动力系统V:分歧理论和突变理论

图书标准信息

• 作者 [俄罗斯]阿诺德 著
• 出版社 科学出版社
• 出版时间 2009-01
• 版次 1
• ISBN 9787030234933
• 定价 68.00元
• 装帧 精装
• 开本 16开
• 纸张 胶版纸
• 页数 271页
• 字数 341千字
• 正文语种 英语

【内容简介】

　　Both bifurcation theory and catastrophe theory are studies of smooth systems，tbcusing on properties that seem manifestly non-smooth. Bifurcations are sudden changes that occur in a system as one or more parameters are varied.Catastrophe theory is accurately described as singularity theory and its applications.

　　These two theories are important tools in the study of differential equations and of related physical systems.Analyzing the bifurcations or singularities of a system provides useful qualitative information about its behaviour. The authors have written this book with reffeshing clarity.Theexposition is masterful，with penetrating insights.

【目录】

Preface
Chapter1.BifurcationsofEquilibria
1.FamiliesandDeformations
1.1.FamiliesofVectorFields
1.2.TheSpaceofJets
1.3.SardsLemmaandTransversalityTheorems
1.4.SimplestApplications：SingularPointsofGenericVectorFields
1.5.TopologicallyVersalDeformations
1.6.TheReductionTheorem
1.7.GenericandPrincipalFamilies

2.BifurcationsofSingularPointsinGenericOne-ParameterFamilies
2.1TypicalGermsandPrincipalFamilies
2.2.SoftandHardLossofStability

3.BifurcationsofSingularPointsinGenericMulti-ParameterFamilieswithSimplyDegenerateLinearParts
3.1.PrincipalFamilies
3.2.BifurcationDiagramsofthePrincipalFamilies(3-+)inTable1
3.3.BifurcationDiagramswithRespecttoWeakEquivalenceandPhasePortraitsofthePrincipalFamilies(4-+)inTable1

4.BifurcationsofSingularPointsofVectorFieldswithaDoubly-DegenerateLinearPart
4.1.AListofDegeneracies
4.2.TwoZeroEigenvalues
4.3.ReductionstoTwo-DimensionalSystems
4.4.OneZeroandaPairofPurelyImaginaryEigenvalues
4.5.TwoPurelyImaginaryPairs
4.6.PrincipalDeformationsofEquationsofDifficultTypeinProblemswithTwoPairsofPurelyImaginaryEigenvalues(FollowingZolitdek)
5.TheExponentsofSoftandHardLossofStability
5.1.Definitions
5.2.TableofExponents

Chapter2.BifurcationsofLimitCycles
1.BifurcationsofLimitCyclesinGenericOne-ParameterFamilies
1.1.MultiplierI
1.2.Multiplier-1andPeriod-DoublingBifurcations
1.3.APairofComplexConjugateMultipliers
1.4.NonlocalBifurcationsinOne-ParameterFamiliesofDiffeomorphisms
1.5.NonlocalBifurcationsofPeriodicSolutions
1.6.BifurcationsResultinginDestructionsofInvariantTori

2.BifurcationsofCyclesinGenericTwo-ParameterFamilieswithan
AdditionalSimpleDegeneracy
2.1.AListofDegeneracies
2.2.AMultiplier+1or-1withAdditionalDegeneracyintheNonlinearTerms
2.3.APairofMultipliersontheUnitCirclewithAdditionalDegeneracyintheNonlinearTerms

3.BifurcationsofCyclesinGenericTwo-ParameterFamilieswithStrongResonancesofOrdersq≠4
3.1.TheNormalFormintheCaseofUnipotentJordanBlocks
3.2.AveragingintheSeifertandtheM6biusFoliations
3.3.PrincipalVectorFieldsandtheirDeformations
3.4.VersalityofPrincipalDeformations
3.5.BifurcationsofStationarySolutionsofPeriodicDifferentialEquationswithStrongResonancesofOrdersq≠4

