Spaces of dynamical systems

Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface......Page 8
Nomenclature......Page 12
Contents......Page 14
1.1 Main definitions......Page 18
1.2 Embedding of a discrete dynamical system into a flow......Page 27
1.3 Local Poincaré diffeomorphism......Page 28
1.4 Time-periodic systems of differential equations......Page 31
1.5 Action of an Abelian group......Page 32
2.1 C0-topology......Page 33
2.2 C1-topology......Page 34
2.3 Metrics on the space of systems of differential equations......Page 35
2.5 Immersions and embeddings......Page 41
3.1 Topological conjugacy......Page 43
3.3 Nonwandering set......Page 47
3.4 Local equivalence......Page 53
4.1 Hyperbolic linear mapping......Page 54
4.2 The Grobman-Hartman theorem......Page 57
4.3 Neighborhood of a hyperbolic fixed point......Page 65
4.4 The stable manifold theorem......Page 70
4.5 Hyperbolic periodic point......Page 82
5.1 Hyperbolic rest point......Page 84
5.2 Hyperbolic closed trajectory......Page 89
6.1 Transversality of mappings and submanifolds......Page 95
6.2 Transversality condition......Page 97
6.3 Palis lemma......Page 99
6.4 Transversality and hyperbolicity for one-dimensional mappings......Page 107
7.1 Definition of a hyperbolic set......Page 109
7.2 Examples of hyperbolic sets......Page 111
7.3 Basic properties of hyperbolic sets......Page 114
7.4 Stable manifold theorem......Page 118
7.5 Axiom A......Page 120
7.6 Hyperbolic sets of flows......Page 129
8 Anosov diffeomorphisms......Page 136
9.1 Smale’s horseshoe......Page 143
9.2 Chaotic sets......Page 148
9.3 Homoclinic points......Page 149
10 Closing Lemma......Page 152
11.1 Hausdorff metric......Page 158
11.2 Semicontinuous mappings......Page 159
11.3 Tolerance stability and Takens’ theory......Page 160
11.4 Attractors of dynamical systems......Page 164
12.1 Definitions and results......Page 176
12.2 Proof of Theorem 12.1......Page 181
12.3 Proof of Theorem 12.2......Page 189
12.4 Proof of Theorem 12.3......Page 192
A Scheme of the proof of the Mane theorem......Page 198
B.1 Differential equations and Newton’s anagram......Page 209
B.2 Development of the general theory......Page 211
B.3 Linear equations and systems......Page 215
B.4 Stability......Page 220
B.5 Nonlocal qualitative theory. Dynamical systems......Page 227
B.6 Structural stability......Page 231
B.7 Dynamical systems with chaotic behavior......Page 234
Bibliography......Page 240
Index......Page 244