A modern introduction to dynamical systems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Oxford University Press
2018
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| Ausgabe: | First edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverzeichnis Seite [401]-403 |
| Beschreibung: | xvi, 408 Seiten Illustrationen, Diagramme (überwiegend farbig) 24 cm |
| ISBN: | 9780198743286 9780198743279 |
Internformat
MARC
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| 245 | 1 | 0 | |a A modern introduction to dynamical systems |c Richard J. Brown |
| 250 | |a First edition | ||
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents 1 What Is a Dynamical System? 1.1 Definitions 1 1 1.1.1 Ordinary Differential Equations (ODEs) 3 1.1.2 1.1.3 1.1.4 1.1.5 6 7 Maps Symbolic Dynamics Billiards Higher-Order Recursions 1.2 The Viewpoint S 9 18 2 Simple Dynamics 21 2.1 Preliminaries 21 2.1.1 2.1.2 2.1.3 2.1.4 A Simple System The Time-է Map Metrics on Sets Lipschitz Continuity 2.2 The Contraction Principle 2.2.1 Contractions on Intervals 2.2.2 Contractions in Several Variables 2.2.3 Application: The Newton-Raphson Method 2.2.4 Application: Existence and Uniqueness of ODE Solutions 2.2.5 Application: Heron ofAlexandria 2.3 Interval Maps 2.3.1 Cobwebbing 2.3.2 Fixed-Point Stability 2.3.3 Monotonie Maps 2.3.4 Homoclinic/Heteroclinic Points 2.4 Bifurcations of Interval Maps 2.4.1 Saddle-Node Bifiircation 2.4.2 Transcriticai Bifurcation 2.4.3 Pitchfork Bifurcation 2.5 First Return Maps 2.6 A Quadratic Interval Map: The Logistic Map 21 23 30 33 37 38 40 42 44 48 50 51 52 56 59 64 67 70 73 76 79
xiv I Contents 3 The Objects of Dynamics 87 3.1 Topology on Sets 3.2 More on Metrics 3.2.1 More on Lipschitz Continuity 3.2.2 Metric Equivalence 3.2.3 Fixed-Point Theorems 3.3 Some Non-Euclidean Metric Spaces 3.3.1 Then-Sphere 3.3.2 The Unit Circle 3.3.3 The Cylinder 3.3.4 The 2-Torus 3.4 A Cantor Set 3.4.1 The Koch Curve 3.4.2 Sierpiński Carpet 3.4.3 The Sponges 87 92 93 94 96 100 101 102 106 106 107 112 113 115 4 Flows and Maps of Euclidean Space 117 4.1 Linear, First-order ODE Systems in the Plane 4.1.1 General Homogeneous, Linear Systems in Euclidean Space 4.1.2 Autonomous Linear Systems 4.1.3 The Matrix Exponential 4.1.4 TWo-Dimensional Classification 4.2 Bifurcations in Linear Planar Systems 4.2.1 Linearized Poincaré-Andronov-Hopf Bifurcation 4.2.2 Saddle-Node Biforcation 4.3 Linear Planar Maps 4.3.1 Nodes: Sinks and Sources 4.3.2 Star or Proper Nodes 4.3.3 Degenerate or Improper Nodes 4.3.4 Spirals and Centers 4.3.5 Saddle Points 4.4 Linear Flows versus Linear Maps 4.5 Local Linearization and Stability of Equilibria 4.6 Isolated Periodic Orbit Stability 4.6.1 The Poincaré-Bendixson Theorem 4.6.2 Limit Sets of Flows 4.6.3 Flows in the Plane 4.6.4 Application: The van der Pol Oscillator 4.6.5 The Poincaré-Andronov-Hopf Biforcation 4.7 Application: Competing Species 4.7.1 The Fixed Points 4.7.2 Type and Stability 117 117 121 123 126 135 135 137 139 144 145 146 148 148 155 160 169 171 175 177 179 184 187 191 192
