Ordinary_Differential_Equations_and_Dynamical_Systems-Teschl.pdf

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Ordinary Differential Equations and Dynamical Systems
Ordinarydierentialequations
and
DynamicalSystems
GeraldTeschl
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GeraldTeschl
Institutf¨urMathematik
Strudlhofgasse4
Universit¨atWien
1090Wien,Austria
E-mail: Gerald.Teschl@univie.ac.at
URL: http://www.mat.univie.ac.at/~gerald/
1991Mathematicssubjectclassification.34-01
Abstract.Thismanuscriptprovidesanintroductiontoordinarydierential
equationsanddynamicalsystems.Westartwithsomesimpleexamples
ofexplicitlysolvableequations.Thenweprovethefundamentalresults
concerningtheinitialvalueproblem:existence,uniqueness,extensibility,
dependenceoninitialconditions.Furthermoreweconsiderlinearequations,
theFloquettheorem,andtheautonomouslinearflow.
ThenweestablishtheFrobeniusmethodforlinearequationsinthecom-
plexdomainandinvestigatesSturm–Liouvilletypeboundaryvalueproblems
includingoscillationtheory.
Nextweintroducetheconceptofadynamicalsystemanddiscusssta-
bilityincludingthestablemanifoldandtheHartman–Grobmantheoremfor
bothcontinuousanddiscretesystems.
WeprovethePoincar´e–Bendixsontheoremandinvestigateseveralex-
amplesofplanarsystemsfromclassicalmechanics,ecology,andelectrical
engineering.Moreover,attractors,Hamiltoniansystems,theKAMtheorem,
andperiodicsolutionsarediscussedaswell.
Finally,thereisanintroductiontochaos.Beginningwiththebasicsfor
iteratedintervalmapsandendingwiththeSmale–Birkhotheoremandthe
Melnikovmethodforhomoclinicorbits.
Keywordsandphrases.Ordinarydierentialequations,dynamicalsystems,
Sturm-Liouvilleequations.
TypesetbyA M S-L A T E XandMakeindex.
Version:February18,2004
Copyrightc2000-2004byGeraldTeschl
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Contents
Preface
vii
Part1.Classicaltheory
Chapter1. Introduction 3
§1.1.Newton’sequations 3
§1.2.Classificationofdierentialequations 5
§1.3.Firstorderautonomousequations 8
§1.4.Findingexplicitsolutions 11
§1.5.Qualitativeanalysisoffirstorderequations 16
Chapter2. Initialvalueproblems 21
§2.1.Fixedpointtheorems 21
§2.2.Thebasicexistenceanduniquenessresult 23
§2.3.Dependenceontheinitialcondition 26
§2.4.Extensibilityofsolutions 29
§2.5.Euler’smethodandthePeanotheorem 32
§2.6.Appendix:Volterraintegralequations 34
Chapter3.Linearequations 41
§3.1.Preliminariesfromlinearalgebra 41
§3.2.Linearautonomousfirstordersystems 47
§3.3.Generallinearfirstordersystems 50
§3.4.Periodiclinearsystems 54
iii
 
iv Contents
Chapter4.Dierentialequationsinthecomplexdomain 61
§4.1.Thebasicexistenceanduniquenessresult 61
§4.2.Linearequations 63
§4.3.TheFrobeniusmethod 67
§4.4.Secondorderequations 70
Chapter5.Boundaryvalueproblems 77
§5.1. Introduction 77
§5.2.Symmetriccompactoperators 80
§5.3.RegularSturm-Liouvilleproblems 85
§5.4.Oscillationtheory 90
Part2.Dynamicalsystems
Chapter6.Dynamicalsystems 99
§6.1.Dynamicalsystems 99
§6.2.Theflowofanautonomousequation 100
§6.3.Orbitsandinvariantsets 103
§6.4.Stabilityoffixedpoints 107
§6.5.StabilityviaLiapunov’smethod 109
§6.6.Newton’sequationinonedimension 110
Chapter7.Localbehaviornearfixedpoints 115
§7.1.Stabilityoflinearsystems 115
§7.2.Stableandunstablemanifolds 118
§7.3.TheHartman-Grobmantheorem 123
§7.4.Appendix:Hammersteinintegralequations 127
Chapter8.Planardynamicalsystems 129
§8.1.ThePoincar´e–Bendixsontheorem 129
§8.2.Examplesfromecology 133
§8.3.Examplesfromelectricalengineering 137
Chapter9.Higherdimensionaldynamicalsystems 143
§9.1.Attractingsets 143
§9.2.TheLorenzequation 146
§9.3.Hamiltonianmechanics 150
§9.4.CompletelyintegrableHamiltoniansystems 154
§9.5.TheKeplerproblem 159

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