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1、Electron.J.Diff.Eqns.,Monograph02,2000http://ejde.math.swt.eduorhttp://ejde.math.unt.eduftpejde.math.swt.eduorejde.math.unt.edu(login:ftp)LinearizationviatheLieDerivative¤CarmenChicone&RichardSwansonAbstractThestandardproofoftheGrobman–Hartmanlinearizationtheoremforaflowatahyperb
2、olicrestpointproceedsbyfirstestablishingtheanalogousresultforhyperbolicfixedpointsoflocaldiffeomorphisms.Inthisexpositionwepresentasimpledirectproofthatavoidsthediscretecasealtogether.WegivenewproofsforHartman’ssmoothnessresults:212ACflowisClinearizableatahyperbolicsink,andaCflowint
3、he1planeisClinearizableatahyperbolicrestpoint.Also,weformulateandprovesomenewresultsonsmoothlinearizationforspecialclassesofquasi-linearvectorfieldswhereeitherthenonlinearpartisrestrictedoradditionalconditionsonthespectrumofthelinearpart(notrelatedtoresonanceconditions)areimpose
4、d.Contents1Introduction22ContinuousConjugacy43SmoothConjugacy73.1HyperbolicSinks...........................103.1.1SmoothLinearizationontheLine.............323.2HyperbolicSaddles..........................344LinearizationofSpecialVectorFields454.1SpecialVectorFields..............
5、...........464.2Saddles................................504.3InfinitesimalConjugacyandFiberContractions..........504.4SourcesandSinks..........................51¤MathematicsSubjectClassifications:34-02,34C20,37D05,37G10.Keywords:Smoothlinearization,Liederivative,Hartman,Grobman,hy
6、perbolicrestpoint,fibercontraction,Dorrohsmoothing.°c2000SouthwestTexasStateUniversity.SubmittedNovember14,2000.PublishedDecember4,2000.12LinearizationviatheLieDerivative1IntroductionThispaperisdividedintothreeparts.Inthefirstpart,anewproofispresentedfortheGrobman–Hartmanlineariz
7、ationtheorem:AC1flowisC0linearizableatahyperbolicrestpoint.ThesecondpartisadiscussionofHartman’sresultsonsmoothlinearizationwheresmoothnessofthelinearizingtransformationisprovedinthosecaseswhereresonanceconditionsarenotrequired.Forexample,wewillusethetheoryofordinarydifferentiale
8、quationstoprovetwomaintheorems:AC2vectorfieldisC1linear