Contents I xv 5 Recurrence 5.1 Rotations of the circle 195 196 5.1.1 Continued Fraction Representation 199 5.2 Equidistribution and Weyl’s Theorem 204 5.2.1 Application: Periodic Function Reconstruction via Sampling 5.3 Linear Flows on the Torus 5.3.1 Application: Lissajous Figures 5.3.2 Application: A Polygonal Billiard 5.4 Toral Translations 5.5 Invertible Circle Maps 6 Phase Volume Preservation 6.1 Incompressibility 6.2 Newtonian Systems of Classical Mechanics 209 214 218 220 225 228 237 237 241 6.2.1 Generating Flows from Functions: Lagrange 6.2.2 Generating Flows from Functions: Hamilton 243 246 6.2.3 6.2.4 6.2.5 6.2.6 250 253 256 259 Exact Differential Equations Application: The Planar Pendulum First Integrals Application: The Spherical Pendulum 6.3 Poincaré Recurrence 6.3.1 Non-Wandering Points 6.3.2 The Poincaré Recurrence Theorem 261 262 265 6.4 Billiards 265 6.4.1 6.4.2 6.4.3 6.4.4 6.4.5 266 267 272 278 279 Circular Billiards Elliptic Billiards General Convex Billiards Poincaré’s Last Geometric Theorem Application: Pitcher Problems 7 Complicated Orbit Structure 7.1 Counting Periodic Orbits 7.2 7.3 7.4 7.5 283 283 7.1.1 The Quadratic Map: Beyond 4 7.1.2 Hyperbolic Toral Automorphisms 7.1.3 Application: Image Restoration 287 293 301 7.1.4 Inverse Limit Spaces 7.1.5 Shift Spaces 7.1.6 Markov Partitions 303 307 316 7.1.7 Application: The Baker’s Transformation 326 Two-Dimensional Markov Partitions: Amol’ds Cat Map Chaos and Mixing Sensitive Dependence on Initial Conditions Quadratic Maps: The Final Interval 329 334 339 342
xvi I Contents 7.5.1 Period-Doubling Bifurcation 7.5.2 The Schwarzian Derivative 7.5.3 Sharkovskiis Theorem 7.6 Two More Examples of Complicated Dynamical Systems 7.6.1 Complex Dynamics 7.6.2 Smale Horseshoe 8 Dynamical Invariants 345 353 357 358 358 362 367 8.1 Topological Conjugacy 368 8.1.1 Conjugate Maps 368 8.1.2 Conjugate Flows 8.1.3 Conjugacy as Classification 8.2 Topological Entropy 8.2.1 Lyapunov Exponents 8.2.2 Capacity 8.2.3 Box Dimension 8.2.4 Bowen-Dinaburg (Metric) Topological Entropy Bibliography Index 378 379 380 380 384 385 388 401 405
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| author | Brown, Richard J. 1962- |
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| discipline | Mathematik |
| edition | First edition |
| format | Book |
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| indexdate | 2024-07-10T08:07:48Z |
| institution | BVB |
| isbn | 9780198743286 9780198743279 |
| language | English |
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| spelling | Brown, Richard J. 1962- (DE-588)1089796943 aut A modern introduction to dynamical systems Richard J. Brown First edition Oxford Oxford University Press 2018 xvi, 408 Seiten Illustrationen, Diagramme (überwiegend farbig) 24 cm txt rdacontent n rdamedia nc rdacarrier Literaturverzeichnis Seite [401]-403 Differenzierbares dynamisches System (DE-588)4137931-7 gnd rswk-swf Differential dynamical systems Differenzierbares dynamisches System (DE-588)4137931-7 s DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030462535&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Brown, Richard J. 1962- A modern introduction to dynamical systems Differenzierbares dynamisches System (DE-588)4137931-7 gnd |
| subject_GND | (DE-588)4137931-7 |
| title | A modern introduction to dynamical systems |
| title_auth | A modern introduction to dynamical systems |
| title_exact_search | A modern introduction to dynamical systems |
| title_full | A modern introduction to dynamical systems Richard J. Brown |
| title_fullStr | A modern introduction to dynamical systems Richard J. Brown |
| title_full_unstemmed | A modern introduction to dynamical systems Richard J. Brown |
| title_short | A modern introduction to dynamical systems |
| title_sort | a modern introduction to dynamical systems |
| topic | Differenzierbares dynamisches System (DE-588)4137931-7 gnd |
| topic_facet | Differenzierbares dynamisches System |
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