关于小数据量法计算最大Lyapunov指数的讨论

Lyapunov(,430033)LyapunovLyapunovLyapunovDiscussionontheMethodforCalculatingLargestLyapunovExponentsfromSmallDataSetsLuZhen-boCaiZhi-mingJiangKe-yu(CollegeofElectronicEngineering,NavyEngineeringUniversity,WuHan430033,China)Abstract:TheproblemsonthemethodforcalculatinglargestLyapunovexponentsfromsmalldatasetsarediscussed.Thealgorithmisrobustthechangesinthefollowingquantities:embeddingdimension,reconstructiondelayandmeanperiodofthetimeseries.Weindicatethatthealgorithmisstillaccuratewithoutconsideringtemporalseparation,ifthemeanperiodisunknown.AnewcriterionofselectinglinearzoneforfittingthelargestLyapunovexponentsispresented.Intheend,theeffectsofadditivenoiseisdiscussed.KeyWords:chaos,Lyapunovexponents,timeseriesanalysis1LyapunovLyapunov1985Wolf[1]LyapunovLyapunov1993Rosenstein[2]WolfLyapunov*(:514450801JB1101)(:51444030105JB1101)(1978-)E-mail:[e-mail];:13476281645(Mobile),027-83444036(Office)1(1)(2)(3)Lyapunov1λ(4)2[2]},,,{21NxxxL},,2,1,],,,,[|{)1(MixxxXXXTmiiiii⋅⋅⋅=⋅⋅⋅==−++ττ(1)τmτ)1(−−=mNMτLiebertSchustere11−[3]Takens[4]G-P[5]mddm2mjXjXˆjjXjXXdjˆˆmin)0(−=(2))0(jdjjXjXˆ⋅jXjXˆpjj−ˆ(3)pLyapunov1λjXjXˆi)(idjijijjXXid++−=ˆ)((4))ˆ,min(,,2,1jMjMi−−=LjXjXˆ1λ)(1)(tijjeCidΛ⋅=λ)0(jjdC=(5)(5))ln(⋅)(ln)(ln1tiCidjj∆⋅+=λ(6)(6)iidj~)(lnt∆1λij)(lnidjt∆)(iy∑=∆=qjjidtqiy1)(ln1)((7)qLyapunov)(idjiiy~)(1λ23(1)~(7)Lyapunov1λ(1)τ(Auto-Correlation,AC)[3](MutualInformation,MI)[6]G-P[5](FalseNearestNeighbor,FNN)[7]Rosenstein[2]mτG-PLyapunovm1λLogisticLorenzxLogistic))(1)((4)1(nxnxnx−=+(8)LorenzRunge-Kutta0.01⎪⎩⎪⎨⎧−=−+−=−−=bzxyzyrxxzyyxx&&&)(σ(9)4,92.45,16===brσLogistic500Lorenzx50001LogisticLorenzxLyapunov1LyapunovSystemNτmpCalculated1λExpected1λ[ref.]%errorLogistic50012(G-P)500.6980.693[8]+0.721(FNN)0.702+1.30Lorenz500012(AC)5(G-P)501.4671.500[1]-2.2011(MI)3(FNN)1.471-1.93Logistic1G-P21Lyapunov1λ0.6980.7020.693Lorenz%3.1xG-P125113Lyapunov1λ1.4671.4711.500%2.2p50(2)(2)(3)[2]jXjXˆ3LogisticLorenzxLyapunov22LyapunovSystemNτmpCalculated1λExpected1λ[ref.]error%Logistic5001210.7030.693[8]+1.44500.698+0.72Lorenz500012511.4541.500[1]-3.07501.467-2.20p150LogisticLorenzxLyapunov1λ%1.31=p(3)Lyapunoviiy~)(1λ)(iyf=f′)1()()()(lim0−−≈∆∆−−==′→∆iyiyiiiyiydidffi(10)iiy~)(f′