Effects of noise on synchronization phenomena exhibited by mean-field coupled limit cycle oscillators with two natural frequencies

0.15 MB·English

by  K. Okumura

Effects of noise on synchronization phenomena exhibited by mean-field coupled limit
cycle oscillators with two natural frequencies
KeijiOkumuraa,AkihisaIchikia,MasatoshiShiinoa
aDepartmentofPhysics,FacultyofScience,TokyoInstituteofTechnology,2-12-1Oh-okayama,Meguro-ku,Tokyo152-8551,Japan
1
1Abstract
0
2Relationships between inter-cluster synchronization phenomena and external noise are studied on the basis of noise level-free
analysis. Weconsideramean-fieldmodelofensemblesofcoupledlimitcycleoscillatorswithtwonaturalfrequencies,whichare
n
asubjectedtoexternalwhiteGaussiannoise. UsinganonlinearFokker-Planckequationapproach,weanalyticallyderivetheorder
Jparameterequationsassociatedwithnonequilibriumphasetransitionsinthethermodynamiclimit. Solvingthemnumerically,we

1systematicallyinvestigatethemodelparameterdependencesoftheappearanceanddisappearanceofsynchronizationphenomena.
Demonstratingbifurcationsfromchaoticattractorsinthedeterministiclimittolimitcycleattractorswithincreasingnoiseintensity,
]
hwe confirmtheoccurrenceof nonequilibriumphase transitionsincludinginter-clustersynchronizationinducedbyexternalwhite
cGaussiannoise.
e
mKeywords: Noiseinducedsynchronization,Mean-fieldmodel,NonlinearFokker-Planckequation,Nonequilibriumphase
transitions,Stochasticlimitcycleoscillators
-
t
a
t
s
1. Introduction withouttakingthethermodynamiclimitasinRef. [9].
.
t To avoid such limitations, one may take nonlinear Fokker-
a
m Effects of noise on synchronization phenomena in oscilla- Planck equationapproaches, which are veryclosely related to
torysystemshaverecentlyattractedmuchattentionfrommany the study of (thermal) equilibrium phase transitions. Present-
-
dresearchers. Synchronization phenomena are ubiquitous ones ingthevalidityofanHtheoremforanonlinearFokker-Planck
n
observed in various fields of natural sciences [1]. In neuro- equation for a stochastic system of mean-field coupled over-
o
sciences, neuronsin the basalgangliaexhibitmoresynchrony damped oscillators, Shiino studied statistical behaviors of the
c
[inParkinson’sdiseasethaninnormalstate,suggestingthatneu- system exhibiting equilibrium phase transitions [10] (see also

ralinformationcodingiscloselyrelatedtothesynchronization Ref. [11, 12]). Furthermore, in the case of nonequilibrium
1
phenomena[2]. Forsucha reason,tostudyhowthenoiseex- phase transitions, one can take advantage of using nonlinear
v
1ertsitsinfluenceonthestructureofsynchronizationwillbeof Fokker-Planckequationstoexactlyderivethetimeevolutionof
4paramountimportancefromtheviewpointofnonlineardynam- the order-parameterequations in the thermodynamiclimit for
3icalcontrolinvolvingchangesinsynchronyofoscillatorysys- systems of nonlinearly mean-field coupled oscillators. There,
0
tems. thenonlineardynamicaspectsofphasetransitionswereinvesti-
.
1 On one hand, someone would intuitively suppose that the gated,yieldingtheoccurrenceofchaos-nonchaosphasetransi-
0presence of noise might deteriorate the degree of synchro- tionsandthoseincludingintra-clustersynchronizationinduced
1
nization of oscillatory systems. Breakdown of synchroniza- by externalnoise [13]. Purely noiseinducedphasetransitions
1
tion due to external noise of an ensemble of limit cycle os- involvingchaos-nonchaosphase transitions in an ensemble of
:
vcillators was reported [3]. On the other hand, the opposite limit cycle oscillators were also explored [14]. A nonlinearly
i
Xphenomena of noise-induced synchronization are becoming coupled system with time delay was found to exhibit various
an active field of the study of nonlinear dynamical systems typesofnonequilibriumphasetransitions[15].
r
a[4,5,6,7,8,9]. Noiseinducedsynchronizationincoupledex- In the present study, we apply the nonlinear Fokker-Planck
citable systems/activerotatorsisinvestigatedbothanalytically approach to noise induced inter-cluster synchronization phe-
[4, 5] and numerically [6]. Among analytical studies on syn- nomenaofcoupledlimitcycleoscillators. Dealingwithasolv-
chronizationphenomenaofensembleoflimitcycleoscillators ablemodelbasedonthemean-fieldconcepttoderiveorderpa-
is the phase reductionanalysis. Such type of studies revealed rameter equations [13, 14, 15], we study the effects of noise
the effects of commonnoise on synchronizationof uncoupled onsynchronizationinthethermodynamiclimitandnonequilib-
oscillators[7]anduncoupledtwopopulationsofoscillators[8]. riumstationarystates,forwhichnumericalapproachestosolv-
These studies, however, are restricted to the case with weak ingstochasticdifferentialequationsdonotproperlywork. Our
noise. Numericalapproachestonoiseinducedsynchronization resultsshowtheappearanceofnonequilibriumphasetransitions
of coupled limit cycle oscillators might include the possibili- involvinginter-clustersynchronization. Sincethemodelofos-
tiesthatwhataresoobtainedaresubjectedtoafinitesizeeffect cillatorsis basedonthatof analogneuralnetworks, behaviors
PreprintsubmittedtoPhysicaE January4,2011
ofnoise-inducedsynchronizationinthesystemwillbeofprac- density P(t,z(x),z(y),z(x),z(y)), the mean-fieldcouplingtermsin
1 1 2 2
ticalinterest. Eqs. (1)and(2)areexpressedintermsof
F(µ) dz(x)dz(y)dz(x)dz(y)F(µ)(b(µ,x)z(x)+b(µ,y)z(y))
h α i≡Z 1 1 2 2 α α α α α
2. ModelandnonlinearFokker-Planckequationapproach
P(t,z(x),z(y),z(x),z(y)), (4)
× 1 1 2 2
Weconsiderasystemofcoupledlimitcycleoscillatorscon- F(µ)2 dz(x)dz(y)dz(x)dz(y)F(µ)2(b(µ,x)z(x)+b(µ,y)z(y))
sistingoftwosubsystems. Theoscillatorsarecoupledvianon- h α i≡Z 1 1 2 2 α α α α α
linear global interactions and subjected to independent addi- P(t,z(x),z(y),z(x),z(y)). (5)
× 1 1 2 2
tiveandmultiplicativeexternalnoise. Themodeldynamicswe
The total number of dynamic variables of the system is con-
areconcernedwithisdescribedbyasetofLangevinequations
sequently reducedfrom 2(n +n ) to 4 in the thermodynamic
[14]: 1 2
limit. ThenonehasasetoftheLangevinequations,
fori N = 1, ,n ,
1 1
∈ { ··· } dz(µ)
dz(1µi) = a(µ)z(µ)+ J(µ) F(µ)(b(µ,x)z(x)+b(µ,y)z(y)) dαt =−a(αµ)z(αµ)+Jα(µα)hFα(µ)i+ǫJα(µ+)1,αhFα(µ+)1i+ζα(µ)(t), (6)
dt − 1 1i 11ij 1 1 1j 1 1j wherethenoiseissubjectedtowhiteGaussiannoise,
+ǫ J(µ) XFj∈N(µ1)(b(µ,x)z(x)+b(µ,y)z(y))+η(µ)(t), (1) ζα(µ)(t) =0,
kX∈N2 21ik 2 2 2k 2 2k 1i hhζα(µ)(t)iζβ(ν)(t′)i=2D(effµα)δµνδαβδ(t−t′),
fori N = n +1, ,n +n , D(µ) = D(µ)+D˜(µ) F(µ)2 +ǫ2D˜(µ) F(µ)2 .
