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1、NormalFormsandUnfoldingsforLocalDynamicalSystemsJamesMurdockSpringerSpringerMonographsinMathematicsJamesMurdockNormalFormsandUnfoldingsforLocalDynamicalSystemsWith15IllustrationsJamesMurdockMathematicsDepartmentIowaStateUniversityAmes,IA50011USAjmurdock@iastate.eduMat
2、hematicsSubjectClassification(2000):37-xx,58F36,34CxxLibraryofCongressCataloging-in-PublicationDataMurdock,JamesA.Normalformsandunfoldingsforlocaldynamicalsystems/JamesMurdock.p.cm.—(Springermonographsinmathematics)Includesbibliographicalreferencesandindex.ISBN0-387-9
3、5464-3(hc:alk.paper)1.Differentiabledynamicalsystems.2.Normalforms(Mathematics)I.Title.II.SeriesQA614.8.M8720022002024169ISBN0-387-95464-3Printedonacid-freepaper.2003Springer-VerlagNewYork,Inc.Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithout
4、thewrittenpermissionofthepublisher(Springer-VerlagNewYork,Inc.,175FifthAvenue,NewYork,NY10010,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysim
5、ilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjecttoproprietaryrights.Print
6、edintheUnitedStatesofAmerica.987654321SPIN10869537www.springer-ny.comSpringer-VerlagNewYorkBerlinHeidelbergAmemberofBertelsmannSpringerScience+BusinessMediaGmbHPrefaceThesubjectoflocaldynamicalsystemsisconcernedwiththefollowingtwoquestions:1.Givenann×nmatrixA,describe
7、thebehavior,inaneighborhoodoftheorigin,ofthesolutionsofallsystemsofdifferentialequationshavingarestpointattheoriginwithlinearpartAx,thatis,allsystemsoftheformx˙=Ax+···,wherex∈Rnandthedotsdenotetermsofquadraticandhigherorder.2.Describethebehavior(neartheorigin)ofallsyst
8、emsclosetoasystemofthetypejustdescribed.Toanswerthesequestions,thefollowingstepsareemployed:1.Anormalformisobtainedforthegen
