Progress of Theoretical Physics Supplement No. 161, 2006 43 Effects of Delayed Feedback on

43

EffectsofDelayedFeedbackonKuramotoTransition

DenisS.GoldobinandArkadyPikovsky

DepartmentofPhysics,UniversityofPotsdam,Potsdam,Germany

(ReceivedJuly4,2005)

WedevelopaweaklynonlineartheoryoftheKuramototransitioninanensembleofgloballycoupledoscillatorsinpresenceofadditionaltime-delayedcouplingterms.Weshowthatalineardelayedfeedbacknotonlycontrolsthetransitionpoint,buteffectivelychangesthenonlineartermsnearthetransition.Apurelynonlineardelayedcouplingdoesnoteffectthetransitionpoint,butcanreduceorenhancetheamplitudeofcollectiveoscillations.

§1.Introduction

AtransitiontocollectivesynchronyinanensembleofgloballycoupledoscillatorsisknownastheKuramototransition.3)Animportantapplicationofthetheoryiscollectivedynamicsofneuronalpopulations.Indeed,synchronizationofindividualneuronsisbelievedtoplaythecrucialroleintheemergenceofpathologicalrhythmicbrainactivityinParkinson’sdisease,essentialtremor,andepilepsies;adetaileddiscussionofthistopicandnumerouscitationscanbefoundinRefs.2),4)and8).Oneapproachtosuppresssuchanactivityistoapplytothesystemanegativefeedbackloop.5)–7)

ThegoalofthispaperistodevelopaweaklynonlineartheoryoftheKuramototransitioninthepresenceoflinearandnonlineartime-delayedcouplingterms.WeheavilyrelyinouranalysisonthecorrespondingtreatmentofthesystemwithoutdelaybyCrawford.1)

§2.

Fromlimitcyclesystemstophasemodels

Hereweintroduceourbasicmodel—anensembleofautonomousoscillatorssubjecttodifferenttypesofglobalcoupling.WetakeindividualoscillatorsasVanderPolonesandwritethemodelas

√2

)x˙+ωx=22ωiξi(t)+ε󰀃F(x,y),(2.1)x¨i−µ(1−x2iiiiwhereξi(t)isaδ-correlatedGaussiannoise:󰀊ξi(t)ξj(t−t󰀃)󰀋=2Dδijδ(t󰀃).The

ensembleaveragesaredefinedas

N1󰀎x=xj,

N

j=1

N

˙j1󰀎x

y=.

Nωj

j=1

Inthereductiontophaseequationsweusethesmallnessofparametersµandε󰀃,

andsupposethenaturalfrequenciesωitobedistributedinarelativelyclosevicinity󰀌ofthemeanfrequencyω0≡N−1Nj=1ωj.Becauseµ󰀄ωi,thesolutionofthe

44D.S.GoldobinandA.Pikovsky

autonomousVanderPoloscillatorcanbewrittenasxi(t)≈Ai(t)cos(ϕi(t))where

˙i=ωi.Becauseε󰀃󰀄µ,couplingdoesnotaffecttheonthelimitcycleAi≈2andϕ

amplitude(whichremains≈2),butonlythephase.Itisconvenienttointroducethecomplexorderparameter

11󰀎iϕj(t)

eR(t)=|R|eiθ(t)=(x+iy)=

2N

j

(2.2)

andtorepresenttheglobalcouplingintermsofR.Theabsolutevalueoftheorder

parameterisclosetozerofornearlyuniform,nonsynchronizeddistributions,andreaches1forstronglysynchronizedstates.

Belowwewillbeinterestedinlinearcouplingwithandwithouttimedelay,6),7)andinanonlinearcoupling:5)

ε󰀃F(x,y)=2ω0εy(t)+2ω0εfy(t−T)+

d2(x(t−T))(Kxx(t)+Kyy(t)).dt

Asaresult,thephaseequationsfortheoscillatorsread

NN

εf󰀎ε󰀎

sin(ϕj(t)−ϕi(t))+sin(ϕj(t−T)−ϕi(t))ϕ˙i=ωi+NN

j=1

j=1

+εof|R|2(t−T)|R|(t)sin[2θ(t−T)−θ(t)−ϕi(t)+ν]+ξi(t),

(2.3)

whereεofeiν=2(Kx+iKy).Herethreecouplingparametersdescribedifferenttypes

ofcoupling:εdescribescollectivelinearcouplingwithoutdelay,asintheoriginalKuramotomodel;εfdescribeslinearcouplingwithdelay,ashasbeenproposedin6)and7);εofdescribesnonlinearcouplingwithdelayashasbeenproposedin5).