iiy~)(iiyiy~)1()(−−i12LogisticLorenzx1Logistic2Lorenzx41(b)6~2=iiiyiy~)1()(−−Logistic2(b)iiy~)(6~2=i50~1=iiiyiy~)1()(−−300~200=i)1()(−−iyiyi0Lorenz200~50=iiiyiy~)1()(−−xii0200~50=ijXjXˆ)(idjiiy~)(iiyiy~)1()(−−iiy(~)1)1(+−τm(4)(4)[2]LogisticLorenzx(SignaltoNoiseRatio,SNR)=SNR(11)33Lyapunov[2]SystemNτmSNRCalculated1λExpected1λ[ref.]error%Logistic5001210.7040.693[8]+1.6100.779+12.4Lorenz500011310.6451.500[1]-57.0101.184-21.1Lyapunov1Lyapunov3Logistic1=SNR10=SNRLogistic1=SNRLyapunov},,,{21NnnnLτmjNjNˆ∑−=++−=−=102ˆˆ2ˆ2)(minmin)0(ˆmkkjkjjjjNjnnNNdjττ(12)i)(idj∑−=++++++−=−=102ˆ2ˆ2)()(mkikjikjijijjnnNNidττ(13))0(2jd)(2τ⋅ldj1,,1,0−=mlL)0(2jd)(2τ⋅ldjlm−52222,minmin)(minbabacbaba+=+=(14))(idjτ⋅=li1,,1,0−=mlL1≠τ))1(()()0(ττ−mdddjjjL(7)iiy~)(iidj~)(}{inNi,,2,1L=τm4=m345=τ1=τ3,4,5==mτ4,4,1==mτ3(a))(iy15,10,5,0=i)15()10()5()0(yyyy4(a)iiy~)(1=τ)3(),2(),1(),0(yyyy)3()2()1()0(yyyy4(a)iiy~)(4~1=i)3()2()1()0(yyyy4~1=im3Logistic5Logistic65Lyapunov1=SNR1=SNR2~1=i1λ0.706[2]0.0026Lyapunov2~1=i1λ0.704[2]Logistic56Logistic1=SNR1=SNRLyapunov6[2]Logisticiiy~)(1=SNR5Logistic,2,1==mτ,1=SNR6,2,1==mτLyapunov1λiiy~)(1)1(+−τmLyapunovLyapunov3(a)4(a)6(a)1)1(+−=τmiiiy~)(4(1)G-PLyapunov1λ(2)(3)iiy~)(iiyiy~)1()(−−i1)1(+−τm(4)LyapunovLyapunov[1]A.WolfJ.B.Swift,H.L.SwinneyandJ.A.Vastano.DeterminingLyapunovexponentsfromatimeseries,Physica16D,71985,285~317.[2]M.T.Rosenstein,J.J.CollinsandC.J.Deluca.ApracticalmethodforcalculatinglargestLyapunovexponentsfromsmalldatasets,PhysicaD,1993,65:117~134.[3]W.Liebert,H.G.Schuster,Properchoiceofthetimedelayfortheanalysisofchaotictimeseries.Phys.Lett.A142(1989)107.[4]F.Takens.Determingstrangeattractorsinturbulence[J].LecturenotesinMath.1981,898:361-381.[5]P.Grassberger,I.Procaccia.Measuringthestrangenessofstrangeattractors[J].PhysicaD,1983,9:189-208[6]A.M.Fraser,H.L.Swinney.Independentcoordinatesforstrangeattractorsformtimeseries[J].Phys.Rev.A.1986,33:1134-1140.[7]M.B.Kennel,R.Brown,H.D.I.Abarbanel.Determiningembeddingdimensionforphase-spacereconstructionusingageometricalconstruction[J].Phys.Rev.A1992,45:3403.[8]J.-P.Eckmann,D.Ruelle,Ergodictheoryofchaosandstrangeattractors[J],Rev.Mod.Phys.57(1985)617(1978-)43003313476281645(Mobile),027-83444036(Office)[e-mail]