不连续及连续系统中的分岔和混沌（英文版）
出版时间：2011年版
内容简介
本书利用泛函分析工具来谈论混沌与分岔，并提供简明扼要的数学证明。书中通过许多有趣、经典的例子展示了其具体的应用。本书研究了大量的非线性问题，包括非线性差分方程、常微分方程和偏微分方程、脉冲微分方程、分段光滑微分方程及在无限格上的微分方程等。本书可供对非线性机械系统的振动、弦或梁的摆动以及应用动力系统中分岔方法来研究电路等问题感兴趣的数学家、物理学家、工程师及相关专业研究生等参考。
目录
1 Introduction
References
2 Preliminary Results
2.1 Linear Functional Analysis
2.2 Nonlinear Functional Analysis
2.2.1 Banach Fixed Point Theorem
2.2.2 Implicit Function Theorem
2.2.3 Lyapunov-Schmidt Method
2.2.4 Brouwer Degree
2.2.5 Local Invertibility
2.2.6 Global Invertibility
2.3 Multivalued Mappings
2.4 Differential Topology
2.4.1 Differentiable Manifolds
2.4.2 Vector Bundles
2.4.3 Tubular Neighbourhoods
2.5 Dynamical Systems
2.5.1 Homogenous Linear Equations
2.5.2 Chaos in Diffeomorphisms
2.5.3 Periodic ODEs
2.5.4 Vector Fields
2.5.5 Global Center Manifolds
2.5.6 Two-Dimensional Flows
2.5.7 Averaging Method
2.5.8 Carath6odory Type ODEs
2.6 Singularities of Smooth Maps
2.6.1 Jet Bundles
2.6.2 Whitney C~O Topology
2.6.3 Transversality
2.6.4 Malgrange Preparation Theorem
2.6.5 Complex Analysis
References
3 Chaos in Discrete Dynamical Systems
3.1 Transversal Bounded Solutions
3.1.1 Difference Equations
3.1.2 Variational Equation
3.1.3 Perturbation Theory
3.1.4 Bifurcation from a Manifold of HomoclinicSolutions
3.1.5 Applications to Impulsive DifferentialEquations
3.2 Transversal Homoclinic Orbits
3.2.1 Higher Dimensional DifferenceEquations
3.2.2 Bifurcation Result
3.2.3 Applications to McMillan TypeMappings
3.2.4 Planar Integrable Maps withSeparatrices
3.3 Singular Impulsive ODEs
3.3.1 Singular ODEs with Impulses
3.3.2 Linear Singular ODEs with Impulses
3.3.3 Derivation of the Melnikov Function
3.3.4 Examples of Singular Impulsive ODEs
3.4 Singularly Perturbed Impulsive ODEs
3.4.1 Singularly Perturbed ODEs withimpulses
3.4.2 Melnikov Function
3.4.3 Second Order Singularly Perturbed ODEswith Impulses
3.5 Inflated Deterministic Chaos
3.5.1 Inflated Dynamical Systems
3.5.2 Inflated Chaos
References
4 Chaos in Ordinary Differential Equations
4.1 Higher Dimensional ODEs
4.1.1 Parameterized Higher DimensionalODEs
4.1.2 Variational Equations
4.1.3 Melnikov Mappings
4.1.4 The Second Order Melnikov Function
4.1.5 Application to Periodically PerturbedODEs
4.2 ODEs with Nonresonant Center Manifolds
4.2.1 Parameterized Coupled Oscillators
4.2.2 Chaotic Dynamics on the HyperbolicSubspace
4.2.3 Chaos in the Full Equation
4.2.4 Applications to Nonlinear ODEs
4.3 ODEs with Resonant Center Manifolds
4.3.1 ODEs with Saddle-Center Parts
4.3.2 Example of Coupled Oscillators atResonance
4.3.3 General Equations
4.3.4 Averaging Method
4.4 Singularly Perturbed and Forced ODEs
4.4.1 Forced Singular ODEs
4.4.2 Center Manifold Reduction
4.4.3 ODEs with Normal and Slow Variables
4.4.4 Homoclinic Hopf Bifurcation
4.5 Bifurcation from Degenerate Homoclinics
4.5.1 Periodically Forced ODEs with DegenerateHomoclinics...
4.5.2 Bifurcation Equation
4.5.3 Bifurcation for 2-Parametric Systems
4.5.4 Bifurcation for 4-Parametric Systems
4.5.5 Autonomous Perturbations
4.6 Inflated ODEs
4.6.1 Inflated Carathtodory Type ODEs
4.6.2 Inflated Periodic ODEs
4.6.3 Inflated Autonomous ODEs
4.7 Nonlinear Diatomic Lattices
4.7.1 Forced and Coupled NonlinearLattices
4.7.2 Spatially Localized Chaos
References
5 Chaos in Partial Differential Equations
5.1 Beams on Elastic Bearings
5.1.1 Weakly Nonlinear Beam Equation
5.1.2 Setting of the Problem
5.1.3 Preliminary Results
5.1.4 Chaotic Solutions
5.1.5 Useful Numerical Estimates
5.1.6 Lipschitz Continuity
5.2 Infinite Dimensional Non-Resonant Systems
5.2.1 Buckled Elastic Beam
5.2.2 Abstract Problem
5.2.3 Chaos on the Hyperbolic Subspace
5.2.4 Chaos in the Full Equation
5.2.5 Applications to Vibrating ElasticBeams
5.2.6 Planer Motion with One Buckled Mode
5.2.7 Nonplaner Symmetric Beams
5.2.8 Nonplaner Nonsymmetric Beams
5.2.9 Multiple Buckled Modes
5.3 Periodically Forced Compressed Beam
5.3.1 Resonant Compressed Equation
5.3.2 Formulation of Weak Solutions
5.3.3 Chaotic Solutions
References
6 Chaos in Discontinuous Differential Equations
6.1 Transversal Homoclinic Bifurcation
6.1.1 Discontinuous Differential Equations
6.1.2 Setting of the Problem
6.1.3 Geometric Interpretation of NondegeneracyCondition..
6.1.4 Orbits Close to the Lower HomoclinicBranches
6.1.5 Orbits Close to the Upper HomoclinicBranch
6.1.6 Bifurcation Equation
6.1.7 Chaotic Behaviour
6.1.8 Almost and Quasiperiodic Cases
6.1.9 Periodic Case
6.1.10 Piecewise Smooth Planar Systems
6.1.11 3D Quasiperiodic Piecewise LinearSystems
6.1.12 Multiple Transversal Crossings
6.2 Sliding Homoclinic Bifurcation
6.2.1 Higher Dimensional SlidingHomoclinics
6.2.2 Planar Sliding Homoclinics
6.2.3 Three-Dimensional SlidingHomoclinics
6.3 Outlook
References
7 Concluding Related Topics
7.1 Notes on Melnikov Function
7.1.1 Role of Melnikov Function
7.1.2 Melnikov Function and Calculus ofResidues
7.1.3 Second Order ODEs
7.1.4 Applications and Examples
7.2 Transverse Heteroclinic Cycles
7.3 Blue Sky Catastrophes
7.3.1 Symmetric Systems with FirstIntegrals
7.3.2 D'Alembert and Penalized Equations
References
Index
