Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems

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1、PhysicaD56(1992)185-187North-HollandFundamentallimitationsforestimatingdimensionsandLyapunovexponentsindynamicalsystemsJ.-P.Eckmann1andD.RuelleLH.E.S.,91440Bures-sur-Yvette,FranceReceived2October1991Revisedmanuscriptreceived3December1991Accepted14January1992CommunicatedbyP.E.RappWeshowth

2、atvaluesofthecorrelationdimensionestimatedoveradecadefromtheGrassberger-Procacciaalgorithmcannotexceedthevalue2log~0NifNisthenumberofpointsinthetimeseries.Whenthisboundissaturateditisthusnotlegitimatetoconcludethatlowdimensionaldynamicsispresent.TheestimationofLyapunovexponentsisalsodisc

3、ussed.Thepurposeofthisnoteistoquestionthedata-wrong(toolow)dimensionswillbeob-validityofanumberofrecentlypublishedesti-tained.AsimilaranalysiswillapplytoestimatesmatesofdimensionsofattractorswhicharebasedofLyapunovexponents.onrathershorttimeseries.ThevaluesobtainedLet(ui)bea(scalar)times

4、erieswithi=arelike6or7,andweshallarguethattheyare1,...,N(thechoiceofsamplingtimeunitwillbeprobablyareflectionofthesmallnumberofdatadiscussedbelow).Usinganembeddingdimensionpointsratherthanofthedimensionofahypo-m,wefirstreconstructatrajectoryinW",withtheticalattractor.Ourconclusionsgoin~h

5、esameXn=(Un,Un+1.....Un+,,--1)"(Thismethod,advo-directionasthoseofGrassberger[1]discussingcatedbyoneofus(DR),wasfirstdocumentedworkofNicolisandNicolis[2],andProcaccia[3]inref.[6].)Then,accordingtotheGrassberger-discussingworkofTsonisandEisner[4].OurProcacciaalgorithm(GP)[7],wecountthe.nu

6、m-analysisishowevermoreprecise,andsomewhatberX(r)ofpairsofpointswithmutualdistancemoreoptimisticthanthatofProcaccia(webe-

7、glogA/'(r)versuslogr,ForsmallboundsofSmith[5].r,theslopeofthisplotisanestimateoftheWhileitisobviousthatashorttimeseriesofcorrelationdimensiond[7].(Forlargerr,thelowprecisionmustleadtospuriousresults,weplotisnotexpectedtobelinear.)Thus,thewishtoarguethat-evenwithgoodprecis