§3.

Linearfeedback:thermodynamiclimitandstability

Westartwithaconsiderationofanensembleofoscillatorswithlinearcouplings,i.e.inthisandthenextsectionsweconsider(2.3)withεof=0.Inthethermody-namiclimitN→∞wecanintroduceadistributionofnaturalfrequenciesg(ω)andrewritesystem(2.3)as

󰀏

ϕ˙(ω)=ω+ε

󰀏+εf

+∞

󰀁󰀂

g(ω󰀃)sinϕ(ω󰀃,t)−ϕ(ω,t)dω󰀃

(3.1)

−∞+∞

󰀁󰀂

g(ω󰀃)sinϕ(ω󰀃,t−T)−ϕ(ω,t)dω󰀃+ξ(ω,t).

−∞

Forastatisticaldescriptiononeintroducesadistributiondensityρ(ω,ϕ,t)(nor-󰀍2π

malizedas0ρ(ω,ϕ,t)dϕ=1)thatisgovernedbytheFokker-Planckequation:

∂∂2ρ∂ρ+(ρv)−D2=0,∂t∂ϕ∂ϕ

(3.2)

EffectsofDelayedFeedbackonKuramotoTransition

where

v(ω)=ω+ε

󰀏+εf

0

45

󰀏

2π

󰀏dθ󰀏

+∞

dω󰀃g(ω󰀃)sin(θ−ϕ)ρ(ω󰀃,θ,t)

(3.3)

02π

dθ

−∞+∞

dω󰀃g(ω󰀃)sin(θ−ϕ)ρ(ω󰀃,θ,t−T).

−∞

Theorderparameterintroducedin(2.2)nowtakestheform

󰀏+∞󰀏2π

1󰀎iϕj(t)

e=dωg(ω)dϕρ(ω,ϕ,t)eiϕ.R(t)=N−∞0

j

(3.4)

Hereweshortlydiscussalinearstabilityanalysisoftheabsolutelynonsynchro-nousstateρ0=21π.Infinitesimalperturbationsρ1ofthisstatearegovernedbythe

linearizationofEq.(3.2)

∂v1∂ρ1∂ρ1∂2ρ1+ρ0+v0−D=0∂t∂ϕ∂ϕ∂ϕ2with

∂v1

=−ε∂ϕ

󰀏

02π

(3.5)

󰀏dθdθ

+∞

󰀏

0

−εf

Substitutingρ1=differentck:

󰀍+∞

−∞

2π

󰀏

−∞+∞−∞

dω󰀃g(ω󰀃)cos(θ−ϕ)ρ1(ω󰀃,θ,t)dω󰀃g(ω󰀃)cos(θ−ϕ)ρ1(ω󰀃,θ,t−T).

(3.6)

󰀌

kck(ω)e

ikϕ+λt

(k=0),onefindsindependentequationsfor

(3.7)

ε+εfe−λT

(δk,1+δk,−1)Ck,(λ+ikω+Dk)ck(ω)=

2

2

ε+εfe−λT

C1.(3.8)c1(ω)=

2(λ+D+iω)

Multiplyingthisequationbyg(ω)andintegratingoverω,onefindsthatthespectrumisformedbytherootsofthe“spectralfunction”Λ(λ)

󰀏

ε+εfe−λT+∞g(ω)dω

=0.(3.9)Λ(λ)≡1−

2−∞D+λ+iω

󰀄󰀍󰀅󰀍+∞+∞−1Generally,󰀉−∞g(ω)(D+iω)dω=−∞ωg(ω)(D2+ω2)−1dω=0;there-forerealrootsofΛ(λ)(includingλ=0)arenotadmittedandonlyonecomplexroot

λ=−iΩwiththecorrespondingmodeρ1=α(ω)ei(ϕ−Ωt)+ccdetermineslinearsta-bility.FromthelinearanalysiswethusexpectaHopfbifurcationforthetransitiontosynchrony.󰀍+∞

Inthedegeneratedcase−∞ωg(ω)(D2+ω2)−1dω=0,arelationΛ∗(λ)=Λ(λ∗)holds,thenrealrootsareadmittedandcomplexrootsappearinpairs(λ,λ∗).Weexpectthatinrealapplicationsthedegeneracyofthefrequencydistributionisabsent,sowedonotconsiderthissituationbelow.

whereCk=onefinds

g(ω)ck(ω)dω.Modeswith|k|=1alwaysdecaywhilefork=1

46D.S.GoldobinandA.Pikovsky

§4.

