Differential equations, dynamical systems, and an introduction to chaos:
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Amsterdam [u.a.]
Elsevier
2013
|
| Ausgabe: | 3. ed. |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025290065&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIV, 418 Seiten Illustrationen, Diagramme |
| ISBN: | 9780123820105 |
Internformat
MARC
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| 100 | 1 | |a Hirsch, Morris W. |d 1933- |e Verfasser |0 (DE-588)172139120 |4 aut | |
| 245 | 1 | 0 | |a Differential equations, dynamical systems, and an introduction to chaos |c Morris W. Hirsch ; Stephen Smale ; Robert L. Devaney |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 2013 | |
| 300 | |a XIV, 418 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Algebras, Linear | |
| 650 | 4 | |a Chaotic behavior in systems | |
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| 650 | 0 | 7 | |a Differentialgleichung |0 (DE-588)4012249-9 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Smale, Stephen |d 1930- |e Verfasser |0 (DE-588)119122472 |4 aut | |
| 700 | 1 | |a Devaney, Robert L. |d 1948- |e Verfasser |0 (DE-588)171063988 |4 aut | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-025290065 | |
Datensatz im Suchindex
| _version_ | 1819280285161750528 |
|---|---|
| adam_text | CHAPTER
CHAPTER
Contents
Preface to the Third Edition ix
Preface xi
First-Order Equations 1
1.1 The Simplest Example 1
1.2 The Logistic Population Model 4
1.3 Constant Harvesting and Bifurcations 7
1.4 Periodic Harvesting and Periodic
Solutions 10
1.5 Computing the Poincaré Map 11
1.6 Exploration: A Two-Parameter Family 15
Planar Linear Systems 21
2.1 Second-Order Differential Equations 23
2.2 Planar Systems 24
2.3 Preliminaries from Algebra 26
2.4 Planar Linear Systems 29
2.5 Eigenvalues and Eigenvectors 30
2.6 Solving Linear Systems 33
2.7 The Linearity Principle 36
iii
Contents
IV
CHAPTER 3 Phase Portraits for Planar Systems 39
3.1 Real Distinct Eigenvalues 39
3.2 Complex Eigenvalues 44
3.3 Repeated Eigenvalues 47
3.4 Changing Coordinates 49
CHAPTER 4 Classification of Planar Systems 61
4.1 The Trace-Determinant Plane 61
4.2 Dynamical Classification 64
4.3 Exploration: A 3D Parameter Space 71
CHAPTER 5 Higher-Dimensional Linear Algebra 73
5.1 Preliminaries from Linear Algebra 73
5.2 Eigenvalues and Eigenvectors 82
5.3 Complex Eigenvalues 85
5.4 Bases and Subspaces 88
5.5 Repeated Eigenvalues 93
5.6 Genericity 100
CHAPTER 6 Higher-Dimensional Linear Systems 107
6.1 Distinct Eigenvalues 107
6.2 Harmonic Oscillators 114
6.3 Repeated Eigenvalues 120
6.4 The Exponential of a Matrix 123
6.5 Nonautonomous Linear Systems 130
CHAPTER 7 Nonlinear Systems 139
7.1 Dynamical Systems 140
7.2 The Existence and Uniqueness
Theorem 142
7.3 Continuous Dependence of Solutions 147
7.4 The Variational Equation 149
7.5 Exploration: Numerical Methods 153
7.6 Exploration: Numerical Methods
and Chaos 156
CHAPTER 8 Equilibria in Nonlinear Systems 159
8.1 Some Illustrative Examples 159
8.2 Nonlinear Sinks and Sources 165
Contents
V
8.3 Saddles 168
8.4 Stability 174
8.5 Bifurcations 175
8.6 Exploration: Complex Vector Fields 182
CHAPTER 9 Global Nonlinear Techniques 187
9.1 Nullclines 187
9.2 Stability of Equilibria 192
9.3 Gradient Systems 202
9.4 Hamiltonian Systems 206
9.5 Exploration: The Pendulum with Constant
Forcing 209
CHAPTER 10 Closed Orbits and Limit Sets 213
10.1 Limit Sets 213
10.2 Local Sections and Flow Boxes 216
10.3 The Poincaré Map 218
10.4 Monotone Sequences in Planar Dynamical
Systems 220
10.5 The Poincaré-Bendixson Theorem 222
10.6 Applications of Poincaré-Bendixson 225
10.7 Exploration: Chemical Reactions that
Oscillate 228
CHAPTER 11 Applications in Biology 233
11.1 Infectious Diseases 233
11.2 Predator-Prey Systems 237
11.3 Competitive Species 244
11.4 Exploration: Competition and
Harvesting 250
11.5 Exploration: Adding Zombies to the SIR
Model 251
CHAPTER 12 Applications in Circuit Theory 257
12.1 An RLC Circuit 257
12.2 The Liénard Equation 261
12.3 The van der Pol Equation 263
12.4 A Hopf Bifurcation 270
12.5 Exploration: Neurodynamics 272
VI
Contents
