不連續(xù)及連續(xù)系統(tǒng)中的分岔和混沌

內(nèi)容概要

　　本書(shū)利用泛函分析工具來(lái)談?wù)摶煦缗c分岔，并提供簡(jiǎn)明扼要的數(shù)學(xué)證明。書(shū)中通過(guò)許多有趣、經(jīng)典的例子展示了其具體的應(yīng)用。本書(shū)研究了大量的非線性問(wèn)題，包括非線性差分方程、常微分方程和偏微分方程、脈沖微分方程、分段光滑微分方程及在無(wú)限格上的微分方程等。
　　本書(shū)可供對(duì)非線性機(jī)械系統(tǒng)的振動(dòng)、弦或梁的擺動(dòng)以及應(yīng)用動(dòng)力系統(tǒng)中分岔方法來(lái)研究電路等問(wèn)題感興趣的數(shù)學(xué)家、物理學(xué)家、工程師及相關(guān)專業(yè)研究生等參考。

書(shū)籍目錄

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

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　　系統(tǒng)介紹非線性動(dòng)力系統(tǒng)中的混沌理論及其在力學(xué)與振動(dòng)中的應(yīng)用　　詳細(xì)討論不連續(xù)動(dòng)力系統(tǒng)中的混沌與分岔　　給出了簡(jiǎn)明扼要的數(shù)學(xué)證明　　提供了大量有趣而直觀的例子　　給出stick—slip系統(tǒng)混沌存在性的嚴(yán)格證明　　將smale馬蹄理論推廣到了膨脹動(dòng)力系統(tǒng)

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