4.BifurcationsofLimitCyclesforaPairofMultipliersCrossingthe
UnitCircleat±i
4.1.DegenerateFamilies
4.2.DegenerateFamiliesFoundAnalytically
4.3.DegenerateFamiliesFoundNumerically
4.4.BifurcationsinNondegenerateFamilies
4.5.LimitCyclesofSystemswithaFourthOrderSymmetry

5.Finitely-SmoothNormalFormsofLocalFamilies
5.1.ASynopsisofResults
5.2.DefinitionsandExamples
5.3.GeneralTheoremsandDeformationsofNonresonantGerms
5.4.ReductiontoLinearNormalForm
5.5.DeformationsofGermsofDiffeomorphismsofPoincareType
5.6.DeformationsofSimplyResonantHyperbolicGerms
5.7.DeformationsofGermsofVectorFieldswithOneZeroEigenvalueataSingularPoint
5.8.FunctionalInvariantsofDiffeomorphismsoftheLine
5.9.FunctionalInvariantsofLocalFamiliesofDiffeomorphisms
5.10.FunctionalInvariantsofFamiliesofVectorFields
5.11.FunctionalInvariantsofTopologicalClassificationsofLocalFamiliesofDiffeomorphismsoftheLine

6.FeigenbaumUniversalityforDiffeomorphismsandFlows
6.1.Period-DoublingCascades
6.2.PerestroikasofFixedPoints
6.3.Cascadesofn-foldIncreasesofPeriod
6.4.DoublinginHamiltonianSystems
6.5.ThePeriod-DoublingOperatorforOne-DimensionalMappings
6.6.TheUniversalPeriod-DoublingMechanismforDiffeomorphisms

Chapter3.NonlocalBifurcations
1.DegeneraciesofCodimension1.SummaryofResults
1.1.LocalandNonlocalBifurcations
1.2.NonhyperbolicSingularPoints
1.3.NonhyperbolicCycles
1.4.NontransversalIntersectionsofManifolds
1.5.Contours
1.6.BifurcationSurfaces
1.7.CharacteristicsofBifurcations
1.8.SummaryofResults

2.NonlocalBifurcationsofFlowsonTwo-DimensionalSurfaces
2.1.SemilocalBifurcationsofFlowsonSurfaces
2.2.NonlocalBifurcationsonaSphere：TheOne-ParameterCase.
2.3.GenericFamiliesofVectorFields
2.4.ConditionsforGenericity
2.5.One-ParameterFamiliesonSurfacesdifferentfromtheSphere
2.6.GlobalBifurcationsofSystemswithaGlobalTransversalSectiononaTorus
2.7.SomeGlobalBifurcationsonaKleinbottle
2.8.BifurcationsonaTwo-DimensionalSphere：TheMulti-ParameterCase
2.9.SomeOpenQuestions

3.BifurcationsofTrajectoriesHomoclinictoaNonhyperbolicSingularPoint
3.1.ANodeinitsHyperbolicVariables
3.2.ASaddleinitsHyperbolicVariables：OneHomoclinicTrajectory
3.3.TheTopologicalBernoulliAutomorphism
3.4.ASaddleinitsHyperbolicVariables：SeveralHomoclinicTrajectories
3.5.PrincipalFamilies

4.BifurcationsofTrajectoriesHomoclinictoaNonhyperbolicCycle
4.1.TheStructureofaFamilyofHomoclinicTrajectories
4.2.CriticalandNoncriticalCycles
4.3.CreationofaSmoothTwo-DimensionalAttractor
4.4.CreationofComplexInvariantSets(TheNoncriticalCase)...
4.5.TheCriticalCase
4.6.ATwo-StepTransitionfromStabilitytoTurbulence
4.7.ANoncompactSetofHomoclinicTrajectories
4.8.Intermittency
4.9.AccessibilityandNonaccessibility
4.10.StabilityofFamiliesofDiffeomorphisms
4.11.SomeOpenQuestions

