Spaces of Dynamical Systems

Spaces of Dynamical Systems

by Sergei Yu. Pilyugin

Hardcover

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Product Details

ISBN-13: 9783110255959
Publisher: De Gruyter
Publication date: 03/15/2012
Series: De Gruyter Studies in Mathematical Physics Series , #3
Pages: 244
Product dimensions: 6.69(w) x 9.45(h) x 0.03(d)
Age Range: 18 Years

About the Author

Sergei Yu. Pilyugin, St. Petersburg State University, Russia.

Table of Contents

Preface vii

Nomenclature xi

1 Dynamical systems 1

1.1 Main definitions 1

1.2 Embedding of a discrete dynamical system into a flow 10

1.3 Local Poincare diffeomorphism 11

1.4 Time-periodic systems of differential equations 14

1.5 Action of an Abelian group 15

2 Topologies on spaces of dynamical systems 16

2.1 C0-topology 16

2.2 C1-topology 17

2.3 Metrics on the space of systems of differential equations 18

2.4 Generic properties 24

2.5 Immersions and embeddings 24

3 Equivalence relations 26

3.1 Topological conjugacy 26

3.2 Topological equivalence of flows 30

3.3 Nonwandering set 30

3.4 Local equivalence 36

4 Hyperbolic fixed point 37

4.1 Hyperbolic linear mapping 37

4.2 The Grobman-Hartman theorem 40

4.3 Neighborhood of a hyperbolic fixed point 48

4.4 The stable manifold theorem 53

4.5 Hyperbolic periodic point 65

5 Hyperbolic rest point and hyperbolic closed trajectory 67

5.1 Hyperbolic rest point 67

5.2 Hyperbolic closed trajectory 72

6 Transversality 78

6.1 Transversality of mappings and submanifolds 78

6.2 Transversality condition 80

6.3 Palis lemma 82

6.4 Transversality and hyperbolicity for one-dimensional mappings 90

7 Hyperbolic sets 92

7.1 Definition of a hyperbolic set 92

7.2 Examples of hyperbolic sets 94

7.3 Basic properties of hyperbolic sets 97

7.4 Stable manifold theorem 101

7.5 Axiom A 103

7.6 Hyperbolic sets of flows 112

8 Anosov diffeomorphisms 119

9 Smale's horseshoe and chaos 126

9.1 Smale's horseshoe 126

9.2 Chaotic sets 131

9.3 Homoclinic points 132

10 Closing Lemma 135

11 C0-generic properties of dynamical systems 141

11.1 Hausdorff metric 141

11.2 Semicontinuous mappings 142

11.3 Tolerance stability and Takens' theory 143

11.4 Attractors of dynamical systems 147

12 Shadowing of pseudotrajectories in dynamical systems 159

12.1 Definitions and results 159

12.2 Proof of Theorem 12.1 164

12.3 Proof of Theorem 12.2 172

12.4 Proof of Theorem 12.3 175

A Scheme of the proof of the Mane theorem 181

B Lectures on the history of differential equations and dynamical systems 192

B.1 Differential equations and Newton's anagram 192

B.2 Development of the general theory 194

B.3 Linear equations and systems 198

B.4 Stability 203

B.5 Nonlocal qualitative theory. Dynamical systems 210

B.6 Structural stability 214

B.7 Dynamical systems with chaotic behavior 217

Bibliography 223

Index 227

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