微分方程，动态系统和混沌导论

CHAPTER1First-OrderEquations

1．1TheSimplestExample

1．2TheLogisticPopulationModel

1．3ConstantHarvestingandBifurcations

1．4PeriodicHarvestingandPeriodicSolutions

1．5ComputingthePoincardMap

1．6Exploration：ATwo-ParameterFamily

CHAPTER2PlanarLinearSystems

2．1Second-OrderDifferentialEquations

2．2PlanarSystems

2．3PreliminariesfromAlgebra

2．4PlanarLinearSystems

2．5EigenvaluesandEigenvectors

2．6SolvingLinearSystems

2．7TheLinearityPrinciple

CHAPTER3PhasePortraitsforPlanarSystems

3．1RealDistinctEigenvalues

3．2ComplexEigenvalues

3．3RepeatedEigenvalues

3．4ChangingCoordinates

CHAPTER4ClassificationofPlanarSystems

4．1TheTrace-DeterminantPlane

4．2DynamicalClassification

4．3Exploration：A3DParameterSpace

CHAPTER5HigherDimensionalLinearAlgebra

5．1PreliminariesfromLinearAlgebra

5．2EigenvaluesandEigenvectors

5．3ComplexEigenvalues

5．4BasesandSubspaces

5．5RepeatedEigenvalues

5．6Genericity

CHAPTER6HigherDimensionalLinearSystems

6．1DistinctEigenvalues

6．2HarmonicOscillators

6．3RepeatedEigenvalues

6．4TheExponentialofaMatrix

6．5NonautonomousLinearSystems

CHAPTER7NonlinearSystems

7．1DynamicalSystems

7．2TheExistenceandUniquenessTheorem

7．3ContinuousDependenceofSolutions

7．4TheVariationalEquation

7．5Exploration：NumericalMethods

CHAPTER8EquilibriainNonlinearSystems

8．1SomeNustrativeExamples

8．2NonlinearSinksandSources

8．3Saddles

8．4Stability

8．5Bifurcations

8．6Exploration：ComplexVectorFields

CHAPTER9GlobalNonlinearTechniques

9．1Nullclines

9．2StabilityofEquilibria

9．3GradientSystems

9．4HamiltonianSystems

9．5Exploration：ThePendulumwithConstantForcing

CHAPTER10ClosedOrbitsandLimitSets

10．1LimitSets

10．2LocalSectionsandFlowBoxes

10．3ThePoincareMap

10．4MonotoneSequencesinPlanarDynamicalSystems

10．5ThePoincare-BendixsonTheorem

10．6ApplicationsofPoincare-Bendixson

10．7Expl0ration：ChemicalReactionsThatOscillate

CHAPTER11ApplicationsinBiology

11．1InfectiousDiseases

11．2Predator／PreySystems

11．3CompetitiveSpecies

11．4Exploration：CompetitionandHarvesting

CHAPTER12ApplicationsinCircuitTheory

12．1AnRLCCircuit

12．2TheLienardEquation

12．3ThevanderPolEquation

12．4AHopfBifurcation

12．5Exploration：Neurodynamics

CHAPTER13ApplicationsinMechanics

13．1Newton’SSecondLaw

13．2ConservativeSystems

13．3CentralForceFields

13．4TheNewtonianCentralForceSystem

13．5Kepler’sFirstLaw

13．6TheTwo-BodyProblem

13．7BlowingUptheSingularity

13．8Exploration：OtherCentralForceProblems

13．9Exploration：ClassicalLimitsofQuantumMechanicalSystems

CHAPTER14TheLorenzSystem

14．1IntroductiontotheLorenzSystem

14．2ElementaryPropertiesoftheLorenzSystem

14．3TheLorenzAttractor

14．4AModelfortheLorenzAttractor

14．5TheChaoticAttractor

14．6Exploration：TheRosslerAttractor

CHAPTER15DiscreteDynamicalSystems

15．1IntroductiontoDiscreteDynamicalSystems

15．2Bifurcations

15．3TheDiscreteLogisticModel

15．4Chaos

15．5SymbolicDynamics

15．6TheShiftMap

15．7TheCantorMiddle-ThirdsSet

15．8Exploration：CubicChaos

15．9Exploration：TheOrbitDiagram

CHAPTER16HomoclinicPhenomena

16．1TheShil’nikovSystem

16．2TheHorseshoeMap

16．3TheDoubleScrollAttractor

16．4HomoclinicBifurcations

16．5Exploration：TheChuaCircuit

CHAPTER17ExistenceandUniquenessRevisited

17．1TheExistenceandUniquenessTheorem

17．2ProofofExistenceandUniqueness

17．3ContinuousDependenceonInitialConditions

17．4ExtendingSolutions

17．5NonautonomousSystems

17．6DifferentiabilityoftheFlow

Bibliography

Index

　　本书是30年前世界著名的动力系统专家赫希(M．Hirsch)和斯梅尔(S．Smale)合著的“DifferentialEquations，DynamicalSystemsandLinearAlgebra”一书的修订本，原书初版后被许多高校作为动力系统入门的标准教材，多年来在国际上产生较大影响。　　30年来，动力系统的数学理论与应用有了很大发展。30多年前还没有高速的台式计算机和计算机图像，“混沌”一词也没有在数学界使用，而对于微分方程与动力系统的研究兴趣主要仅限于数学界中比较小的范围。到今天，处处有计算机，求微分方程近似解的软件包已得到广泛运用，使人们从图形中就能看到结果。对于非线性微分方程的分析已为广大学者所接受，一些复杂的动力学行为，如马蹄映射、同宿轨、Lorenz系统中揭示出来的复杂现象，以及数学方面的分析，使学者们确信简单的稳定运动，如平衡态和周期解己不总是微分方程解的最重要的行为，而混沌现象揭示出来的美妙性态正促使各个领域的科学家与工程师细心关注在他们自己领域中提出的重要的微分方程及其混沌特性。动力系统现象在今天已出现在几乎每个科学领域中，从化学中的振荡Belousov-Zhabotinsky反应到电子工程中的混沌Chua电路，从天体力学中的复杂运动到生态系统中的分岔。