Study of chaos in hamiltonian systems via converge

arXiv:chao-dyn/9606002v17Jun1996StudyofchaosinhamiltoniansystemsviaconvergentnormalformsWernerM.Vieira†,AlfredoM.O.deAlmeida‡(February5,2008)AbstractWeuseMoser’snormalformstostudychaoticmotionintwo-degreehamil-toniansystemsnearasaddlepoint.Besidesbeingconvergent,theyprovideasuitabledescriptionofthecylindricaltopologyofthechaoticflowinthatvicin-ity.Bothaspectscombinedallowedaprecisecomputationofthehomoclinicinteractionofstableandunstablemanifoldsinthefullphasespace,ratherthanjustthePoincar´esection.TheformalismwasappliedtotheH´enon-Heileshamiltonian,producingstrongevidencethattheregionofconvergenceofthesenormalformsextendsoverthatoriginallyestablishedbyMoser.PACSindex:05.45.+b†DepartamentodeMatem´aticaAplicada,UniversidadeEstadualdeCampinas,CP6065,13083-970Campinas,SP,Brazil.e-mail:[e-mail]‡CentroBrasileirodePesquisasF´ısicas,RuaXavierSigaud,150,2290-180RiodeJaneiro,RJ,Brazil.1I.INTRODUCTIONNormalForms(NF)areamongthesuccessfulmethodsforeitheranalyticornumericstudiesofdynamicalsystems;byperformingasuitablecoordinatetransformation,weeven-tuallyobtainamoresimpleordynamically“transparent”versionoftheoriginalsystem.Thisapproachcanbeformulatedeitherforgenericsystemsincludingtwo-dimensionalmaps[1]orforhamiltoniansystems[2]andconservativemaps[3].So,evenwhenagivenNFdoesnotconverge,duetosmalldenominatorsorexactressonances,itisofimportancefornumericalpurposes.ThisisjustwhatoccurswhentheNFisobtainedaroundastablepointororbit.Then,inspiteofthewellknowndivergenceinthiscase,atruncationallowsustofollowthemotionforalongtimewithgreatprecision.Infact,thenonconvergentcaseisthemostconsideredintheliterature[4–6].ThepresentworkconcernstheNFaroundaunstablepointororbitofaconservativesystem.Forthat,Moserdemonstratedtheirconvergenceforbothmaps[7]andhamiltoniansystems[8].Althoughconvergent,thiscasedidnotreceivemuchattentionuntilrecently,presumablybecausetheparticleremainsaveryshorttimeinthatregion.Nevertheless,wewillseethattheMosernormalforms(MNF)arebothconvergentandapowerfulltoolforthesearchforthebasicstructuresofthechaoticmotionratherthanjustfollowingaspecificorbitforalongtime.TheusefulnessoftheMNFforthestudyofconservativechaoticmapsisalreadyknownintheliterature.Itallowedpreciseanalyticalcomputationsofhomoclinicpointsandtheperiodicpointswithlongperiod,whichaccumulateinthehomoclinicones[9,10].Additionalgoodnumericalresultswereobtainedevenifsmalldissipativeperturbationswereadded[11].Area-preservingmapsareusuallyonlysimplifiedreductions(appropriatePoincar´esec-tions)ofautonomoushamiltoniansystemsoftwodegreesoffreedom.Inparticular,theverycomplextwo-dimensionalhomoclinictangle[12]isalreadyareductionofthemuchmoreinvolvedchaoticmotioninthefullphasespace.However,extendingtheuseoftheMNFtothehamiltoniancaseallowsustostudydirectlytheproperfour-dimensionalchaoticflow.2Thishaspreviouslybeenattemptedintheliterature[13],butwithouttakingfulladvantageofthemethodthatweshalldevelophere.AnotheruseofMNFsisfoundin[14]andtakesadvantageofitsconvergencetoascertainstabilitytransitionsoffamiliesofperiodicorbitsnearhamiltonians’saddlepoints.InsectionIIwewilldeveloptheMNFapproachforthecaseofagenericautonomoushamiltonianoftwodegreesoffreedomaroundasaddlepoint,encompassedbyMoser’sconvergenceproof.Inthatvicinity,theflow’stopologyiscylindricalratherthantoroidal,inthecaseofachaoticregime[15].Firstly,weconstructtherelationslinkingtheoriginalsystemtothecorrespondingmore“transparent”normalizedsystem.Thisisdonethroughanearidentitypolynomialcoordinatetransformation.Wewillseethat,besidesconvergent,thattransformationalsorevealsinanaturalway,thecylindricalcharacterofthetopology.BothfeaturesturntheMNFintoapowerfulltool.So,itwaspossibletocompute,preciselyforthefirsttime,thecontinuousstructuresinthefullphasespace,underlyingthehomoclinictangleinaPoincar´esection:the(un)stablemanifoldswhichoriginateatthesaddlepointandateachneighbouringunstableperiodicorbit,thehomoclinicorbitsassociatedwiththelatterandtheperiodicorbitswithlongperiodwhichaccumulateonthehomoclinicorbits.InsectionIIIweobtaintherecurrencerelationsforthecoefficientsinvolvedinthetheory.InsectionsIVandVweapplytheformalismtothespecificcaseoftheH´enon-Heileshamiltonian.ThenumericalresultsexhibitedinsectionVfullyconfirmedtheexpectationsabouttheMNFasatoolforthestudyandcharacterizationofchaoticmotions.Moreover,theyalsopointtosomekindofextensionoftheregionofconvergenceinitiallyassumedforMoser’stheorem.Infact,thisissueisjustbeingconsideredbytheauthorspresently.Finally,insectionVIwesummarizetheresultsandpossibleextensionsofthepresentwork.3II.MOSER’SNORMALFORMItisessentialthatthehamiltonianbeinthecomplexifiedform,forthemethod’simple-mentation:h(x1,x3,x2,x4)=λ1x1x3+λ2x2x4+∞Xℓ=3H(ℓ)xℓ.(1)Here,theoriginisassumedtobeasaddlepointandweusethenotation:ℓ=(ℓ1,ℓ3,ℓ2,ℓ4),x=(x1,x3,x2,x4),ℓ=ℓ1+ℓ3+ℓ2+ℓ4andxℓ=xℓ11xℓ33xℓ22xℓ44.Thepositionsarex1andx2andtheconjugatemomentaare,respectively,x3andx4.Theeigenvaluesofthesystem’slinearpartareλ1=iωandλ2=−λ,withωandλreal(theothertwobeingofcourse−iωandλ).H(ℓ)isthecoefficientofxℓandℓisitsorder.Theusualnoncomp