∈ 2 { 1 ··· 1 2} effα h α i h α+1i
The nonlinear Fokker-Planck equation [16] corresponding to
dz(µ)
2i = a(µ)z(µ)+ J(µ) F(µ)(b(µ,x)z(x)+b(µ,y)z(y)) Eqs. (6)reads
dt − 2 2i 22ij 2 2 2j 2 2j
Xj∈N2 ∂P(t,z(x),z(y),z(x),z(y))= ∂
+ǫ J1(µ2)ikF1(µ)(b1(µ,x)z1(xk)+b(1µ,y)z(1yk))+η(2µi)(t), (2) ∂t 1 1 2 2 −αX=1,2µX=x,y∂z(αµ)
kX∈N1 a(µ)z(µ)+J(µ) F(µ) +ǫJ(µ) F(µ) D(µ) ∂ P. (7)
switheeir,eaz((αµµ)i),(bµ(µ=,ν)xa,rye)c(oαns=ta1n,ts2a(3nd=F1(µ)))(a)reartehbeo2uDn-doesdciflulantcotriosnast AG−ausαsiaαnprobαabhiliαtyidensitαy+1shatiαs+fi1eis−Eq.eff(α7∂)za(αµs)a special
α α α
·
specifying the nonlinear couplings. The mean-field coupling solution. SincetheHtheorem[13]ensuresthattheprobability
strengthsJ(µ) (t)aregivenby density satisfying Eq. (7) convergesto the Gaussian-formfor
αβij
sufficiently large times, we representthe Gaussian probability
J(µ) densityas
Jα(µβ)ij(t)= nααβ +ξα(µβ)ij(t). (3) PG(t,z1(x),z(1y),z2(x),z(2y))=
1 1
Itenrsclausstae(rµ)1=anad(µ2),+wδe(µp)o,swtuitlhateδ(tµh)ebdeiinffgereesnstecnotinasltlryairnetsppoanrasmibele- (2π)2√detCG(t)exp"−2sTGCG−1(t)sG#,
2 1
forthedifferenceofnaturalfrequenciesbetweentheoscillators sT = (z(x) z(x) , z(y) z(y) ,
inthetwosubsystems. Theinter-clustercouplingstrengthsare G 1 −h 1 iG 1 −h 1 iG
controlledby ǫ. This parametertakesanyreal constantvalue, z(x) z(x) , z(y) z(y) )
2 −h 2 iG 2 −h 2 iG
withoutconstrainingto weak connections. The externalnoise (u(x), u(y), u(x), u(y)),
η(µ)(t), ξ(µ) (t)areofwhiteGaussiantype, ≡ 1 1 2 2
αi αβij
C (t)= ss ,
η(µ)(t) =0, Gij h i jiG
h αi i whereh·iG denotesexpectationoverPG. NowthatEqs. (4)and
η(µ)(t)η(ν)(t ) =2D(µ)δ δ δ δ(t t ), (5)aredescribedintermsofthefirstandsecondmoments,one
h αi βj ′ i ij µν αβ − ′ obtainsasetofclosedordinarydifferentialequationsinvolving
ξ(µ) (t) =0,
h αβij i atmostsecondmoments. Thetimeevolutionofeachmoment
2D˜(µ) iscalculatedfromEq. (7)andwewritethemomentequations
ξ(µ) (t)ξ(ν) (t ) = δ δ δ δ δ δ(t t ).