Weaklynonlinearanalysis

Inthissectionweperformaweaklynonlinearanalysisofthesynchronizationtransition,consideringεasabifurcationparameter.Wewriteε=ε0+κ2ε2whereε0isthecriticalvalueofεandκisasmallparameter,andrepresenttheprobabilitydistributionρ(x,t)asρ0+κρ1+κ2ρ2+κ3ρ3+....Assumingρ1=α1(ω,t2,t4,...)ei(ϕ−Ωt0)+cc(heretkare“slowtimes”)andsubstitutingthisinEq.(3.2)weobtainintheorderκ2(therearenoseculartermsinthisorder):

∂∂2ρ2∂v2∂ρ2∂ρ2

+(ρ1v1)+v0−D+ρ0=0,

∂t0∂ϕ∂ϕ∂ϕ∂ϕ2where

󰀏

2π

(4.1)

󰀁󰀂i(θ−Ωt)

iΩT0

dθdωg(ω)sin(θ−ϕ)α1(ω)ε0+εfe+ccv1=e0−∞󰀁󰀂

=iπε0+εfeiΩTA1ei(ϕ−Ωt0)+cc,(4.2)

󰀍+∞

andwehaveintroducedAj≡−∞αj(ω)g(ω)dω.Notethatfrom(3.8)itfollowsthat

ε+εfe−λT

A1.(4.3)α1(ω)=

2(λ+D+iω)

Thisgivesthe“drivingterm”in(4.1):

󰀅󰀂∂󰀄󰀁∂iΩTi2(ϕ−Ωt0)

(ρ1v1)=iπε0+εfe+cc+...α1(ω)A1e∂ϕ∂ϕ

󰀃

󰀃

󰀃

󰀏

+∞

󰀂󰀁

=−2πε0+εfeiΩTα1(ω)A1ei2(ϕ−Ωt0)+cc.

SearchingforsolutionofEq.(4.1)intheformρ2=α2(ω,t2,t4,...)ei2(ϕ−Ωt0)+cc,weobtain,using(4.3),

󰀁󰀁󰀂󰀂2πε0+εfeiΩTA22πε0+εfeiΩTA1α1(ω)1

=.(4.4)α2(ω)=

−i2Ω+i2ω+4D2(D+i(ω−Ω))(2D+i(ω−Ω))

Intheorderκ3ofEq.(3.2),seculartermsappear:

∂v3∂ρ3∂∂2ρ3∂ρ3∂ρ1++(ρ1v2+ρ2v1)+v0−D+ρ0=0.(4.5)2∂t∂t2∂ϕ∂ϕ∂ϕ∂ϕ

󰀍2π

Notethatv2=0because0eiϕρ2(ω,ϕ,t)dϕ=0.Calculationofotherseculartermsyields

󰀏2π󰀏+∞

dθdω󰀃g(ω󰀃)sin(θ−ϕ)ρ1(ω󰀃,θ,t)v3=vρ3+ε20−∞

󰀆󰀇󰀏2π󰀏+∞󰀃,t,...)∂α(ω12

dθdω󰀃g(ω󰀃)sin(θ−ϕ)(−T)ei(θ−Ω(t−T))+cc+εf

∂t20−∞

󰀆󰀆󰀇󰀇

∂A1

ei(ϕ−Ωt0)+cc,=vρ3+iπε2A1−εfeiΩTT∂t2

EffectsofDelayedFeedbackonKuramotoTransition

wherevρ3=

󰀍2π

0

47

dθ

󰀍+∞

−∞

dω󰀃g(ω󰀃)sin(θ−ϕ)(ε0ρ3(ω󰀃,θ,t)+εfρ3(ω󰀃,θ,t−T))and

󰀅󰀁󰀂∗∂󰀄∂−iΩTi(ϕ−Ωt0)

