ode toeschl g ordinary differential equations and dynamic system 2004外语英文电子书

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1、OrdinarydifferentialequationsandDynamicalSystemsGeraldTeschlGeraldTeschlInstitutf¨urMathematikNordbergstraße15Universit¨atWien1090Wien,AustriaE-mail:Gerald.Teschl@univie.ac.atURL:http://www.mat.univie.ac.at/~gerald/1991Mathematicssubjectclassification.34-01Ab

2、stract.Thismanuscriptprovidesanintroductiontoordinarydifferentialequationsanddynamicalsystems.Westartwithsomesimpleexamplesofexplicitlysolvableequations.Thenweprovethefundamentalresultsconcerningtheinitialvalueproblem:existence,uniqueness,extensibility,depen

3、denceoninitialconditions.Furthermoreweconsiderlinearequations,theFloquettheorem,andtheautonomouslinearflow.ThenweestablishtheFrobeniusmethodforlinearequationsinthecom-plexdomainandinvestigatesSturm–Liouvilletypeboundaryvalueproblemsincludingoscillationtheory

4、.Nextweintroducetheconceptofadynamicalsystemanddiscusssta-bilityincludingthestablemanifoldandtheHartman–Grobmantheoremforbothcontinuousanddiscretesystems.WeprovethePoincar´e–Bendixsontheoremandinvestigateseveralex-amplesofplanarsystemsfromclassicalmechanics

5、,ecology,andelectricalengineering.Moreover,attractors,Hamiltoniansystems,theKAMtheorem,andperiodicsolutionsarediscussedaswell.Finally,thereisanintroductiontochaos.BeginningwiththebasicsforiteratedintervalmapsandendingwiththeSmale–BirkhofftheoremandtheMelniko

6、vmethodforhomoclinicorbits.Keywordsandphrases.Ordinarydifferentialequations,dynamicalsystems,Sturm-Liouvilleequations.TypesetbyAMS-LATEXandMakeindex.Version:October21,2004Copyrightc2000-2004byGeraldTeschlContentsPrefaceviiPart1.ClassicaltheoryChapter1.Introd

7、uction3§1.1.Newton’sequations3§1.2.Classificationofdifferentialequations6§1.3.Firstorderautonomousequations8§1.4.Findingexplicitsolutions11§1.5.Qualitativeanalysisoffirstorderequations17Chapter2.Initialvalueproblems23§2.1.Fixedpointtheorems23§2.2.Thebasicexist

8、enceanduniquenessresult26§2.3.Dependenceontheinitialcondition29§2.4.Extensibilityofsolutions32§2.5.Euler’smethodandthePeanotheorem34§2.6.Appendix:Volterraintegralequations37Chapter3.Linearequations43§3