Geometric Interpretation of Chaos in Two-Dimension

arXiv:astro-ph/9707114v19Jul1997toappearinPhysicalReviewEGeometricInterpretationofChaosinTwo-DimensionalHamiltonianSystemsHenryE.Kandrup1DepartmentofAstronomyandDepartmentofPhysicsandInstituteforFundamentalTheory,UniversityofFlorida,Gainesville,FL32611ThispaperexploitsthefactthatHamiltonianflowsassociatedwithatime-independentHcanbeviewedasgeodesicflowsinacurvedmanifold,sothattheproblemofstabilityandtheonsetofchaoshingeonpropertiesofthecurvatureKabenteringintotheJacobiequation.Attentionfocusesonensemblesoforbitsegmentsevolvedinrepresentativetwo-dimensionalpotentials,examininghowsuchpropertiesasorbittype,valuesofshorttimeLyapunovexponentsχ,complexitiesofFourierspectra,andlocationsofinitialconditionsonasurfaceofsectioncorrelatewiththemeanvalueanddispersion,h˜Kiandσ˜K,ofthe(suitablyrescaled)traceofKab.Mostanalysesofchaosinthiscontexthaveexploredtheeffectsofnegativecurvature,whichimpliesadivergenceofnearbytrajectories.TheaimhereistoexploitinsteadapointstressedrecentlybyPettini(Phys.Rev.E47,828[1993]),namelythatgeodesicscanbechaoticevenifKiseverywherepositive,chaosinthiscasearisingasaparametericinstabilitytriggeredbyregularvariationsinKalongtheorbit.Forensemblesoffixedenergy,containingbothregularandchaoticsegments,simplepatternsexistconnectingh˜Kiandσ˜Kfordifferentsegmentsbothwitheachotherandwiththeshorttimeχ:Often,butnotalways,thereisanearlyone-to-onecorrelationbetweenh˜Kiandσ˜K,aplotofthesetwoquantitiesapproximatingasimplecurve.Overallχvariessmoothlyalongthecurve,certainregionscorrespondingtoregularand“confined”chaoticorbitswhereχisespeciallysmall.Chaoticsegmentslocatedfurthestfromtheregularregionstendsystematicallytohavethelargestχ’s.Thevaluesofh˜Kiandσ˜K(andinsomecasesχ)forregularorbitsalsovarysmoothlyasafunctionofthe“distance”fromthechaoticphasespaceregions,asprobed,e.g.,bythelocationoftheinitialconditiononasurfaceofsection.Manyoftheseobservedpropertiescanbeunderstoodqualitativelyintermsofaone-dimensionalMathieuequation,inwhichparametricinstabilityisintroducedinthesimplestpossibleway.PACSnumber(s):05.45.+b,03.20.+i,02.40.+m1Electronicaddress:kandrup@astro.ufl.edu1I.INTRODUCTIONANDMOTIVATIONItiswellknown[1]thattheflowassociatedwithatime-independentHamiltonianH=12δabpapb+V(xa)canbereformulatedasageodesicflowinacurved,butconformallyflat,manifold.Specifically,letds2=W(xa)δabdxadxa,(1)whereEistheconservedenergyassociatedwiththetime-independentHandtheconformalfactorW(xa)=E−V(xa)(2)isequalnumericallytothekineticenergyassociatedwithatrajectoryatthepointxa.Itthenfollowsthat,withthefurtheridentificationds=√2Wdt,(3)thegeodesicequationformotioninthemetricgab=WδabiscompletelyequivalenttotheHamiltonequationsdxadt=∂H∂paanddpadt=−∂H∂xa.(4)Thisimpliesthattheconfluenceordiverenceofnearbytrajectoriesxa(s)and[x+ξ]a(s)isdeterminedbytheJacobiequation,i.e.,theequationofgeodesicdeviation,whichtakestheformD2ξaDs2=−Rabcdubudξc≡−Kacξc,(5)whereRabcdistheRiemanntensorassociatedwithgabandD/Ds=ua∇adenotesadirec-tionalderivativealongua=dxa/ds.Linearstabilityorlackthereofforthetrajectoryxa(s)isthusrelatedtoRabcdor,moreprecisely,tothecurvatureKac.If,e.g.,Rabcdiseverywherenegative,sothatKacalwayshasoneormorenegativeeigenvalues,thetrajectorymustbelinearlyunstable.Itwouldseemintuitivethat,ifthecurvatureofgabiseverywherenegative,sothatnearbytrajectoriesalwaystendtodiverge,everygeodesicwillbehaveinafashionthatismanifestlychaotic.Ifoneassumesthatthemanifoldiscompact,sothattrajectoriesarerestrictedtoaregionoffinitevolume,thisintuitioncanbeelevatedtoatheorem.Forexample,geodesicflowsonacompactmanifoldwithconstantnegativecurvaturearenecessarilyK-flows,wheregenericensemblesofinitialconditionsevolvetowardsamicrocanonicaldistributionataratesetbythemagnitudeofthecurvature[2].Ifthecurvatureiseverywherenegativebutnotconstant,theflowismorecomplex,butonecanstillinfer[3]achaoticevolutiontowardsamicrocanonicaldistributionatarateboundedfrombelowbytheleastnegativevalueofthecurvature.Whenthecurvatureisnoteverywherenegative,muchlessisknown.Nevertheless,thepreceedinghasmotivatedtheexpectationthat,inmanydynamicalsystems,chaosshouldbeassociatedwith(regionsof)negativecurvature.Inparticular,severalauthors(cf.[4,5])havesoughttousenegativecurvaturetoexplainthefactthatthegravitationalN-bodyproblemforalargenumberofobjectsofcomparablemassischaoticinthesensethattheevolutionmanifestsanexponentiallysensitivedependenceoninitialconditions.However,asstressedrecentlybyPettini[6],notallchaoscanbeassociatedwithnegativecurvature.Inparticular,onecanhavelargemeasuresofchaoticorbitsevenforsystemsandenergieswhereKabiseverywherepositive.Inretrospect,thisiseasytounderstand.Viewing2theJacobiequationasamatrixequation,onecansolveatanygivenpointinspacetoderiveeigenvectors{Xi}andeigenvalues{λi},eachpairsolvingalinearequationoftheformD2XiDs2=−λiXi.(6)Whenthecurvatureiseverywherepositive,λi≥0,sothatthesolutionsareoscillatoryratherthanexponential.Iftheλi’swereconstantalongthetrajectory,onecouldthusinferstableoscillations.Ingeneral,however,theλi’sarenotconstant,dependinginsteadontheunperturbedxi(s)sinceRabcdanduabot