Chaotic-Dynamics-TAU混沌動力學(xué)的頭-課件

Chaosvs.RandomnessDonotconfusechaoticwithrandom:Random:irreproducibleandunpredictableChaotic:deterministic-sameinitialconditionsleadtosamefinalstate…butthefinalstateisverydifferentforsmallchangestoinitialconditions

difficultorimpossibletomakelong-termpredictionsClockwork(Newton)vs.Chaotic(Poincaré)UniverseSupposetheUniverseismadeofparticlesofmatterinteractingaccordingtoNewtonlaws

→thisisjustadynamicalsystemgovernedbya(verylargethough)setofdifferentialequationsGiventhestartingpositionsandvelocitiesofallparticles,thereisauniqueoutcome→P.Laplace’s

ClockworkUniverse(XVIIICentury)!CanChaosbeExploited?BriefChaoticHistory:PoincaréBriefChaoticHistory:LorenzChaosintheBraveNewWorldofComputersPoincarécreatedanoriginalmethodtounderstandsuchsystems,anddiscoveredaverycomplicateddynamics,but:

"ItissocomplicatedthatIcannotevendrawthefigure."

AnExample…APendulumstartingat1,1.001,and1.000001rad:ChangingtheDrivingForcef=1,1.07,1.15,1.35,1.45ChaosinPhysicsChaosisseeninmanyphysicalsystems:Fluiddynamics(weatherpatterns),somechemicalreactions,Lasers,Particleaccelerators,…Conditionsnecessaryforchaos:systemhas3independentdynamicalvariablestheequationsofmotionarenon-linearWhyNonlinearityand3DPhaseSpace?DynamicalSystemsAdynamicalsystemisdefinedasadeterministicmathematicalprescriptionforevolvingthestateofasystemforwardintimeExample:AsystemofNfirst-order,autonomousODEDampedDrivenPendulum:PartIThissystemdemonstratesfeaturesofchaoticmotion:Convertequationtoadimensionlessform:)cos(

sin

22fwqqq+=++tAmgdtdcdtdmlD)cos(

sin0tfdtdqdtdDwqqw=++DampedDrivenPendulum:PartII3dynamicvariables:

,,tthenon-linearterm:sinthissystemischaoticonlyforcertainvaluesofq,f0,andwDIntheseexamples:

wD=2/3,q=1/2,andf0near1DampedDrivenPendulum:PartIIItowatchtheonsetofchaos(asf0isincreased)welookatthemotionofthesysteminphasespace,oncetransientsdieawayPaycloseattentiontotheperioddoublingthatprecedestheonsetofchaos...07.10=f15.10=ff0=1.35f0=1.48f0=1.45f0=1.49f0=1.47f0=1.50ForgetAboutSolvingEquations!NewLanguageforChaos:Attractors(DissipativeChaos)KAMtorus(HamiltonianChaos)PoincaresectionsLyapunovexponentsandKolmogoroventropyFourierspectrumandautocorrelationfunctionsPoincaréSectionPoincaréSection:ExamplesPoincaréSection:PendulumThePoincarésectionisasliceofthe3Dphasespaceatafixedvalueof:Dtmod2Thisisanalogoustoviewingthephasespacedevelopmentwithastrobelightinphasewiththedrivingforce.Periodicmotionresultsinasinglepoint,perioddoublingresultsintwopoints...PoincaréMovieTovisualizethe3Dsurfacethatthechaoticpendulumfollows,amoviecanbemadeinwhicheachframeconsistsofaPoincarésectionatadifferentphase...PoincareMap:Continuoustimeevolutionisreplacebyadiscretemapf0=1.07f0=1.48f0=1.50f0=1.15q=0.25AttractorsThesurfacesinphasespacealongwhichthependulumfollows(aftertransientmotiondecays)arecalledattractorsExamples:foradampedundrivenpendulum,attractorisjustapointat=0.(0Din2Dphasespace)foranundampedpendulum,attractorisacurve(1Dattractor)StrangeAttractorsChaoticattractorsofdissipativesystems(strangeattractors)arefractals

