常微分方程基础理论

Preface

ChapterⅠ.FundamentalTheoremsofOrdinaryDifferentialEquations

Ⅰ-1.ExistenceanduniquenesswiththeLipschitzcondition

Ⅰ-2.ExistencewithouttheLipschitzcondition

Ⅰ-3.Someglobalpropertiesofsolutions

Ⅰ-4.Analyticdifferentialequations

ExercisesⅠ

ChapterⅡ.DependenceonData

Ⅱ-1.Continuitywithrespecttoinitialdataandparameters

Ⅱ-2.Differentiability

ExercisesⅡ

ChapterⅢ.Nonuniqueness

Ⅲ-1.Examples

Ⅲ-2.TheKnesertheorem

Ⅲ-3.SolutioncurvesontheboundaryofR(A)

Ⅲ-4.Maximalandminimalsolutions

Ⅲ-5.Acomparisontheorem

Ⅲ-6.Sufficientconditionsforuniqueness

ExercisesⅢ

ChapterⅣ.GeneralTheoryofLinearSystems

Ⅳ-1.Somebasicresultsconcerningmatrices

Ⅳ-2.Homogeneoussystemsoflineardifferentialequations

Ⅳ-3.Homogeneoussystemswithconstantcoefficients

Ⅳ-4.Systemswithperiodiccoefficients

Ⅳ-5.LinearHamiltoniansystemswithperiodiccoefficients

Ⅳ-6.Nonhomogeneousequations

Ⅳ-7.Higher-orderscalarequations

ExercisesⅣ

ChapterⅤ.SingularitiesoftheFirstKind

Ⅴ-1.Formalsolutionsofanalgebraicdifferentialequation

Ⅴ-2.Convergenceofformalsolutionsofasystemofthefirstkind

Ⅴ-3.TheS-Ndecompositionofamatrixofinfiniteorder

Ⅴ-4.TheS-Ndecompositionofadifferentialoperator

Ⅴ-5.Anormalformofadifferentialoperator

Ⅴ-6.Calculationofthenormalformofadifferentialoperator

Ⅴ-7.Classificationofsingularitiesofhomogeneouslinearsystems

ExercisesⅤ

ChapterⅥ.Boundary-ValueProblemsofLinearDifferentialEquationsoftheSecond-Order

Ⅵ-1.Zerosofsolutions

Ⅵ-2.Sturm-Liouvilleproblems

Ⅵ-3.Eigenvalueproblems

Ⅵ-4.Eigenfunctionexpansions

Ⅵ-5.Jostsolutions

Ⅵ-6.Scatteringdata

Ⅵ-7.Refiectionlesspotentials

Ⅵ-8.Constructionofapotentialforgivendata

Ⅵ-9.Differentialequationssatisfiedbyreflectionlesspotentials

Ⅵ-10.Periodicpotentials

ExercisesⅥ

ChapterⅦ.AsymptoticBehaviorofSolutionsofLinearSystems

Ⅶ-1.Liapounoffstypenumbers

Ⅶ-2.Liapounoffstypenumbersofahomogeneouslinearsystem

Ⅶ-3.CalculationofLiapounoffstypenumbersofsolutions

Ⅶ-4.Adiagonalizationtheorem

Ⅶ-5.Systemswithasymptoticallyconstantcoefficients

Ⅶ-6.AnapplicationoftheFloquettheorem

ExercisesⅦ

ChapterⅧ.Stability

Ⅷ-1.Basicdefinitions

Ⅷ-2.Asufficientconditionforasymptoticstability

Ⅷ-3.Stablemanifolds

Ⅷ-4.Analyticstructureofstablemanifolds

Ⅷ-5.Two-dimensionallinearsystemswithconstantcoefficients

Ⅷ-6.AnalyticsystemsinR2

Ⅷ-7.Perturbationsofanimpropernodeandasaddlepoint

Ⅷ-8.Perturbationsofapropernode

Ⅷ-9.Perturbationofaspiralpoint

Ⅷ-10.Perturbationofacenter

ExercisesⅧ

ChapterⅨ.AutonomousSystems

Ⅸ-1.Limit-invariantsets

Ⅸ-2.Liapounoffsdirectmethod

Ⅸ-3.Orbitalstability

Ⅸ-4.ThePoincare-Bendixsontheorem

Ⅸ-5.IndicesofJordancurves

ExercisesⅨ

ChapterⅩ.TheSecond-OrderDifferentialEquation(d2x)/(dt2)+h(x)*(dx)/(dt)+g(x)=0

Ⅹ-1.Two-pointboundary-valueproblems

Ⅹ-2.ApplicationsoftheLiapounofffunctions

Ⅹ-3.Existenceanduniquenessofperiodicorbits

Ⅹ-4.MultipliersoftheperiodicorbitofthevanderPolequation

Ⅹ-5.ThevanderPolequationforasmallε>0

Ⅹ-6.ThevanderPolequationforalargeparameter

Ⅹ-7.AtheoremduetoM.Nagumo

Ⅹ-8.Asingularperturbationproblem

ExercisesⅩ

ChapterⅪ.AsymptoticExpansions

Ⅺ-1.AsymptoticexpansionsinthesenseofPoincare

Ⅺ-2.Gevreyasymptotics

Ⅺ-3.FlatfunctionsintheGevreyasymptotics

Ⅺ-4.BasicpropertiesofGevreyasymptoticexpansions

Ⅺ-5.ProofofLemmaⅪ-2-6

ExercisesⅪ

ChapterⅫ.AsymptoticExpansionsinaParameter

Ⅻ-1.Anexistencetheorem

Ⅻ-2.Basicestimates

Ⅻ-3.ProofofTheoremⅫ-1-2

Ⅻ-4.Ablock-diagonalizationtheorem

Ⅻ-5.Gevreyasymptoticsolutionsinaparameter

Ⅻ-6.Analyticsimplificationinaparameter

ExercisesⅫ

ChapterⅩⅢ.SingularitiesoftheSecondKind

ⅩⅢ-1.Anexistencetheorem

ⅩⅢ-2.Basicestimates

ⅩⅢ-3.ProofofTheoremⅩⅢ-1-2

ⅩⅢ-4.Ablock-diagonalizationtheorem

ⅩⅢ-5.Cyclicvectors(AlemmaofP.Deligne)

ⅩⅢ-6.TheHukuhara-Turrittintheorem

ⅩⅢ-7.Ann-th-orderlineardifferentialequationatasingularpointofthesecondkind

ⅩⅢ-8.Gevreypropertyofasymptoticsolutionsatanirregularsingularpoint

ExercisesⅩⅢ

References

Index

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