(ρ2v1)=−iπε0+εfe+cc+...A1α2(ω)e∂ϕ∂ϕ

󰀂󰀁i(ϕ−Ωt0)

+cc+....=πε0+εfe−iΩTA∗1α2(ω)e

(Here“...”denotesnon-secularterms.)Collectingallsecularterms,wecanwrite

themas

󰀉󰀈

󰀂󰀁εfeiΩT∂A1∂α1(ω)ε2i(ϕ−Ωt0)

T−A1++πε0+εfe−iΩTA∗+cc.1α2(ω)e∂t222∂t2

(4.6)

Nowwehavetowriteouttheconditionoforthogonalityofthesetermstothesolu-tionsoftheconjugatedproblem,i.e.theconditionthatthesecularpartof“driving”vanishes.Assoonasthescalarproductofτ-time-periodicfieldss(ω,ϕ,t)andc(ω,ϕ,t)isdefinedby

󰀏󰀏2π󰀏+∞

dϕτdt∗

s(ω,ϕ,t)c(ω,ϕ,t),dωg(ω)(4.7)󰀊s,c󰀋≡

2π0τ−∞0theconjugatedproblemreads

󰀇󰀆󰀏+∞

ε0+εfe−λT∂2

(δk,1+δk,−1)g(ω󰀃)ck(ω󰀃)dω󰀃(4.8)−−ikω+Dkck(ω)=

∂t2−∞andhasasolution

ei(ϕ−Ωt)

.

D−i(ω−Ω)

(4.9)

Finally,theorthogonalitycondition,i.e.thevanishingofthescalarproductof(4.9)and(4.6),yieldstheweaklynonlinearamplitudeequation:

󰀈󰀉󰀏+∞

󰀁󰀂εfeiΩT∂A1∂α1(ω)ε2g(ω)dω

T−A1++πε0+εfe−iΩTA∗1α2(ω)D+i(ω−Ω)∂t22∂t22−∞

=0.

󰀍+∞g(ω)dωi

SubstitutinghereforαjandintroducingafunctionG(z)=2π−∞ω−zweobtain

˙1=λ2(ε0,Ω)A1−P(ε0,Ω)A1|A1|2,A

whereλ2isthelineargrowthrate

λ2(ε,Ω)=

andP(ε,Ω)=

iπ(ε+εf󰀃

2eiΩT)G󰀃(Ω

(4.10)

ε−ε0

+iD)+εfTeiΩT

(4.11)

π2󰀃ε

󰀃

+iD)−G(Ω+i2D)+G(Ω+iD))+εf

󰀄󰀅.(4.12)

−2

DiDG󰀃(Ω+iD)+π−1DεfeiΩTT(ε+εfeiΩT)

2

eiΩT󰀃(iDG󰀃(Ω

48D.S.GoldobinandA.Pikovsky

Equation(4.10)andtheexpressions(4.11),(4.12)arethemainresultofouranalysis.Theygiveafulldescriptionoftheeffectofthedelayedglobalfeedbackonthesynchronizationtransitionintheensembleofoscillators.Thelinearpart(4.11)hasalreadybeendiscussedin6),andtheexpression(4.12)completesthedescriptionofthesynchronizationtransition.HavingdeterminedtheamplitudeA1from(4.10),onecanfindtheestablishingprobabilitydistribution

󰀊󰀁󰀂iΩTπε+εe10f1+A1(t)ei(ϕ−Ωt)+ccρ(ω,ϕ,t)=2πD+i(ω−Ω)

󰀋󰀁󰀂2

iΩTε0+εfei2(ϕ−Ωt)

A2+cc+O(A3+1(t)e1),(D+i(ω−Ω))(2D+i(ω−Ω))

(4.13)

π2

andtheorderparameter

iΩt

R(t)=2πA∗+O(A31e1).

§5.Anexample:Lorentzdistributionofnaturalfrequencies

Thegeneralexpressions(4.11)and(4.12)abovecanbeconsiderablysimplified

fortheLorentziandistribution

γ

g(ω)=,(5.1)22π((ω−ω0)+γ)whereγisacharacteristicwidthofthedistributionandω0isthemeanfrequency.

