不连续及连续系统中的分岔和混沌

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 Equatio

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-Dime ional 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 Tra ve ality

2.6.4 Malgrange Preparation Theorem

2.6.5 Complex Analysis

References

3 Chaos in Discrete Dynamical Systems

3.1 Tra ve al Bounded Solutio

3.1.1 Difference Equatio

3.1.2 Variational Equation

3.1.3 Perturbation Theory

3.1.4 Bifurcation from a Manifold of Homoclinic Solutio

3.1.5 Applicatio to Impulsive Differential Equatio

3.2 Tra ve al Homoclinic Orbits

3.2.1 Higher Dime ional Difference Equatio

3.2.2 Bifurcation Result

3.2.3 Applicatio 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 Equatio

4.1 Higher Dime ional ODEs

4.1.1 Parameterized Higher Dime ional ODEs

4.1.2 Variational Equatio

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 Oscillato

4.2.2 Chaotic Dynamics on the Hyperbolic Subspace

4.2.3 Chaos in the Full Equation

4.2.4 Applicatio to Nonlinear ODEs

4.3 ODEs with Resonant Center Manifolds

4.3.1 ODEs with Saddle-Center Parts

4.3.2 Example of Coupled Oscillato at Resonance

4.3.3 General Equatio

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 Perturbatio

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 Equatio

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 Solutio

5.1.5 Useful Numerical Estimates

5.1.6 Lipschitz Continuity

5.2 Infinite Dime ional 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 Applicatio to Vibrating Elastic Beams

5.2.6 Planer Motion with One Buckled Mode

5.2.7 Nonplaner Symmetric Beams

5.2.8 Nonplaner No ymmetric 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 Solutio

5.3.3 Chaotic Solutio

References

6 Chaos in Discontinuous Differential Equatio

6.1 Tra ve al Homoclinic Bifurcation

6.1.1 Discontinuous Differential Equatio

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 Tra ve al Crossings

6.2 Sliding Homoclinic Bifurcation

6.2.1 Higher Dime ional Sliding Homoclinics

6.2.2 Planar Sliding Homoclinics

6.2.3 Three-Dime ional 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 Applicatio and Examples

7.2 Tra ve e Heteroclinic Cycles

7.3 Blue Sky Catastrophes

7.3.1 Symmetric Systems with Fi t Integrals

7.3.2 D'Alembert and Penalized Equatio

References

Index

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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

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将smale马蹄理论推广到了膨胀动力系统