h αβij γδkl ′ i n ik jl µν αγ βδ − ′ as
α
d z(µ)
Notethatnotcommonbutindependentnoiseisintroducedand h α iG = a(µ) z(µ) +J(µ) F(µ) +ǫJ(µ) F(µ) (8)
η(µ)(t), ξ(µ) (t)arealsoindependenteachother. dt − α h α iG ααh α iG α+1,αh α+1iG
oαniIanntyheoαntβheiejormscoildlyatnoarmfircomlimthitenoαth→ers∞a,lminoflsutesnucreelwyicthonEvqe.rg(3es) dhud(αµt)2iG =−2a(αµ)hu(αµ)2iG+2D(effµα) (9)
accordingtothelawoflargenumbers,satisfyingthevalidityof d u(µ)u(ν)
the self-averagingproperty. Introducingempirical probability h αdtβ iG =−(a(αµ)+a(βν))hu(αµ)u(βν)iG, (10)
2
where vary in time. In this case, the order parameters of the system
may oscillate periodically in time and we can define the syn-
(α,β,µ,ν) = (1,1,x,y),(2,2,x,y), (1,2,x,x),
chronizationphenomenaaccordingto their behaviors. In par-
(1,2,y,y),(1,2,x,y),(1,2,y,x). ticular, the oscillations of the mean values have an important
Notethat u(µ)u(ν) 0(t ), implyingthatthecovariant meaningfromtheviewpointofanalogywith thespontaneous
h α β iG → → ∞ magnetizationsinspinmodelsandpossiblyresearchesinneu-
componentsofC takezerointhestationarystates.
G rosciences.
For observing qualitative as well as quantitative dynamical
For these reasons, we consider synchronizationphenomena
behaviors of the system, we proceed to solve numerically the
of the oscillators in each cluster α, i.e., intra-cluster synchro-
above set of equations. We specify the coupling function as
F(µ)(x) = sin(x), forsimplicity. Then, F(µ) and F(µ)2 are nizationandbetweencluster1and2,i.e.,inter-clustersynchro-
α h α iG h α iG nization. We define intra-cluster synchronization phenomena
calculatedas
as (i) the case that ”allof the meanvaluesoverthe oscillators
F(µ) =sin b(µ,x) z(x) +b(µ,y) z(y) inaclusterperiodicallyoscillatetoformlimitcycleattractors,
h α iG α h α iG α h α iG
(cid:16) (cid:17) regardlessof the behaviorsof the variances,” and inter-cluster
b(µ,x)2 b(µ,y)2
exp α u(x)2 α u(y)2 , (11) synchronizationphenomenaas (ii) the case that “(i) occursas
× − 2 h α iG− 2 h α iG a whole system.” We focus our attention on the inter-cluster
F(µ)2 = 1 1cos 2b(µ,x) z(x) +2b(µ,y) z(y)  synchronizationphenomena.
h α iG 2 − 2 α h α iG α h α iG For simplicity, we only treat the Langevin noise case, i.e.,
exp 2b((cid:16)µ,x)2 u(x)2 2b(µ,y) u(y)2 (cid:17). (12) D˜x = D˜y = 0, and do not study inter-cluster chaotic synchro-
× (cid:18)− α h α iG− α h α iG(cid:19) nization in this paper. From Eqs. (9), the variances turn out
to be constant for sufficiently large times. To investigate the
3. Nonequilibrium phase transitions and noise induced effectsoftheLangevinnoiseontheappearanceanddisappear-
inter-clustersynchronization anceoftheinter-clustersynchronizationphenomenainthether-
modynamiclimitandnonequilibriumstationarystates,wecon-
Thedynamicsystemoftheorderparameterequationswritten fine ourselvesto the bifurcationsof limit cycle attractors with
byEqs. (8)-(10)togetherwithEqs. (11)and(12)isnonlinear, changesinthenoiseintensity. Todeterminewhetheranattrac-
allowing varioustypesof bifurcationswith changesin the pa- torisoflimitcycletypeornot,weemploytwocriteria: (a)the
rameters. Forthedefinitionofsynchronization,letusconsider largestLyapunovexponent(LLE)oftheattractorestimatednu-
relationshipsbetween behaviorsofstatistical variablesin Eqs. mericallynearlyequalszeroand(b)allofthemeanvalueshave
(8)-(10)andthoseofindividualoscillatorsinthesystem. acommontimeperiod.Notethatbothofthemexcludethecase
First,weaddresstheissueofsynchronizationphenomenaof wheretheattractorisofergodictorustype.
