23-New-approach-to-synchronization-analysis-of-lin

PhysicaD213(2006)214–230∗KeyLaboratoryofMathematicsforNonlinearSciences,FudanUniversity,MinistryofEducation,Shanghai,200433,PRChinaReceived17October2004;receivedinrevisedform25June2005;accepted29November2005CommunicatedbyC.K.R.T.JonesAbstractInthispaper,ageneralframeworkispresentedforanalyzingthesynchronizationstabilityofLinearlyCoupledOrdinaryDifferentialEquations(LCODEs).Theuncoupleddynamicalbehaviorateachnodeisgeneral,andcanbechaoticorotherwise;thecouplingconfigurationisalsogeneral,withthecouplingmatrixnotassumedtobesymmetricorirreducible.Onthebasisofgeometricalanalysisofthesynchronizationmanifold,anewapproachisproposedforinvestigatingthestabilityofthesynchronizationmanifoldofcoupledoscillators.Inthisway,criteriaareobtainedforbothlocalandglobalsynchronization.Thesecriteriaindicatethattheleftandrighteigenvectorscorrespondingtoeigenvaluezeroofthecouplingmatrixplaykeyrolesinthestabilityanalysisofthesynchronizationmanifold.Furthermore,therolesoftheuncoupleddynamicalbehavioroneachnodeandthecouplingconfigurationinthesynchronizationprocessarealsostudied.c2005ElsevierB.V.Allrightsreserved.Keywords:Linearlycoupledordinarydifferentialsystems;Synchronizationmanifold;Transversespace;Stabilityofsynchronizationmanifold1.IntroductionManynaturalandsyntheticsystems,suchasneuralsystems,socialsystems,fields,forinstance,sociology,biology,mathematicsandphysics.LinearlyCoupledOrdinaryDifferentialEquations(LCODEs)arealargeclassofdynamicalsystemswithcontinuoustimeandstate,aswellasdiscretespace,fordescribingcoupledoscilla-tors.ThisclassofdynamicalsystemshasbeeninvestigatedasIThisworkwassupportedbytheNationalScienceFoundationofChina60374018,60574044andtheGraduateStudentInnovationFoundationofFudanUniversity.∗Correspondingaddress:FudanUniversity,DepartmentofMathematics,HangdanRoad220,200433Shanghai,PRChina.Tel.:[phone];fax:[phone].E-mailaddresses:[e-mail],[e-mail](W.Lu),[e-mail](T.Chen).1Presentaddress:Max-Planck-Institutf¨urMathematikindenNaturwis-senschaftenInselstraße22D-04103Leipzig,Germany.theoreticalmodelsforspatiotemporalphenomenaofcomplexnetworks(forexample,see[1]andthe“Specialissueonnonlin-earwaves,pattern,andspatiotemporalchaosindynamicalar-rays”,IEEETrans.CAS-I.42(10),1995,editedbyL.O.Chua).Thedynamicalmodelisgovernedbythefollowingtwomech-anisms:theintrinsicnonlineardynamicsateachnodeanddif-fusionduetothespatialcouplingbetweennodes.Suchamodelcomesfromthespace-discretereaction–diffusionequationwithnspecies[3]:dxi(t)dt=f(xi)+c(xi+1+xi−1−2xi)=f(xi)+c[(xi+1−xi)+(xi−1−xi)]withperiodicboundaryxn+1=x1.Similarequationsarealsointroducedin[2]toexplainsymmetricbreakinginthedevelopmentoflivingorganisms.Ingeneral,theLCODEscanbedescribedasfollows:dxi(t)dt=f(xi(t),t)+mXj=1,j6=iaijΓ[xj(t)−xi(t)],i=1,2,...,m0167-2789/$-seefrontmatterc2005ElsevierB.V.Allrightsreserved.doi:10.1016/j.physd.2005.11.009W.Lu,T.Chen/PhysicaD213(2006)214–230215whereaij≥0fori,j=1,...,m,andΓ=diag[γ1,...,γn].Sincexi(t)−xi(t)=0,foralli=1,2,...,m,wecanchooseaiiasanyvalueintheequationabove.So,lettingaii=−Pmj=1,j6=iaij,theequationabovecanberewrittenasfollows:dxi(t)dt=f(xi(t),t)+mXj=1aijΓxj(t)i=1,2,...,m(1)wherexi(t)∈Rnisthestatevariableofthei-thnode,t∈[0,+∞)isacontinuoustime,f:R×[0,+∞)→Rniscontinuousmap,A=(aij)∈Rn×nisthecouplingmatrixwithzero-sumrowsandaij≥0,fori6=j,whichisdeterminedbythetopologicalstructureoftheLCODEs,andΓ=diag{γ1,γ2,...,γn}withγi≥0,fori=1,2,...,n.TheLCODEmodeliswidelyusedtodescribethemodelsinnatureandengineering.Inparticular,in[4],theauthorsstudiedspike-burstneuralactivityandthetransitionstoasynchronizedstateusingamodeloflinearlycoupledburstingneurons;in[5],theauthorspresentedanextensionofananalysistechniquedescribedforanecologicallymotivated,n-patchmeta-populationmodeldescribinganinteractionbetweendifferentspecies;in[13],thedynamicsoflinearlycoupledChuacircuitswasstudiedwithapplicationtoimageprocessing,andmanyothercases.Manycomplicateddynamicalbehaviorsofcoupledchaoticoscillatorshavebeenwidelystudied(fordetails,see[6–8,18,21]).Amongthem,thesynchronizationofLCODEshasbeenanactiveareafordecades(see[9]).Theword“synchronization”comesfromaGreekword,whichmeans“sharetime”.Today,inscienceandtechnology,ithascometobeconsideredas“timecoherenceofdifferentprocess”.Sincesynchronizationofinteractingsystemswasobservedin[10],thisphenomenonhasalsoappearedinawiderangeofrealsystems.Intheoreticalfields,therearevariouskindsofconceptsofsynchronization,forexample,phasesynchronization,imperfectsynchronization,lagsynchronization,andalmostsynchronization.Inthispaper,theconceptofcompletesynchronizationisconsidered;itcanbemathematicallydefinedasfo