ordinary differential equations and dynamical systems - g. teschl

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1、OrdinarydifferentialequationsandDynamicalSystemsGeraldTeschlGeraldTeschlInstitutf¨urMathematikStrudlhofgasse4Universit¨atWien1090Wien,AustriaE-mail:Gerald.Teschl@univie.ac.atURL:http://www.mat.univie.ac.at/~gerald/1991Mathematicssubjectclassification.34-

2、01Abstract.Thismanuscriptprovidesanintroductiontoordinarydifferentialequationsanddynamicalsystems.Westartwithsomesimpleexamplesofexplicitlysolvableequations.Thenweprovethefundamentalresultsconcerningtheinitialvalueproblem:existence,uniqueness,extensibil

3、ity,dependenceoninitialconditions.Furthermoreweconsiderlinearequations,theFloquettheorem,andtheautonomouslinearflow.ThenweestablishtheFrobeniusmethodforlinearequationsinthecom-plexdomainandinvestigatesSturm–Liouvilletypeboundaryvalueproblemsincludingosc

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

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

6、khofftheoremandtheMelnikovmethodforhomoclinicorbits.Keywordsandphrases.Ordinarydifferentialequations,dynamicalsystems,Sturm-Liouvilleequations.TypesetbyAMS-LATEXandMakeindex.Version:February18,2004Copyrightc2000-2004byGeraldTeschlContentsPrefaceviiPart1.

7、ClassicaltheoryChapter1.Introduction3§1.1.Newton’sequations3§1.2.Classificationofdifferentialequations5§1.3.Firstorderautonomousequations8§1.4.Findingexplicitsolutions11§1.5.Qualitativeanalysisoffirstorderequations16Chapter2.Initialvalueproblems21§2.1.Fix

8、edpointtheorems21§2.2.Thebasicexistenceanduniquenessresult23§2.3.Dependenceontheinitialcondition26§2.4.Extensibilityofsolutions29§2.5.Euler’smethodandthePeanotheorem32§2.6.Appendix:Volterraintegralequations34Chapter3.Linearequations41§3