the coupled oscillators in the stochastic case. On one hand, Numericalcalculationswereperformedwiththefourth-order
in deterministicsystemsof the coupledlimitcycle oscillators, Runge-Kutta method. The model parameter values are a(x) =
1
thesynchronizationphenomenaarewell-definedbythecondi- 0.5,a(y) = 1.0,b(x,x) = 0.5,b(x,y) = 1.5,b(y,x) = 12.0,b(y,y) =
tion that all of the variables of the system undergo a periodic 1 α α α − α
1.0, J(x) = J(x) = 18.0, J(y) = J(y) = 30.0, a(x) = 0.5+δ(x),
solution with a common time period. On the other hand, in − αβ βα αβ βα 2
stochasticsystemsofafinitenumberofcoupledlimitcycleos- a(y) = 1.0+ δ(y). Nonequilibrium phase transitions involving
2
cillatorswithexternalnoise,eachoscillatorbehavesrandomly the inter-cluster synchronization of the limit cycle oscillators
undertheinfluenceofnoise.SincetheFokker-Planckequations aresystematicallyinvestigatedwithchangesintheinter-cluster
in thiscasearelinear,theprobabilitydensitiesofthesystems, couplingstrengthǫ anddifferenceofconstraintparametersδ(µ)
ingeneral,exhibitequilibrium(i.e.fixedpointtype)probabil- ,andtheLangevinnoiseintensityD(µ).
ity densities for sufficiently large times. This implies that the To showtheappearanceoftheinter-clustersynchronization
orderparametersdonotoscillate evenif individualoscillators duetotheeffectsofnoise, we beginwithinvestigatingbehav-
behaveperiodicallyinthedeterministiccase.Itseemstobedif- iorsofthesysteminthedeterministiclimit,especiallythecase
ficulttoappropriatelydefinethesynchronizationphenomenain whereachaoticattractorappears(Fig. 1). Withincreaseofthe
thissituation. LangevinnoiseintensityD(x),theattractorinthiscasechanges
To consider the synchronization phenomena in systems of intolimit cycletypevia torustype, suggestingthe appearance
stochastic coupled limit cycle oscillators without the problem of the inter-cluster synchronization. Further increase of the
mentionedabove,wehaveintroducedtheconceptoftakingthe noiseintensityleadstothenegativeLLE,implyingfixedpoint
thermodynamiclimitbasedonamean-fieldmodel. Ingeneral, type attractors. These results are qualitatively consistent with
probabilitydensitiesofthelimitcycleoscillatorsystemsmight those observedin Ref. [9], which numericallysolved a set of
be multimodalunderthe influenceof weakexternalnoise. To Langevinequationstobesubjectedtothefinitesizeeffect.
avoidthedifficultyofdealingwithmultimodalprobabilityden- For clearly demonstrating the effects of noise on the inter-
sities, ourmodelis proposedso thatonceexternalnoise isin- cluster synchronization,let us show the temporalevolutionof
troduced to the system, the model constrains the form of the themeanvalues z(x) , z(x) oftheoscillatorsincluster1and
h 1 iG h 2 iG
probabilitydensitiestotheGaussianonesforsufficientlylarge 2 (Fig. 2). It is easily seen that the strength of noise abovea
times. Note that the probability densities may be allowed to certain level causes the synchronization between the clusters,
3
nt 8 εδ (=y) 0=. 742.57 10 D(x) = 0 <<zz1((xx))>>
e 2
on 6 9
p
x
E
v 8
o 4
n
u
p >
st Lya 2 <z(x) 7
e
g 6
ar 0
L
T LC
5
-2
0 0.002 0.004 0.006 0.008 0.01
D(x) 4
100 100.5 101 101.5 102 102.5 103 103.5 104
time
Figure1: ThelargestLyapunovexponentsplottedagainsttheLangevinnoise
intensityD(x). Inthedeterministiclimit,theLLEhasapositivevalue,imply- 10
ingachaoticattractor. Withanincreaseofthenoiseintensity,theLLEstake DD((xx)) == 00.0022 <<zz1((xx))>>
nearlyequalzero,andtheattractorschangesintolimitcycletypeviatorustype 9 2
(denotedbyLCandT,respectively). WesetD(y)=δ(x)=0.