5.HyperbolicSingularPointswithHomoclinicTrajectories
5.1.PreliminaryNotions：LeadingDirectionsandSaddleNumbers
5.2.BifurcationsofHomoclinicTrajectoriesofaSaddlethatTakePlaceontheBoundaryoftheSetofMorse-SmaleSystems
5.3.RequirementsforGenericity
5，4，PrincipalFamiliesinR3andtheirProperties
5.5.VersalityofthePrincipalFamilies
5.6.ASaddlewithComplexLeadingDirectioninR3
5.7.AnAddition：BifurcationsofHomoclinicLoopsOutsidetheBoundaryofaSetofMorse-SmaleSystems
5.8.AnAddition：CreationofaStrangeAttractoruponBifurcationofaTrajectoryHomoclinictoaSaddle

6.BifurcationsRelatedtoNontransversalIntersections
6.1.VectorFieldswithNoContoursandNoHomoclinicTrajectories
6.2.ATheoremonInaccessibility
6.3.Moduli
6.4.SystemswithContours
6.5.DiffeomorphismswithNontrivialBasicSets
6.6，VectorFieldsinR3withTrajectoriesHomoclinictoaCycle
6.7.SymbolicDynamics
6.8.BifurcationsofSmaleHorseshoes
6.9.VectorFieldsonaBifurcationSurface
6.10.DiffeomorphismswithanInfiniteSetofStablePeriodicTrajectories

7.InfiniteNonwanderingSets
7.1.VectorFieldsontheTwo-DimensionalTorus
7.2.BifurcationsofSystemswithTwoHomoclinicCurvesofaSaddle
7.3.SystemswithFeigenbaumAttractors
7.4.BirthofNonwanderingSets
7.5.PersistenceandSmoothnessofInvariantManifolds
7.6.TheDegenerateFamilyandItsNeighborhoodinFunctionSpace
7.7.BirthofToriinaThree-DimensionalPhaseSpace

8.AttractorsandtheirBifurcations
8.1.TheLikelyLimitSetAccordingtoMilnor(1985)
8.2.StatisticalLimitSets
8.3.InternalBifurcationsandCrisesofAttractors
8.4.InternalBifurcationsandCrisesofEquilibriaandCycles
8.5.BifurcationsoftheTwo-DimensionalTorus

Chapter4.RelaxationOscillations
1.FundamentalConcepts
1.1.AnExample：vanderPorsEquation
1.2.FastandSlowMotions
1.3.TheSlowSurfaceandSlowEquations
1.4.TheSlowMotionasanApproximationtothePerturbedMotion
1.5.ThePhenomenonofJumping

2.SingularitiesoftheFastandSlowMotions
2.1.SingularitiesofFastMotionsatJumpPointsofSystemswithOneFastVariable
2.2.SingularitiesofProjectionsoftheSlowSurface
2.3.TheSlowMotionforSystemswithOneSlowVariable
2.4.TheSlowMotionforSystemswithTwoSlowVariables
2.5.NormalFormsofPhaseCurvesoftheSlowMotion
2.6.ConnectionwiththeTheoryofImplicitDifferentialEquations
2.7.DegenerationoftheContactStructure

3.TheAsymptoticsofRelaxationOscillations
3.1.DegenerateSystems
3.2.SystemsofFirstApproximation
3.3.NormalizationsofFast-SlowSystemswithTwoSlowVariablesfor
3.4.DerivationoftheSystemsofFirstApproximation
3.5.InvestigationoftheSystemsofFirstApproximation
3.6.Funnels
3.7.PeriodicRelaxationOscillationsinthePlane

4.DelayedLossofStabilityasaPairofEigenvaluesCrosstheImaginaryAxis
4.1.GenericSystems
4.2.DelayedLossofStability
4.3.HardLossofStabilityinAnalyticSystemsofType2
4.4.Hysteresis
4.5.TheMechanismofDelay
4.6.ComputationoftheMomentofJumpinginAnalyticSystems
4.7.DelayUponLossofStabilitybyaCycle
4.8.DelayedLossofStabilityand"Ducks".
5.DuckSolutions
5.1.AnExample：ASingularPointontheFoldoftheSlowSurface
5.2.ExistenceofDuckSolutions
5.3.TheEvolutionofSimpleDegenerateDucks
5.4.ASemi-localPhenomenon：DuckswithRelaxation
5.5.DucksinR3andRn
RecommendedLiterature
References
AdditionalReferences