李雅普诺夫指数谱的研究与仿真

:2004-08-292212200512:1006-9348(2005)12-0285-041,1,2,1(1.,3001302.,200237):,Matlab,,,,:;;;;:O322:AResearchandSimulationofLyapunovpsExponentsLUOLi-jun1,LIYin-shan1,LITong2,DONGQing-tian1(1.InstituteofEngineeringMechanicsDepartment,HebeiUniversityofTechnology,Tianjin300130,China;2.InstituteofMechanicalEngineering,EastChinaUniversityofScienceandTechnology,Shanghai200237,China)ABSTRACT:Lyapunovpsexponentsplaysanimportantroleinresearchingthecharactersofbifurcationandchaosmotionofdynamics.Inthepaperthedefinition,calculationprincipleandthemethodofnumericalcalculationofLyapunovexponentsareintroduced.TheprogramsforcalculatingtheLyapunovpsexponentsarecompiledbytheMATLABsoftware.Therelationshipbetweenthebifurcation,chaosandLyapunovpsexponentsareanalyzedinnonlineardynamicalsystemofone,twoandthreedimensions.Thenasexamples,thebifurcationchartsforLogisticmapping,HenonmappingandLorenzsystemareprotractedandtheLyapunovpsexponentscorrespondingtothesesystemsarecalculatednumerically.TheexamplesofcomputersimulationindicatethatLyapunovpsexponentisoneoftheefficientmethodsforresearchingthebifurcationandchaos,aswellasforsolvingthepracticalengineeringproblems.KEYWORDS:Nonlinearsystem;Chaos;Lyapunovpsexponents;Bifurcationchart;Strangeattractor1[1,2,3,4]Lyapunov,,Lyapunov:Lyapunov,,,,n,n,,;,()(),Lyapunov,Lyapunov,et,,LyapunovLyapunov,1nLyapunov,,LyapunovHamilton,LyapunovLyapunov,582Hamilton,Lyapunov;,Lyapunov,Lyapunovm(m=1m=2),,mLyapunov,Lyapunov,10nkLyapunov1++k(k=1k=2),k,10,,,1+20,,,,201+201+2+30,,,1++k0,1++k+10,kLyapunov,J.KaplanJ.Yorke1++k01++k+10,k+10,d=k+(1++k)/|k+1|(1),d,,,d2Lyapunov[5,6]:(x0)=limn1nlndfn(x0)dx(2)xn+1=f(xn)(3)Lyapunov,fC1[,],x0,(x0)=limn1nn-1i=0ln|f(xi)|(4)x1=f(x0),xi=f(xi-1):N:Ûx=F(x)xRN(5)x(0)=x0(5)x(t)=Ttx0(6)(5)Ûx=5F5x(Ttx0)x(7)(7)x(t)=Utx0x0(8)(7)Utx0Ut+sx0=UtTsx0.Usx0(9)(ek,x0)=limt1tlnUtx0e1Utx0e2Utx0eke1e2ek(k=1,2,,N)(10)ekkx0Ex0,{ei}(i=1,2,,k)ek(10)kLyapunovLyapunov(e1,x0),(i)1iN,12NkLyapunov(ek,x0),N=3(e1,x0){1,2,3}(e2,x0){1+2,2+3,3+1}(e3,x){1+2+3}{ei}(i=1,2,,N),x0,Lyapunov,{ei}x03Lyapunov(),(),,xi,,Lyapunov,,(1,2)=(-,-),xi,,0;xi,,,=0,()Lyapunov,Lyapunov(1,2)=(0,-)(1,2,3)=(-,-,-):;(1,2,3)=(0,-,-):;(1,2,3)=(0,0,-):;(1,2,3)=(+,+,0):;(1,2,3)=(+,0,0):;(1,2,3)=(+,0,-):44.1Logisticxn+1=a(1-xn)xn(11)682,xn[0,1],a[1,4]Lyapunov:(a=1),;:;,;,Lyapunov,11Logistic,a[1,4]LogisticLyapunovMatlab:Le=zeros(401,2);i=0;foru=0:0.01:4i=i+1;x=0.5;y=0;forj=1:100x=u3x3(1-x);endfork=1:200x=u3x3(1-x);y=y+log(abs(1-23x));endLe(i,1)=u;Le(i,2)=log(u)+y/200;endplot(Le(:,1),Le(:,2));holdon;n=0:0.001:4;plot(n,0,’k’);4.2HenonT:xn+1=1-ax2n+ynyn+1=bxn(12),xn[0,1],yn[0,1]Jacobiej=-2axn1b0(13)2HenonJacobie|J|=-b(14)b=1,Henon;b(-,-1)(1,+),Henon;b[-1,+1],Henon2(a)Henona(a[0,1.4],b=0.3),Henona2(b)HenonLyapunov1,2a4.3LorenzÛx=(y-x)Ûy=x(r-z)-yÛz=xy-bz(15)3(a)Lorenzr(=10,b=8/3,r[0,500])zx=y,,Lorenzr,r3103(b)LorenzLyapunov1,2,3r3Lorenz5,LyapunovLyapunov,Lyapunov,;Lyapunov,,,,,,782Lyapunov:1),,Lyapunov,2)()Lyapunov1;1=0,13),LyapunovN3,,:[1]ENLorenz.TheEssenceofchaos[R].theUniversityofWashingtonPress,1993.[2]LIYin-shan,ZhangNian-mei,YANGGui-tong,1/3Subharmonicsolutionofellipticalsandwichplates[J].AppliedMathematicsandMechanics,2003,24(10):1147-1157.[3]LIYin-shan,YANGGui-tong,ZHANGShan-yuan,WEIJian-wei.Experimentsonsuperharmonicbifurcationandchaoticmotionofacircularplateoscillator[J].JournalofExperimentalMechanics,2001,16(4):347-358.[4]LIYin-shan,ChenYu-shu,XUEYu-sheng.Stabilitymarginofunbalanceelasticrotorinshortbearingsunderanon-linearoil-filmForceModel[J].JournalofMechanicalEngineering,2002,38(9):27-32.[5]AWolf,JBSwift,HLSwinneyandJAVastano.DeterminingLyapunovpexponentsfromatimeseries[J].Physica1985,16D:285-317.[6],,.[J].,2004,32(3):80-83.[7],,,.[J].,2003,20(4):74-77.[8],,.DNA[J].,2003,20(2):65-68.[9],,,.[J].,2002,24(4):505509.[10],,,.[J].,2002,9(1):6271.[](1980-),(),,,:(1961.8-),(),,,,:;;;,60,,(1962-),(),,,,:Simulink,20(1978-),(),,,:(254)5,RTEMS;,;Msg-Test,RTEMS:[1].[M].,2002-10.[2].RTEMS[J].,2001-9.[3]WEricNorum.GettingstartedwithEPICSonRTEMS[R].ArgonneNationalLaboratoryoftheUniversityofChicago.2003-9.[4]SalmanTaherian.OpenSourceReal-TimeOS(RTEMS)onSCIbasedComputeClusters[R].UniversityofDublinTrinityCollegeSchoolofEngineering.2003.[5].RTEMS[D].,2002-2.[6].DeltaCORE2.1[R].2002-12.[7]OARCorporation.RTEMSCUserpsGuide[M].2003.01.[8].[M].,2002-7.[9]QingLi.Real-TimeConceptsforEmbeddedSystems[M].CMPBooks(USA).2003.[10]OARCorporation.RTEMSSourceCode[DB].[11]()ErichGamma.DesignPatternsElementsofReusableObject-OrientedSoftware[M].,2002-3.[12].RTEMS[R].,2004-7-16.[](1977-),(),,,:/(1961-),(),,,,://882