关于Lyapunov指数计算方法的比较-张海龙

Lyapunov210042Lyapunov、、wolfLorenzLorenzLyapunovLyapunov、Lyapunov．、．LyapunovLorenzO415A1672-1292201201-0005-05TheComparisonforLyapunovExponentsCalculationMethodsZhangHailongMinFuhongWangEnrongSchoolofElectricalandAutomationEngineeringNanjingNormalUniversityNanjing210042ChinaAbstractInthispapertheseveralcomputationalmethodsofLyapunovexponentsarecomparedi．e．thedefinitionmethodtheorthogonalmethodthewolfmethodandthesmalldatasets．TheLyapunovexponentpowerandthemax-LyapunovexponentarecomputedthroughtheabovemethodsforLorenzsystem．Fromtheresultstheaccuraciesandthecomplexityoftheabovemethodsareinvestigated．Furthermorethemax-Lyapunovexponentsarealsocalculatedforthechaotictimeseriesincludingthenoise．Finallynumericalresultsdemonstratethattheperformancesofdifferentcompu-tationalmethodshavedifferencesandsomesummarieswillbepresented．KeywordsLyapunovexponentsLorenzsystemchaossystem2011-09-01．51075275、[e-mail]、．LyapunovLE12．、LELEYanWen3logisticlyapunovLLE、Liao4wolfLLE、Wang5wolflogisticLLE、Xie6．LE．Lyapunov、wolf、LorenzLyapunovLyapunov、．Lyapunov．1LE1.1LE—5—12120123JOURNALOFNANJINGNORMALUNIVERSITYENGINEERINGANDTECHNOLOGYEDITIONVol．12No．1Mar2012．λεnε·enλx0=|Fnx0+ε－Fnx0|1ε→0n→∞λ=lim1n∑n－1i=0lndFxdxx=xi2λLE．1.2wolfx1x2…xnmτYti=xtixti+τ…xti+m－1τi=12…N．Yt0Y0t0L0t1ε＞0L'0=|Yt1－Y0t1|＞εYt1Yt1Y1t1L1=|Yt1－Y1t1|＜εYtNMLLEλ=1tM－t0∑Mi=0lnL'iLi．31.3GSRGram-SchmidtLyapunovt=t0x0mv11v21…vm1kAkt0k≤mt=t1x1mv'11v'21…v'm1x1=Fx0v'l1=Jx0vl1l=12…mA'kt0GSRv'11v'21…v'm1v12v22…vm2kA'kt1t=t2x2mv'12v'22…v'm2x2=Fx1v'l2=Jx1vl2l=12…m．kA'kt2λ1+λ2+…+λk=lim1tn－t0∑n－1i=0lnA'ktiAktik=12…m．41.41kxtii=12…Nmτ2mτYj|j=12…MYjY'jdj0=min‖Yj－Y'j‖|j－j'|＞P5P．3Yjidjidji=min‖Yj+i－Y'j+i‖i=12…minM－jM－j'64ijlndjiyiyi=1qΔt∑qj=1lndji7qdjiLLE．2LorenzLyapunovLorenz7x=－ax－yy=－xz+rx－y{z=xy－bz．8—6—1212012a=10b=8/3x0=1y0=1z0=0．Matlab0.0110001Lorenz．r1500Lorenzrr=24r=310．a=10b=8/3r1500LorenzLEr2．44LLErr=24λmax＞0r=220λmax≈01LE4rLElyapunov、．a=10b=8/3r=283LELLEλ1λ2λ3λmaxmτC－C8m=5τ=12、wolfLLE3Lyapunov1．LLE0．0.93s12.4swolf2min．—7—Lyapunov1a=10b=8/3r=28LyapunovTable1ThecomparisonofLyapunovExponentswitha=10b=8/3r=28λ1λ2λ3λmax\\\1.3671Wolf\\\0.02290.8564－0.0011－14.51850.8564\\\0.024LE59mτmτ．LEmτ．C－CmτmτLE2mτwolfmτC－Cmτ．2LyapunovTable2ThecomparisonofLyapunovexponentsfromembeddingdimensionsanddelaytimes、Wolfm=5τ=121.36710.02290.85640.024m=5τ=141.36710.01980.85640.0087m=8τ=121.36710.02410.85640.04500.002200.018210．Lorenzxii=12…Nm=5τ=124LE3wolf．3LyapunovTable3ThecomparisonofLyapunovexponentsinnoisytimeseriesyapunovlyapunov/%1.36711.42344.1Wolf0.02290.0223－2.60.85640.8514－0.510.0240.02410.423LorenzLLELE、3、wolf、14lyapunov、．2、KLyapunovLyapunov．Lyapunov、、．3、LLEmτC－CC－Cmτwolf．lyapunov、—8—1212012Lya-punov．References1．M．2005．LüJinhuLuJunanChenShihua．TheAnalysisandApplicationforChaoticTimeSeriesM．WuhanWuhanUniversityPress2005．inChinese2．J．20052212285-288．LuoLijunLiYinshanLiTongetal．ResearchandsimulationofLyapunov’sexponentsJ．ComputerSimulation20052212285-288．inChinese3．LyapunovJ．20105524354．YanWenLuZongsheng．ThenumericalcalculationmethodofLyapunovindexJ．China’sForeignTrade20105524354-356．inChinese4．LyapunovJ．20088439-41．LiaoDeweiZhuWeiqiang．ResearchonlyapunovexponentsalgorithmanditsapplicationJ．JournalofWenzhouVocationalandThechnicalCollege20088439-41．inChinese5．J．200535689-94．WangYanXuWeiQuJisheng．Thealgorithmandchickofphase-spacereconstructionbasedonthetimeseriesJ．JournalofShangdongUniversityEngineeringScienceEdition200535689-94．inChinese6XieZhongyuWangKejunZhangLing．ImprovedalgorithmforcalculatingLyapunovexponentanddistinguishingchaosfromnoiseJ．JournalofHarbinInstituteofTechnology2010171101-104．7．MatlabLorenzJ．200431640-42．ZhangZhengwei．AnalysisofLorenzsystemfamilywithMATLABJ．ContemporaryElectricTechnique200431640-42．inChinese8LuZhenboCaiZhiming．DeterminationofembeddingparametersforphasespacereconstructionbasedonimprovedC－CmethedJ．JournalofSystemSimulation200711192527-2538．9．J．200030116-21．ChenGuohuaMaJunhaiShengShaohan．Anewalgorithmofthephasespacereconstructionforthedataobtainedindynami-calsystemsJ．JournalofSoutheastUniversityNaturalScienceEdition200030116-21．inChinese10．LyapunovJ．2008255898-900．HanMinWangYijie．ComputationofthelargestLyapunovexponentinnoisychaoticsignalJ．ControlTheoryandApplica-tions2008255898-900．inChinese—9—Lyapunov