不连续及连续系统中的分岔和混沌

出版时间：2011-3  出版社：高等教育出版社  作者：费坎  页数：378  

内容概要

　　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。

书籍目录

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 Homoclinic
Solutions
 3.1.5 Applications to Impulsive Differential
Equations
 3.2 Transversal Homoclinic Orbits
 3.2.1 Higher Dimensional Difference
Equations
 3.2.2 Bifurcation Result
 3.2.3 Applications to McMillan Type
Mappings
 3.2.4 Planar Integrable Maps with
Separatrices
 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 with
impulses
 3.4.2 Melnikov Function
 3.4.3 Second Order Singularly Perturbed ODEs
with 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 Dimensional
ODEs
 4.1.2 Variational Equations
 4.1.3 Melnikov Mappings
 4.1.4 The Second Order Melnikov Function
 4.1.5 Application to Periodically Perturbed
ODEs
 4.2 ODEs with Nonresonant Center Manifolds
 4.2.1 Parameterized Coupled Oscillators
 4.2.2 Chaotic Dynamics on the Hyperbolic
Subspace
 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 at
Resonance
 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 Degenerate
Homoclinics...
 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 Nonlinear
Lattices
 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 Elastic
Beams
 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 Nondegeneracy
Condition..
 6.1.4 Orbits Close to the Lower Homoclinic
Branches
 6.1.5 Orbits Close to the Upper Homoclinic
Branch
 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 Linear
Systems
 6.1.12 Multiple Transversal Crossings
 6.2 Sliding Homoclinic Bifurcation
 6.2.1 Higher Dimensional Sliding
Homoclinics
 6.2.2 Planar Sliding Homoclinics
 6.2.3 Three-Dimensional Sliding
Homoclinics
 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 of
Residues
 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 First
Integrals
 7.3.2 D'Alembert and Penalized Equations
 References
Index

编辑推荐

　　系统介绍非线性动力系统中的混沌理论及其在力学与振动中的应用　　详细讨论不连续动力系统中的混沌与分岔　　给出了简明扼要的数学证明　　提供了大量有趣而直观的例子　　给出stick—slip系统混沌存在性的严格证明　　将smale马蹄理论推广到了膨胀动力系统

用户评论 (总计1条)

•   这本书是一本非常好的专业书籍，对力学工作者来说是很难得的好书！