8
althoughbothoftheoscillatorschaoticallybehaveinthedeter- >
ministiclimit. <z(x) 7
6
4. Concludingremarks
5
Usinganoiselevel-freeanalysis,wehaveshowntherelation-
shipsbetweeninter-clustersynchronizationphenomenaandex- 4
100 100.5 101 101.5 102 102.5 103 103.5 104
ternalnoise,especiallytheoccurrenceoftheinter-clustersyn-
time
chronization induced by the Langevin noise. Dealing with a
mean-fieldmodelof ensemblesof coupledlimitcycle oscilla-
torswithtwonaturalfrequenciesundertheinfluenceofexternal Figure2:Thetimeevolutionofthemeanvalueshz(1x)iG,hz(2x)iGoftheoscillators
oftheoscillators inthetwo clusters, corresponding tothechanges fromthe
white Gaussian noise, we have analytically derived the order
chaotictolimitcycleattractorsinFig. 1. Whiletheoscillatorsdonotbehave
parameter equations in the stationary states, taking advantage assynchronousinthedeterministiclimit(theupperpanel),anappropriatelevel
oftheingredientsoftheself-averagingpropertyinthethermo- oftheLangevinnoiseintensityinduces theinter-cluster synchronization (the
dynamiclimittogetherwithanHtheoremthatensuresthecon- lowerpanel).
vergence of the probability density to the Gaussian-form for
sufficientlylargetimes. Solvingtheorderparameterequations
References
numerically, we have systematically investigatedthe nonequi-
libriumphasetransitionswithchangesintheinter-clustercou- [1] A.Pikovsky,M.Rosenblum,J.Kurths,Synchronization:Auniversalcon-
plingstrengthǫanddifferenceofconstraintparametersδ(µ),and ceptinnonlinearsciences,UniversityPress,Cambridge,2001.
[2] C.Hammond,H.Bergman,P.Brown,TrendsinNeurosci.30(2007)357.
theLangevinnoiseintensityD(µ). Theresultshaveshownvari-
[3] M.Shiino,Phys.Lett.A111(1985)396.
ousinterestingbifurcationsincludingtheinter-clustersynchro- [4] C.Kurrer,K.Schulten,Phys.Rev.E51(1995)6213.
nizationandchaoticattractorsinducedbynoise. [5] T.Kanamaru,M.Sekine,Phys.Rev.E67(2003)031916.
[6] Y.Wang,D.T.W.Chik,Z.D.Wang,Phys.Rev.E61(2000)740.
Details of the relationship between the parameters involv-
[7] J.Teramae,D.Tanaka,Phys.Rev.Lett.93(2004)204103.
ingtheinter-clustercouplingstrengthǫ aswellasdifferenceof
[8] Y.Kawamura,H.Nakao,K.Arai,H.Kori,Y.Kuramoto,Phys.Rev.Lett.
constraintparametersδ(µ) andthebehaviorsoftheinter-cluster 101(2008)024101.
synchronization,andfurthermoretheeffectsofthemultiplica- [9] V.Hakim,W.J.Rappel,Europhys.Lett.27(1994)637.
[10] M.Shiino,Phys.Rev.A36(1987)2393.
tivenoiseontheinter-clustersynchronizationwillbereported
[11] R.C.Desai,R.Zwanzig,J.Stat.Phys.19(1978)1.
elsewhere.
[12] D.A.Dawson,J.Stat.Phys.31(1983)29.
[13] M.Shiino,K.Yoshida,Phys.Rev.E63(2001)026210.
[14] A.Ichiki,H.Ito,M.Shiino,PhysicaE40(2007)402.
Acknowledgment [15] M.Shiino,K.Doi,Proceedingsofthe2007IEEESymposiumonFoun-
dationsofComputationalIntelligence,IEEEPress,NewJersey,2007,p.
K.O. acknowledges the financial support from the Global 100.
Center of Excellence Program by MEXT, Japan through the [16] T.D.Frank,NonlinearFokker-PlanckEquations: FundamentalsandAp-
plications,Springer,Berlin,2005.
“NanoscienceandQuantumPhysics”Projectofthe TokyoIn-
stituteofTechnology.Oneoftheauthors(A.I.)issupportedby
theGrant-in-AidforJSPSFellowsNo. 20.9513.
4