非线性振动，动力学系统和矢量场的分叉

Problems in dynamics have fascinated physical scientists (and mankind in general) for thousands of years. Notable among such problems are those of celestial mechanics, especially the study of the motions of the bodies in the solar system. Newton's attempts to understand and model their observed motions incorporated Kepler's laws and led to his development of the calculus. With ...

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Problems in dynamics have fascinated physical scientists (and mankind in general) for thousands of years. Notable among such problems are those of celestial mechanics, especially the study of the motions of the bodies in the solar system. Newton's attempts to understand and model their observed motions incorporated Kepler's laws and led to his development of the calculus. With this the study of models of dynamical problems as differential equations began.

　　本书为英文版。

Contents
CHAPTER1
Introduction: Differential Equations and Dynamical Systems
1.0. Existence and Uniqueness ofSolutions
1.1. The Linear System x = Ax
1.2. Flows and Invariant Subspaces
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Contents
CHAPTER1
Introduction: Differential Equations and Dynamical Systems
1.0. Existence and Uniqueness ofSolutions
1.1. The Linear System x = Ax
1.2. Flows and Invariant Subspaces
1.3. The Nonlinear System x = f(x)
1.4. Linear and Nonlinear Maps
1.5. Closed Orbits. Poincare Maps and Forced Oscillations
1.6. Asymptotic Behavior
1.7. Equivalence Relations and Structural Stability
1.8. Two-Dimensional Flows
1.9. Peixoto's Theorem for Two-Dimensional Flows
CHAPTER 2
An Introduction to Chaos: Four Examples
2.1. Van der Pol's Equation
2.2. Duffing's Equation
2.3. The Lorenz Equations
2.4. The Dynamics ofa Bouncing Ball
2.5. Conclusions: The Moral ofthe Tales
CHAPTER 3
Local Bifurcations
3.1. Bifurcation Problems
3.2. Center Manifolds
3.3. Normal Forms
3.4. Codimension One Bifurcations of Equilibria
3.5. Codimension One Bifurcations ofMaps and Periodic Orbits
CHAPTER 4
Averaging and Perturbation from a Geometric Viewpoint
4.1. Averaging and Poincare Maps
4.2. Examples of Averaging
4.3. Averaging and Local Bifureations
4.4. Averaging, Hamiltonian Systems, and Global Behavior:
Cautionary Notes
4.5. Melnikov's Method: Perturbations ofPlanar Homoclinic Orbits
4.6. Melnikov's Method: Perturbations of Hamiltonian Systems and
Subharmonic Orbits
4.7. Stability of Subharmonic Orbits
4.8. Two Degree of Freedom Hamiltonians and Area Preserving Maps
of the Plane
CHAPTER 5
Hyperbolic Sets, Symbolic Dynamics, and Strange Attractors
5.0. Introduction
5.1. The Smale Horseshoe: An Example ofa Hyperbolic Limit Set
5.2. Invariant Sets and Hyperbolicity
5.3. Markov Partitions and Symbolic Dynamics
5.4. Strange Attractors and the Stability Dogma
5.5. Structurally Stable Attractors
5.6. One-Dimensional Evidence for Strange Attractors
5.7. The Geometric Lorenz Attractor
5.8. Statistical Propenies: Dimension. Entropy and Liapunov Exponents
CHAPTER 6
Global Bifurcations
6.1. Saddle Connections
6.2. Rotation Numbers
6.3. Bifurcations of One-Dimensional Maps
6.4. The Lorenz Bifurcations
6.5. Homoclinic Orbits in Three-Dimensional Flows: Silnikov's Example
6.6. Homoclinic Bifurcations of Periodic Orbits
6.7. Wild Hyperbolic Sets
6.8. Renormalization and Universality
CHAPTER7
Local Codimension Two Bifurcations of Flows
7.1. Degeneracy in Higher-Order Terms
7.2. A Note on k-Sels and Determinacy
7.3. The Double Zero Eigenvalue
7.4. A Pure Imaginary Pair and a Simple Zero Eigenvalue
7.5. Two Pure Imaginary Pairs of Eigenvalues without Resonance
7.6. Applications to Large Systems
APPENDIX
Suggestions for Further Reading
Postscript Added at Second Printing
Glossary
References
Index
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