Spaces of Dynamical Systems (eBook)

lt;!doctype html public "-//w3c//dtd html 4.0 transitional//en">

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

Preface 8
Nomenclature 12
1 Dynamical systems 18
1.1 Main definitions 18
1.2 Embedding of a discrete dynamical system into a flow 27
1.3 Local Poincaré diffeomorphism 28
1.4 Time-periodic systems of differential equations 31
1.5 Action of an Abelian group 32
2 Topologies on spaces of dynamical systems 33
2.1 C0-topology 33
2.2 C1-topology 34
2.3 Metrics on the space of systems of differential equations 35
2.4 Generic properties 41
2.5 Immersions and embeddings 41
3 Equivalence relations 43
3.1 Topological conjugacy 43
3.2 Topological equivalence of flows 47
3.3 Nonwandering set 47
3.4 Local equivalence 53
4 Hyperbolic fixed point 54
4.1 Hyperbolic linear mapping 54
4.2 The Grobman-Hartman theorem 57
4.3 Neighborhood of a hyperbolic fixed point 65
4.4 The stable manifold theorem 70
4.5 Hyperbolic periodic point 82
5 Hyperbolic rest point and hyperbolic closed trajectory 84
5.1 Hyperbolic rest point 84
5.2 Hyperbolic closed trajectory 89
6 Transversality 95
6.1 Transversality of mappings and submanifolds 95
6.2 Transversality condition 97
6.3 Palis lemma 99
6.4 Transversality and hyperbolicity for one-dimensional mappings 107
7 Hyperbolic sets 109
7.1 Definition of a hyperbolic set 109
7.2 Examples of hyperbolic sets 111
7.3 Basic properties of hyperbolic sets 114
7.4 Stable manifold theorem 118
7.5 Axiom A 120
7.6 Hyperbolic sets of flows 129
8 Anosov diffeomorphisms 136
9 Smale’s horseshoe and chaos 143
9.1 Smale’s horseshoe 143
9.2 Chaotic sets 148
9.3 Homoclinic points 149
10 Closing Lemma 152
11 C0-generic properties of dynamical systems 158
11.1 Hausdorff metric 158
11.2 Semicontinuous mappings 159
11.3 Tolerance stability and Takens’ theory 160
11.4 Attractors of dynamical systems 164
12 Shadowing of pseudotrajectories in dynamical systems 176
12.1 Definitions and results 176
12.2 Proof of Theorem 12.1 181
12.3 Proof of Theorem 12.2 189
12.4 Proof of Theorem 12.3 192
A Scheme of the proof of the Mane theorem 198
B Lectures on the history of differential equations and dynamical systems 209
B.1 Differential equations and Newton’s anagram 209
B.2 Development of the general theory 211
B.3 Linear equations and systems 215
B.4 Stability 220
B.5 Nonlocal qualitative theory. Dynamical systems 227
B.6 Structural stability 231
B.7 Dynamical systems with chaotic behavior 234
Bibliography 240
Index 244

lt;P>"A book based on 30 years of teaching experience and therefore didactically and logically composed."
In: Enseignement Mathematique 2/2012

"In the reviewer's opinion, in view of the numerical study of nonlinear systems, this book would be very useful to, in particular, engineers, physicists and economists who want to study the systems in a numerical way." Mathematical Reviews

"The main strength of this book is that the author includes complete proofs of results that other publications frequently leave to the reader. […] Altogether, this text is for serious readers that would like a clean and rigorous introduction to the results and methods of proof in dynamical systems. It would make a valuable addition as a reference text to the collection of any academician working in related fields." Zentralblatt für Mathematik