1,and 2005,respectively. Currently,
he is
woTkingtowardtheD,E,degreeinMeijiUniversity. His researchinterests
ure nonlinear

oscil-

lationand synchrenization phenomena in electroniccircuits and biologicalsystems.

[[etsuroEndo
received the B.E,,M.E., andDr.Eng, degrees
inElectrical
EnglneeringfromYOkohama, Japan,in 1972,
Keio University,
1974, and 1977, respectively. From 1977 to[992, he was with the Depaitment of E]ecnicalEngineering,NationalDefenseAcademy,Ybkosuka,

Japanwhere he was an AssistantProfes-sor,

Leeturer,andaan

Associate Prefesser, In

April1992,he joined
theDepament of Electronics andCommunication, Faculty of Science

Tlechnology,
MeijiUniversity,
Kawasaki,include theanalysis of couJapan,as a Prefessor.
His researeh interests
loops.
pledoscillators, and nonlinear dynumicsand chaos of phase-locked
andFrom 1986 to 1989, he served as aSecretaryof the Tbchnical Group pn
Nonlinear Problems of IEICE. Frem 1995 to l997, he was an Associate
Editorof IEEE Transactionson ClrcuLts
and Systems.
From 1996 to L998,
he served as a Vice Chairperson
andChairperson of the 1lechnictdGroupon Nonlinear
ftoblemsof IEICE.From 2006 to 20e8,he serves as a Vice
Presidentof the Fundamenta1s Society of IEICE, Dr. Endo is a mcmber ofIEEE.

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