Study of chaos in hamiltonian systems via convergent normal

We use Moser's normal forms to study chaotic motion in two-degree hamiltonian systems near a saddle point. Besides being convergent, they provide a suitable description of the cylindrical topology of the chaotic flow in that vicinity. Both aspects combined

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aStudyofchaosinhamiltoniansystemsviaconvergentnormalformsWernerM.Vieira ,AlfredoM.O.deAlmeida (February5,2008)AbstractWeuseMoser’snormalformstostudychaoticmotionintwo-degreehamil-toniansystemsnearasaddlepoint.Besidesbeingconvergent,theyprovideasuitabledescriptionofthecylindricaltopologyofthechaotic owinthatvicin-ity.Bothaspectscombinedallowedaprecisecomputationofthehomoclinicinteractionofstableandunstablemanifoldsinthefullphasespace,ratherthanjustthePoincar´esection.TheformalismwasappliedtotheH´enon-Heileshamiltonian,producingstrongevidencethattheregionofconvergenceofthesenormalformsextendsoverthatoriginallyestablishedbyMoser.PACSindex:05.45.+b

We use Moser's normal forms to study chaotic motion in two-degree hamiltonian systems near a saddle point. Besides being convergent, they provide a suitable description of the cylindrical topology of the chaotic flow in that vicinity. Both aspects combined

I.INTRODUCTION

NormalForms(NF)areamongthesuccessfulmethodsforeitheranalyticornumericstudiesofdynamicalsystems;byperformingasuitablecoordinatetransformation,weeven-tuallyobtainamoresimpleordynamically“transparent”versionoftheoriginalsystem.Thisapproachcanbeformulatedeitherforgenericsystemsincludingtwo-dimensionalmaps[1]orforhamiltoniansystems[2]andconservativemaps[3].So,evenwhenagivenNFdoesnotconverge,duetosmalldenominatorsorexactressonances,itisofimportancefornumericalpurposes.ThisisjustwhatoccurswhentheNFisobtainedaroundastablepointororbit.Then,inspiteofthewellknownpergenceinthiscase,atruncationallowsustofollowthemotionforalongtimewithgreatprecision.Infact,thenonconvergentcaseisthemostconsideredintheliterature[4–6].

ThepresentworkconcernstheNFaroundaunstablepointororbitofaconservativesystem.Forthat,Moserdemonstratedtheirconvergenceforbothmaps[7]andhamiltoniansystems[8].Althoughconvergent,thiscasedidnotreceivemuchattentionuntilrecently,presumablybecausetheparticleremainsaveryshorttimeinthatregion.Nevertheless,wewillseethattheMosernormalforms(MNF)arebothconvergentandapowerfulltoolforthesearchforthebasicstructuresofthechaoticmotionratherthanjustfollowingaspeci corbitforalongtime.

TheusefulnessoftheMNFforthestudyofconservativechaoticmapsisalreadyknownintheliterature.Itallowedpreciseanalyticalcomputationsofhomoclinicpointsandtheperiodicpointswithlongperiod,whichaccumulateinthehomoclinicones[9,10].Additionalgoodnumericalresultswereobtainedevenifsmalldissipativeperturbationswereadded[11].

Area-preservingmapsareusuallyonlysimpli edreductions(appropriatePoincar´esec-tions)ofautonomoushamiltoniansystemsoftwodegreesoffreedom.Inparticular,theverycomplextwo-dimensionalhomoclinictangle[12]isalreadyareductionofthemuchmoreinvolvedchaoticmotioninthefullphasespace.However,extendingtheuseoftheMNFtothehamiltoniancaseallowsustostudydirectlytheproperfour-dimensionalchaotic ow.

We use Moser's normal forms to study chaotic motion in two-degree hamiltonian systems near a saddle point. Besides being convergent, they provide a suitable description of the cylindrical topology of the chaotic flow in that vicinity. Both aspects combined

Thishaspreviouslybeenattemptedintheliterature[13],butwithouttakingfulladvantageofthemethodthatweshalldevelophere.AnotheruseofMNFsisfoundin[14]andtakesadvantageofitsconvergencetoascertainstabilitytransitionsoffamiliesofperiodicorbitsnearhamiltonians’saddlepoints.

InsectionIIwewilldeveloptheMNFapproachforthecaseofagenericautonomoushamiltonianoftwodegreesoffreedomaroundasaddlepoint,encompassedbyMoser’sconvergenceproof.Inthatvicinity,the ow’stopologyiscylindricalratherthantoroidal,inthecaseofachaoticregime[15].Firstly,weconstructtherelationslinkingtheoriginalsystemtothecorrespondingmore“transparent”normalizedsystem.Thisisdonethroughanearidentitypolynomialcoordinatetransformation.Wewillseethat,besidesconvergent,thattransformationalsorevealsinanaturalway,thecylindricalcharacterofthetopology.BothfeaturesturntheMNFintoapowerfulltool.So,itwaspossibletocompute,preciselyforthe rsttime,thecontinuousstructuresinthefullphasespace,underlyingthehomoclinictangleinaPoincar´esection:the(un)stablemanifoldswhichoriginateatthesaddlepointandateachneighbouringunstableperiodicorbit,thehomoclinicorbitsassociatedwiththelatterandtheperiodicorbitswithlongperiodwhichaccumulateonthehomoclinicorbits.InsectionIIIweobtaintherecurrencerelationsforthecoe cientsinvolvedinthetheory.InsectionsIVandVweapplytheformalismtothespeci ccaseoftheH´enon-Heileshamiltonian.ThenumericalresultsexhibitedinsectionVfullycon rmedtheexpectationsabouttheMNFasatoolforthestudyandcharacterizationofchaoticmotions.Moreover,theyalsopointtosomekindofextensionoftheregionofconvergenceinitiallyassumedforMoser’stheorem.Infact,thisissueisjustbeingconsideredbytheauthorspresently.

Finally,insectionVIwesummarizetheresultsandpossibleextensionsofthepresentwork.

We use Moser's normal forms to study chaotic motion in two-degree hamiltonian systems near a saddle point. Besides being convergent, they provide a suitable description of the cylindrical topology of the chaotic flow in that vicinity. Both aspects combined

II.MOSER’SNORMALFORM

Itisessentialthatthehamiltonianbeinthecomplexi edform,forthemethod’simple-mentation:

h(x1,x3,x2,x4)=λ1x1x3+λ2x2x4+∞ =3H(

=( 1, 3, 2, 4),

x

)isthecoe cientofxand isitsorder.

Theusualnoncomplexi edformofthequadraticparth2of(1),aroundthesaddlepoint,is

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