| Preface | 8 |
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| Nomenclature | 12 |
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| 1 Dynamical systems | 18 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 5.1 Hyperbolic rest point | 84 |
| 5.2 Hyperbolic closed trajectory | 89 |
| 6 Transversality | 95 |
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| 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 |
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| 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 |
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| 9 Smale’s horseshoe and chaos | 143 |
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| 9.1 Smale’s horseshoe | 143 |
| 9.2 Chaotic sets | 148 |
| 9.3 Homoclinic points | 149 |
| 10 Closing Lemma | 152 |
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| 11 C0-generic properties of dynamical systems | 158 |
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| 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 |
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| 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 |
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| B Lectures on the history of differential equations and dynamical systems | 209 |
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| 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 |
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| Index | 244 |
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