This book is devoted to the comprehensive bifurcation theory of chaos in nonlinear dynamical systems with applications to mechanics and vibrations. Precise and complete proofs of derived mathematical results are presented with many stimulating and illustrative examples. I study bifurcations of chaotic solutions for perturbed problems from either homoclinic or heteroclinic orbits of unperturbed ones. This method is also known as the Melnikov-type approach. Certainly there are many interesting books in this direction, but all results of this book have not yet been published in any book, since I have collected some results of mine together with my coauthors appeared only in articles and manuscripts. So I hope that this book is a useful contribution to a rapidly developing theory of chaos and it is a good continuation of my recently published book in Springer with similar topics.

费坎编著的《不连续及连续系统中的分岔和混沌》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。

《不连续及连续系统中的分岔和混沌》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。

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