The modern study of Differentiable Dynamical Systems began in the early sixties of this century. This began with the articles of M. Peixoto in 1959 and 1962, which deal with structural stability of 2-dimensional differential systems. The articles retreat the results announced by A. Andronov and L. Pontrjagin in 1937, extend them to closed surfaces, and add to them a new content concerning density properties. This attracted people's attention. It was mentioned in the introductionof his first article that "-- .Therefore it seems natural to expect that a fruitful field of research lies in this direction". A natural question hence arisen: How about the case of dimensions higher than two? Since then, early or late, a number of mathematicians in the world, especially S. Smale, started their important research and probe on this topic.

 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.

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 wk(a) 

1.3 The Function log sk(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 Ra(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 Mn 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 ? of Vector Fields

5.3 Contractible Periodic Orbits

5.4 The Case?

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 ? and the Hunched Set ? of ?     

6.5 Non-Simply Minimal Rambling Sets of ? and

   Proofs of Theorem 6.1.1 and 6.1.2

6.6 On the Sets R(s,p) and L(s,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 2'* of Differential Systems

7.6 Obstruction Sets 

7.7 Simple and Non-Simple Minimal Rambling Sets

7.8 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          

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 ? 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