Inthiscase󰀏+∞

i1g(ω)dωi

=,G(z)=

2π−∞ω−z2πω0−iγ−zwhere󰀉zisassumedtobepositive(thisholdsforD>0).

Firstweobtainexplicitexpressionsforspectrumofthelinearproblem.Equa-tion(3.9)takestheform

󰀂󰀁

iε+εfe−βT+iΩT

1+=0,

2(ω0−Ω−i(γ+D+β))wherewehavesubstitutedλ=β−iΩ,βandΩbeingtherealgrowthrateandthefrequency.Separatingrealandimaginaryparts,onecanfind

εf

ε=2(γ+D+β)−εfe−βTcosΩT.(5.2)Ω=ω0−e−βTsinΩT,

2

Thethresholdvalueε0isdeterminedbyβ=0.Substitutingtheexpressionsabovein(4.11)and(4.12)weobtain

ε2

,(5.3)λ2(ε0,Ω)=iΩT2+εfTe

4

.(5.4)P(ε0,Ω)=iΩTiΩT(ε0+εfe+2D)(2+εfTe)

EffectsofDelayedFeedbackonKuramotoTransition

|R||R0|49

10 1 0.1-0.1-0.05εf 0 0.05 0.1 0 1 2 3 4 5 6 7 8TFig.1.Effectofdelayedfeedbackontheorderparameterforω0=1,γ=D=0.01.

λ2ThestationaryamplitudeA1iscalculatedaccordingto(4.10)|A1|2=󰀅󰀅P.Todemonstrate,howthedelayedfeedbackaffectstheamplitude,wepresentinFig.1

R|

whereR0istheorderparameterintheabsenceofdelayedfeedbacktheratio||R0|forthesameclosenesstothetransitionpointε2.

§6.Nonlineardelayedfeedback

Inthissectionweconsiderapurelynonlineardelayedfeedbackintheensembleofoscillators.Wesetεf=0inEq.(2.3)andwritethebasicmodelas

Nε󰀎

sin(ϕj(t)−ϕi(t))ϕ˙i=ωi+N

+εof|R|(t−T)|R|(t)sin(2θ(t−T)−θ(t)−ϕi(t)+ν)+ξi(t).

j=12

(6.1)

Similarlytothepreviouscase,inthethermodynamicallimitN→∞onecanwrite

∂∂2ρ∂ρ+(ρv)−D2=0,∂t∂ϕ∂ϕ

where

󰀏

v(ω)=ω+ε

2π

(6.2)

dϕ

󰀃

󰀏

+∞

+εof|R|(t−T)|R|(t)sin(2θ(t−T)−θ(t)−ϕ+ν).

02

−∞

dω󰀃g(ω󰀃)sin(ϕ󰀃−ϕ)ρ(ω󰀃,ϕ󰀃,t)

(6.3)

50D.S.GoldobinandA.Pikovsky

Thelinearproblemisthesameasinthepreviouscasewhereonesetsεf=0.Thereforeassoonasg(ω0+∆ω)=g(ω0−∆ω),criticalperturbationseitherhavethefrequencyω0oraredegenerate:theyappearinpairsω0−∆ω,ω0+∆ω(seediscussionbyCrawford1)).Werestrictourselvestonon-degeneratecaseonly.Consideringnearlycriticalbehaviorofsmallperturbationρ1=α1(ω,t2,t4,...)ei(ϕ−Ωt0)+cc,onecanwritedownfromEq.(6.2)intheorderκ2(thereisnoseculartermsinthisorder)Eq.(4.1)with

(6.4)v1=iπε0A1ei(ϕ−Ωt0)+cc.NowfromEq.(3.8),α1(ω)=

ε0A1

.Therefore

2(D+i(ω−Ω))

󰀋󰀊

2i2(ϕ−Ωt)0A1e∂

(ρ1v1)=−πε2+cc.0∂ϕD+i(ω−Ω)

Searchingforρ2intheformρ2=α2(ω,t2,t4,...)ei2(ϕ−Ωt0)+ccwefind

2πε20A1.α2(ω)=

2(D+i(ω−Ω))(2D+i(ω−Ω))

Intheorderκ3Eq.(4.5)withv2=0and

v3=iπ(ε0A3+ε2A1)ei(ϕ−Ωt0)+cc

+εof|R|2(t−T)|R|(t)sin(2θ(t−T)−θ(t)−ϕ+ν)