CHAPTER 13 Applications in Mechanics 277
13.1 Newton s Second Law 277
13.2 Conservative Systems 280
13.3 Central Force Fields 282
13.4 The Newtonian Central Force
System 285
13.5 Kepler s First Law 290
13.6 The Two-Body Problem 293
13.7 Blowing Up the Singularity 294
13.8 Exploration: Other Central Force
Problems 298
13.9 Exploration: Classical Limits of Quantum
Mechanical Systems 299
13.10 Exploration: Motion of a Glider 301
CHAPTER 14 The Lorenz System 305
14.1 Introduction 306
14.2 Elementary Properties of the Lorenz
System 308
14.3 The Lorenz Attractor 312
14.4 A Model for the Lorenz Attractor 316
14.5 The Chaotic Attractor 321
14.6 Exploration: The Rössler Attractor 326
CHAPTER 15 Discrete Dynamical Systems 329
15.1 Introduction 329
15.2 Bifurcations 334
15.3 The Discrete Logistic Model 337
15.4 Chaos 340
15.5 Symbolic Dynamics 344
15.6 The Shift Map 349
15.7 The Cantor Middle-Thirds Set 351
15.8 Exploration: Cubic Chaos 354
15.9 Exploration: The Orbit Diagram 355
CHAPTER 16 Homoclinic Phenomena 361
16.1 The Shilnikov System 361
16.2 The Horseshoe Map 368
16.3 The Double Scroll Attractor 375
Contents
16.4 Homoclinic Bifurcations 377
16.5 Exploration: The Chua Circuit 381
CHAPTER 17 Existence and Uniqueness Revisited 385
17.1 The Existence and Uniqueness
Theorem 385
17.2 Proof of Existence and Uniqueness 387
17.3 Continuous Dependence on Initial
Conditions 394
17.4 Extending Solutions 397
17.5 IMonautonomous Systems 401
17.6 Differentiability of the Flow 404
Bibliography 411
Index 415
|
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| author | Hirsch, Morris W. 1933- Smale, Stephen 1930- Devaney, Robert L. 1948- |
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| discipline | Mathematik |
| edition | 3. ed. |
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| id | DE-604.BV040442254 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T16:15:05Z |
| institution | BVB |
| isbn | 9780123820105 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025290065 |
| oclc_num | 795554533 |
| open_access_boolean | |
| owner | DE-384 DE-20 DE-83 DE-19 DE-BY-UBM DE-11 DE-473 DE-BY-UBG DE-898 DE-BY-UBR DE-739 DE-29T |
| owner_facet | DE-384 DE-20 DE-83 DE-19 DE-BY-UBM DE-11 DE-473 DE-BY-UBG DE-898 DE-BY-UBR DE-739 DE-29T |
| physical | XIV, 418 Seiten Illustrationen, Diagramme |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Elsevier |
| record_format | marc |
| spellingShingle | Hirsch, Morris W. 1933- Smale, Stephen 1930- Devaney, Robert L. 1948- Differential equations, dynamical systems, and an introduction to chaos Differential equations Algebras, Linear Chaotic behavior in systems Dynamisches System (DE-588)4013396-5 gnd Differentialgleichung (DE-588)4012249-9 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4012249-9 (DE-588)4035811-2 |
| title | Differential equations, dynamical systems, and an introduction to chaos |
| title_auth | Differential equations, dynamical systems, and an introduction to chaos |
| title_exact_search | Differential equations, dynamical systems, and an introduction to chaos |
| title_full | Differential equations, dynamical systems, and an introduction to chaos Morris W. Hirsch ; Stephen Smale ; Robert L. Devaney |
| title_fullStr | Differential equations, dynamical systems, and an introduction to chaos Morris W. Hirsch ; Stephen Smale ; Robert L. Devaney |
| title_full_unstemmed | Differential equations, dynamical systems, and an introduction to chaos Morris W. Hirsch ; Stephen Smale ; Robert L. Devaney |
| title_short | Differential equations, dynamical systems, and an introduction to chaos |
| title_sort | differential equations dynamical systems and an introduction to chaos |
| topic | Differential equations Algebras, Linear Chaotic behavior in systems Dynamisches System (DE-588)4013396-5 gnd Differentialgleichung (DE-588)4012249-9 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Differential equations Algebras, Linear Chaotic behavior in systems Dynamisches System Differentialgleichung Lineare Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025290065&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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