Alan WOLF etc.Determining Lyapunov Exponents from a Time Series.pdf

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1、Physica16D(1985)285-317North-Holland,AmsterdamDETERMININGLYAPUNOVEXPONENTSFROMATIMESERIESAlanWOLF~-,JackB.SWIFT,HarryL.SWINNEYandJohnA.VASTANODepartmentofPhysics,UniversityofTexas,Austin,Texas78712,USAReceived18October1984Wepresentthefirstalgorithmsthatallow

2、theestimationofnon-negativeLyapunovexponentsfromanexperimentaltimeseries.Lyapunovexponents,whichprovideaqualitativeandquantitativecharacterizationofdynamicalbehavior,arerelatedtotheexponentiallyfastdivergenceorconvergenceofnearbyorbitsinphasespace.Asystemwit

3、honeormorepositiveLyapunovexponentsisdefinedtobechaotic.Ourmethodisrootedconceptuallyinapreviouslydevelopedtechniquethatcouldonlybeappliedtoanalyticallydefinedmodelsystems:wemonitorthelong-termgrowthrateofsmallvolumeelementsinanattractor.Themethodistestedonm

4、odelsystemswithknownLyapunovspectra,andappliedtodatafortheBelousov-ZhabotinskiireactionandCouette-Taylorflow.Contentsconvergenceofnearbyorbitsinphasespace.Since1.Introductionnearbyorbitscorrespondtonearlyidenticalstates,2.TheLyapunovspectrumdefined3.Calculat

5、ionofLyapunovspectrafromdifferentialequationsexponentialorbitaldivergencemeansthatsystems4.Anapproachtospectralestimationforexperimentaldatawhoseinitialdifferenceswemaynotbeableto5.Spectralalgorithmimplementation*resolvewillsoonbehavequitedifferently-predic-

6、6.Implementationdetails*7.Datarequirementsandnoise*tiveabilityisrapidlylost.Anysystemcontaining8.ResultsatleastonepositiveLyapunovexponentisdefined9.Conclusionstobechaotic,withthemagnitudeoftheexponentAppendices*reflectingthetimescaleonwhichsystemdynamicsA.L

7、yapunovspectrumprogramforsystemsofdifferentialbecomeunpredictable[10].equationsB.Fixedevolutiontimeprogramfor~'1Forsystemswhoseequationsofmotionareex-plicitlyknownthereisastraightforwardtechnique[8,9]forcomputingacompleteLyapunovspec-1.Introductiontrum.Thism

8、ethodcannotbeapplieddirectlytoexperimentaldataforreasonsthatwillbedis-Convincingevidencefordeterministicchaoshascussedlater.Wewilldescribeatechniquewhichcomefromavarietyofrecentexperiments[1-6]f