󰀁󰀂

=iπ(ε0A3+ε2A1)ei(ϕ−Ωt0)+cc+εof󰀉R2(t−T)R∗(t)eν−ϕ

iΩtwegetisvalid.SubstitutingR1=2πA∗1e

2∗2i2Ω(t−T)iΩ(t−2T)

R1(t−T)R1(t)=8π3A∗A1e−iΩt=8π3|A1|2A∗.1e1e

Thereforeand

v3=...−8πεof|A1|󰀉A1e

32

󰀐

i(ϕ−Ωt−ν+2ΩT)

󰀑

󰀐󰀑∂v322i(ϕ−Ωt−ν+2ΩT)

ρ0=...−4πεof|A1|󰀈A1e,∂ϕ

where“...”denotesthetermswhichdonotcontributetothesecularpartofthe

∂

(ρ2v1)canbetakenfrom§4,anditscontributiontoP(ε,Ω)equation.Theterm∂ϕisgivenbytheformula(4.12)withεf=0.Summinguptheseresults,onecanfindthatEq.(4.10)holdswith

λ2(ε,Ω)=

ε2

,(6.5)2󰀃iπεG(Ω+iD)

󰀈󰀉

i4πεofei(2ΩT−ν)G(Ω+iD)−G(Ω+2iD)π2ε2

1++.(6.6)P(ε,Ω)=󰀃󰀃DiDG(Ω+iD)εG(Ω+iD)

EffectsofDelayedFeedbackonKuramotoTransition

Theresultingprobabilitydensityreads

󰀈1πε0A1(t)

ρ(ω,ϕ,t)=1+ei(ϕ−Ωt)+cc

2πD+i(ω−Ω)

2π2ε20A1(t)ei2(ϕ−Ωt)+cc+O(A3+1)(D+i(ω−Ω))(2D+i(ω−Ω))

51

󰀉

(6.7)

iΩt+O(A3).andtheorderparameterisR(t)=2πA∗1e1

Asaparticularexampleweconsider,likein§2,theLorentziandistributionofnaturalfrequencies(5.1).ThecharacteristicequationΛ(λ)=0takestheform

ε−2(γ+D)=2(λ−iω0)

(6.8)

andhasonlyoneroot.Thebifurcationofthenon-synchronousstateisaHopfoneatε0=2(γ+D)withthefrequencyΩ=ω0(seediscussionbyCrawford1)).SettingΩ=ω0in(6.5)and(6.6),wefind

λ2(ε0,ω0)=

ε2,2

P(ε0,ω0)=

1

−4π2εofe−iν(γ+D).

2D+γ

(6.9)

TherealpartofPdetermines,accordingto(4.10),theamplitudeoftheestablishingcollectivemode|A1|2=λ2(󰀈P)−1,with

󰀈P(ε0,ω0)=

1

−4π2εof(γ+D)cos(ν).

2D+γ

Onecanseethatdependingonthevalueofν,theamplitudedecreasesorincreasesduetoadditionalnonlinearfeedback.Moreover,forstrongenoughfeedback󰀈Pcanbecomenegative,whatmeansasubcriticalKuramototransition.Also,anonlinearshiftoftherotationfrequencyofRinthecounterclockwisedirectionappears

ω2=󰀉(P)|A1|2=

ε2tanνε2󰀉(P)=.

2󰀈(P)2[4π2εof(2D−γ)(D+γ)cosν]−1−1

§7.

(6.10)

Conclusion

Inthispaperwehavedevelopedaweaklynonlinearanalysisoftheeffectof

delayedfeedbackontheKuramototransition.WehaverestrictedourattentiontothemostgeneralcaseofHopfbifurcationandhavenotconsideredothertypesoftransitionthatoccurundercertainsymmetries.Theanalysisis,ofcourse,restrictedtoavicinityofthetransitionpoint,moreover,thebasicphase-couplingmodelas-sumesthatalltypesofcouplingareweak.Astrongcouplingcaseshouldbestudiednumerically.

Acknowledgements

WewouldliketothankM.Rosenblum,O.PopovychandP.Tassforusefuldiscussions.

52D.S.GoldobinandA.Pikovsky

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