Control of Chaos by Time{Delayed Feedback a Survey

ControlofChaosbyTime{DelayedFeedbak:aSurveyofTheoretialandExperimentalAspetsWolframJust1,HartmutBenner2,andEkehardSholl31InstituteofPhysis,ChemnitzUniversityofTehnology,D{09107Chemnitz,Germany2InstituteofSolidStatePhysis,DarmstadtUniversityofTehnology,Hohshulstrae6,D{64289Darmstadt,Germany3InstituteofTheoretialPhysis,BerlinUniversityofTehnology,Hardenbergstrae36,D{10623Berlin,GermanyAbstrat.Time{delayedfeedbakontrolhasbeenintroduedasapowerfultoolforontrolofunstableperiodiorbitsindynamialsystems.Fromtheexperimentalpointofviewitsstrengthisbasedonthefatthattheappliationofthismethodrequiresjustthemeasurementofsimplesignals,andithasbeenappliedinphysis,hemistryandbiology.Wepresentanoverviewofthetheoretialfoundationsoftime{delayedfeedbakmethodsandexplainindetailtheimpliationsforrealex-periments.1IntrodutionDuringthelastdeadeontrolofhaoshasdevelopedintooneofthemostprominenteldsinappliednonlinearsiene[1℄.Itsbeginningwastriggeredbytheobservationthatontheonehandhaotimotionprovidesahugenumberofunstablestatesandthatontheotherhandeahofthesestatesanbestabilisedbytinyontrolfores[2℄.Theontrolshemethatwasoriginallydevelopedin[2℄isbasedontheloalphasespaestrutureintheviinityofthetargetorbit.Ithasbeenappliedindierentexperimentalontextswheresuhastrutureisexper-imentallyaessible(f.e.g.[3℄).Infat,suhapproahesallowformoresophistiatedgoalsliketrakingandtargetingofpartiulartrajetories.Inthewakeofthesedevelopmentsvariantsoftheontrolmethodhavebeenredisovered[4℄whiharesimplertohandleformtheexperimentalpointofviewandwhihareoftenonsideredinontroltheory.Mostappliationsinthephysiistsontextonernthestabilisationofxedpoints,eitherintheoriginalphasespaeorwithrespettoasuitablePoinarerosssetion.Aboveallitisimportanttostressthatontrolshemesmentionedaboveaimatnoninvasiveontrol,sothattheontrolforenallyvanishesandthetargetstateisnotalteredbytheontrolloop.Noninvasiveshemesopenthepossibilitytouseontrolmethodsasakindofnonlinear2WolframJustetal.spetrosopysinedierentpropereigenmotionsofthesystemanbede-tetedandreorded.Controltheoryisawelldevelopeddisiplineinappliedmathematisandengineeringsiene,andthereexistsahugeamountofliteraturerangingfromelementarytosophistiatedpresentations(f.e.g.[5,6℄).Controlproblemsarefrequentlyformulatedintermsofboundaryvalueandoptimisationproblems.Thusalsoglobalpropertieslikeontrollabilityandobservabilityofatargetstateanbeaddressed.However,suhoneptsaretosomeextentrestritedtosystemswhereatleastpartialinformationabouttheinternalstateisa-essibleortosystemswhihareeitherlinearoronjugatetoalinearsystemsinpartsoftheirphasespae.Whetheraontrolshemeisinvasivedoesnotplayaruialrolefromthepointofviewofontroltheory.Itisawellestablishedfatfordeades[7℄andtosomeextentommonwisdominontroltheorythattimedelayreduestheeÆienyofaontrolsheme.Thereforeitwasquiteasurpriseandhasbeenpointedoutreently[8℄thattime{delayedfeedbakmaybesuitabletogenerateontrolforesforstabilisingtime{periodistates.Themainideaofsuhfeedbakshemesisquitesimpleandanbeappliedindierentexperimentalontexts.Supposeasignals(t)isaessibleformeasurementandweintendtostabiliseanunstableperiodistateofperiod.Fromthetime{delayeddierenes(t)s(t)onegeneratesaontrolfore,e.g.bylinearampliation,andonefeedsitbaktothesystem.Obviouslysuhasheme(f.Fig.1)isnoninvasivesinetheforevanishesprovidedthe{perioditargetstateisreahed.Whethersuhapresriptionworksrequiresamoredetailedinvestigation.Nevertheless,theshemehasbeenappliedsuessfullyinoptialexperiments,e.g.forstabil-isinglasers[9℄ordishargegastubes[10℄,inhydrodynamiexperimentse.g.forstabilisationofturbulentTaylor{Couetteows[11℄,inhemialsetupse.g.foreletrohemialreations[12℄,inmagnetisystems,e.g.forontrol-linghighpowerferromagnetiresonaneexperiments[13℄andinbiologialsystems,e.g.forontrollingarrhythmiardia[14℄.Ofoursedemonstra-tionexperimentslikemehanialpendula[15℄oreletroniiruits[16℄havebeenperformedalso.Thelatterareveryusefultostudydetailsoftheontrolshemefromanexperimentalperspetive.Despiteitsexperimentalsuessadeepertheoretialunderstandingoftime{delayedfeedbakontrolhasbeengainedonlyreently.Herewepresentasummaryofthesedevelopments.Wemainlyfousonessentialtheoretialfeatureswithoutgivingallthetehnialdetails.Therelevaneoftheseitemsisillustratedbyresultsfromeletroniiruitexperiments.Setion2isde-votedtotheanalysisoftheoriginalPyragassheme.Morespeialisedtopislikemultipledelaysortheintrodutionofadditionaltimesalesintheontrolproessareaddressedinsetion3.Thelakofapropertheoretialunder-standingofsomeaspetsisrelatedtothefatthattimedelaysystemsatonaninnite{dimensionalphasespae.Thus,dynamiswithtimedelayisofControlofhaosbytime{delayedfeedbak3h(t)s(t)=g[x(t)]x(t)s(t_τ)s(t)ΔKFig.1.Diagrammativiewoftime{delayedfeedbakontrol.Internaldegreesoffreedomaredenotedbyx(t).Themeasuredsignals(t)=g[x(t)℄dependsontheinternaldegreesoffreedom.TheontrolforeF(t)=K[s(t)s(t)℄isgeneratedfromthetime{delayeddierenebylinearampliation.Theforeisoupledtosomeexternalparameterh(t)oftheexperimentalsetupinterestbyitself.Wepointoutsomereenttrendsintheoutlookinsetion4.2PropertiesoftheOriginalPyra