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Qualitative Theory of Differentiable Dynamical Systems书籍详细信息
- ISBN：9787030054371
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内容简介：
Professor Liao Shantao is an outstanding mathematician in China. As a result of his profound and systematic research work, he won the Mathematics Prize in 1985 which was the first time awarded by the Third World Academy of Sciences for his contributions to differentiable dynamical systems and other fields. He also won first-class award of National Natural Science Prize of China in 1987, The Chinese edition of this book collects eight of his major articles on differentiable dynamical systems in the period of 1963 - 1984, arranged in the order according to the dates of submission for publications, together with a preface and an appendix written also by him. Other two appendices A and B, which are original articles by Liao, are added to the translation to the English edition. .
This book introduces systematically the derivation of the two fundamental notions of "standard systems of differential equations" and "obstruction sets" and exhibits their important properties and applications to stability problems.
书籍目录：
Preface to the English Edition
Preface to the Chinese Edition.
Chapter 1 Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold
1.1 One-parameter Transformation Groups on Frame Bundles
1.2 Covariant Differentiation, the Function ωk(α)
1.3 The Function logζαk(t)
1.4 The Style Number k*(F)
1.5 On the Determination of the Style Number
1.6 Comparison of Certain Functions
1.7 3-dimensional Systems with Degenerate Style Numbers
1.8 Square Matrix Rα(t) and Divergence divS
References
Chapter 2 Standard Systems of Differential Equations
2.1 A Review of Standard Systems of Differential Equations
2.2 Another Class of Standard Systems of Differential Equations
2.3 Families Adp of Differential Equations
2.4 An Application
References
Chapter 3 Obstruction Sets and Strong Transversality Condition
3.1 Introduction
3.2 The Obstruction Set Ob(S)
3.3 Statement of Results
3.4 Trough Sets
References
Chapter 4 Obstruction Sets (I)
4.1 Trough Sets
4.2 Obstruction Set Ob(S)
4.3 Singularities
4.4 Linear Theory of Normal Sets
4.5 Linear Theory of Normal Sets (continued)
References.
Chapter 5 On the Stability Conjecture
5.1 Introduction and Statements of the Main Results
5.2 The Class X*(Mn) of Vector Fields
5.3 Contractible Periodic Orbits
5.4 The Case S ∈ X*(M3)
5.5 The "Sifting" Lemma and the Proof of Theorem 5.4.1
5.6 Proofs of Theorems 5.1.1 and 5.1.2
Appendix
References
Chapter 6 Obstruction Sets (II)
6.1 Introduction
6.2 Obstruction Sets and Minimal Rambling Sets
6.3 Simply Minimal Rambling Sets
6.4 The Set M(η,T;p) and the Hunched Set R(ζ,p) ∪ L(ζ,p) of S ∈ X*
6.5 Non-Simply Minimal Rambling Sets of S ∈ X* and Proofs of Theorem 6.1.1 and 6.1.2
6.6 On the Sets R(ζ,P) and L(ζ,p)
References
Chapter 7 Standard Systems of Differential Equations and Obstruction Sets with Applications to Structural Stability Problems
7.1 Global Linearization of Differential Systems and Expression by Linear Systems
7.2 Standard Systems of Equations
7.3 A Reduction to Lower Dimension by 1
7.4 Examples of Applications
7.5 The Family X* of Differential Systems
7.6 Obstruction Sets
7.7 Simple and Non-Simple Minimal Rambling Sets
7.8 Ω-Stability and Structural Stability
References
Chapter 8 On Characterizations of Structural Stability
8.1 Introduction
8.2 Preliminaries. Obstruction Sets and Minimal Rambling Sets
8.3 A Key Step
8.4 Applications
References
Appendix to the Chinese Edition
References
Supplement (October 1990)
Appendix A An Extension of the C1 Closing Lemma
A.1 Introduction
A.2 An Abstract of the Proof
A.3 Some Preliminaries
A.4 The Family of Squared Matrices θ(t) (0 ≤ t ＜ ∞o)
A.5 The Proof of Theorem I
A.6 The Proof of Theorem II
A.7 Standard Systems of Differential Equations
A.8 The Proof of the Main Theorem
A.9 The Case that ωa ∪ αa is Non-empty
References
Appendix B Obstruction Sets, Minimal Rambling Sets and Their Applications
B.1 Obstruction Sets
B.2 Relations between Obstruction Sets and Transversality Conditions
B.3 A Filtration for Normal Sets
B.4 Minimal Rambling Sets
B.5 The Families X**(Mn) and X(Mn)
B.6 On Stability of Differential Systems
References
Editor's Postscripts
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