OurPendulum:2<dim<3Thefinestructureisquitecomplexandsimilartothegrossstructure:self-similarity.non-integerdimensionWhatisDimension?Capacitydimensionofalineandsquare:1L2L/24L/48L/82nL/2nN1L4L/216L/422nL/2nN)/1log(

/)(log

0lim)/1()(eeeeeNdLNcdd?==TrivialExample:Point,Line,Surface,…Non-TrivialExample:CantorSetTheCantorsetisproducedasfollows:N1121/341/981/2713log

/2log3log

/2log

lim<==cnncdd￥?nLyapunovExponents:PartIThefractionaldimensionofachaoticattractorisaresultoftheextremesensitivitytoinitialconditions.Lyapunovexponentsareameasureoftheaveragerateofdivergenceofneighbouringtrajectoriesonanattractor.LyapunovExponents:PartIIConsiderasmallsphereinphasespace…afterashorttimethespherewillevolveintoanellipsoid:e2te1tLyapunovExponents:PartIIITheaveragerateofexpansionalongtheprincipleaxesaretheLyapunovexponentsChaosimpliesthatatleastoneis>0Forthependulum:i=-

q(dampcoeff.)nocontractionorexpansionalongtdirectionsothatexponentiszerocanbeshownthatthedimensionoftheattractoris:d=2-1/2DissipativevsHamiltonianChaosAttractor:

Anattractorisasetofstates(pointsinthephasespace),invariantunderthedynamics,towardswhichneighboringstatesinagivenbasinofattractionasymptoticallyapproachinthecourseofdynamicevolution.Anattractorisdefinedasthesmallestunitwhichcannotbeitselfdecomposedintotwoormoreattractorswithdistinctbasinsofattraction.Thisrestrictionisnecessarysinceadynamicalsystemmayhavemultipleattractors,eachwithitsownbasinofattraction.

Conservativesystemsdonothaveattractors,sincethemotionisperiodic.Fordissipativedynamicalsystems,however,volumesshrinkexponentiallysoattractorshave0volumeinn-dimensionalphasespace.StrangeAttractors:

Boundedregionsofphasespace(correspondingtopositiveLyapunovcharacteristicexponents)havingzeromeasureintheembeddingphasespaceandafractaldimension.TrajectorieswithinastrangeattractorappeartoskiparoundrandomlyDissipativevsConservativeChaos:LyapunovExponentPropertiesForHamiltoniansystems,theLyapunovexponentsexistinadditiveinversepairs,whileoneofthemisalways0.Indissipativesystemsinanarbitraryn-dimensionalphasespace,theremustalwaysbeoneLyapunovexponentequalto0,sinceaperturbationalongthepathresultsinnodivergence.LogisticMap:PartIThelogisticmapdescribesasimplersystemthatexhibitssimilarchaoticbehaviorCanbeusedtomodelpopulationgrowth:Forsomevaluesof,xtendstoafixedpoint,forothervalues,xoscillatesbetweentwopoints(perioddoubling)andforothervalues,xbecomeschaotic….)1(

11---=nnnxxxmLogisticMap:PartIITodemonstrate…)1(

11---=nnnxxxmxn-1xnBifurcationDiagrams:PartIBifurcation:achangeinthenumberofsolutionstoadifferentialequationwhenaparameterisvariedToobservebifurcatons,plotlongtermvaluesof,atafixedvalueofDtmod2asafunctionoftheforcetermf0BifurcationDiagrams:PartIIIfperiodicsinglevaluePeriodicwithtwosolutions(leftorrightmoving)2valuesPerioddoublingdoublethenumberTheonsetofchaosisoftenseenasaresultofsuccessiveperioddoublings...BifurcationoftheLogisticMapBifurcationofPendulumFeigenbaumNumberTheratioofspacingsbetweenconsecutivevaluesofatthebifurcationsapproachesauniversalconstant,theFeigenbaumnumber.Thisisuniversaltoalldifferentialequations(withincertainlimits)andappliestothependulum.Byusingthefirstfewbifurcationpoints,onecanpredicttheonsetofchaos.￥?==--+-kkkkk...669201.4lim11dmmmmChaosinPHYS306/638Aperiodicmotionconfinedtostrangeattractorsinth