Dynamical systems: differential equations, maps and chaotic behaviour
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London u.a.
Chapman & Hall
1992
|
| Edition: | 1. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 330 S. zahlr. graph. Darst. |
| ISBN: | 0412390701 0412390809 |
Staff View
MARC
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| 100 | 1 | |a Arrowsmith, David K. |e Verfasser |0 (DE-588)113968876 |4 aut | |
| 245 | 1 | 0 | |a Dynamical systems |b differential equations, maps and chaotic behaviour |c D. K. Arrowsmith and C. M. Place |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a London u.a. |b Chapman & Hall |c 1992 | |
| 300 | |a X, 330 S. |b zahlr. graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface ix
1 Introduction 1
1.1 Preliminary ideas 1
1.1.1 Existence and uniqueness 1
1.1.2 Geometrical representation 3
1.2 Autonomous equations 6
1.2.1 Solution curves and the phase portrait 6
1.2.2 Phase portraits and dynamics 11
1.3 Autonomous systems in the plane 12
1.4 Construction of phase portraits in the plane 17
1.4.1 Use of calculus 17
1.4.2 Isoclines 20
1.5 Flows and evolution 23
Exercises 27
2 Linear systems 35
2.1 Linear changes of variable 35
2.2 Similarity types for 2 x 2 real matrices 38
2.3 Phase portraits for canonical systems in the plane 43
2.3.1 Simple canonical systems 43
2.3.2 Non simple canonical systems 46
2.4 Classification of simple linear phase portraits in the plane 48
2.4.1 Phase portrait of a simple linear system 48
2.4.2 Types of canonical system and qualitative equivalence 50
2.4.3 Classification of linear systems 52
2.5 The evolution operator 52
2.6 Affine systems 55
2.7 Linear systems of dimension greater than two 57
2.7.1 Three dimensional systems 57
vi Contents
2.7.2 Four dimensional systems 61
2.7.3 n Dimensional systems 62
Exercises 63
3 Non linear systems in the plane 71
3.1 Local and global behaviour 71
3.2 Linearization at a fixed point 74
3.3 The linearization theorem 77
3.4 Non simple fixed points 81
3.5 Stability of fixed points 84
3.6 Ordinary points and global behaviour 93
3.6.1 Ordinary points 93
3.6.2 Global phase portraits 95
3.7 First integrals 96
3.8 Limit points and limit cycles 101
3.9 Poincare Bendixson theory 105
Exercises 110
4 Flows on non planar phase spaces 120
4.1 Fixed points 120
4.1.1 Hyperbolic fixed points 120
4.1.2 Non hyperbolic fixed points 125
4.2 Closed orbits 129
4.2.1 Poincare maps and hyperbolic closed orbits 129
4.2.2 Topological classification of hyperbolic closed orbits 132
4.2.3 Periodic orbits and quasi periodic motion 136
4.3 Attracting sets and attractors 138
4.3.1 Trapping regions for Poincare maps 140
4.3.2 Saddle points in attracting sets 143
4.4 Further integrals 147
4.4.1 Hamilton s equations 148
4.4.2 Poincare maps of Hamiltonian flows 152
Exercises 155
5 Applications I: planar phase spaces 162
5.1 Linear models 162
5.1.1 A mechanical oscillator 162
5.1.2 Electrical circuits 167
5.1.3 Economics 170
5.1.4 Coupled oscillators 172
5.2 Affine models 175
5.2.1 The forced harmonic oscillator 176
5.2.2 Resonance 177
5.3 Non linear models 179
5.3.1 Competing species 180
Contents vii
5.3.2 Volterra Lotka equations 183
5.3.3 The Holling Tanner model 185
5.4 Relaxation oscillations 188
5.4.1 Van der Pol oscillator 188
5.4.2 Jumps and regularization 192
5.5 Piecewise modelling 195
5.5.1 The jump assumption and piecewise models 196
5.5.2 A limit cycle from linear equations 198
Exercises 202
6 Applications II: non planar phase spaces, families of systems
and bifurcations 212
6.1 The Zeeman models of heartbeat and nerve impulse 212
6.2 A model of animal conflict 218
6.3 Families of differential equations and bifurcations 223
6.3.1 Introductory remarks 223
6.3.2 Saddle node bifurcation 226
6.3.3 Hopf bifurcation 228
6.4 A mathematical model of tumour growth 232
6.4.1 Construction of the model 232
6.4.2 An analysis of the dynamics 233
6.5 Some bifurcations in families of one dimensional maps 240
6.5.1 The fold bifurcation 240
6.5.2 The flip bifurcation 242
6.5.3 The logistic map 245
6.6 Some bifurcations in families of two dimensional maps 251
6.6.1 The child on a swing 251
6.6.2 The Duffing equation 254
6.7 Area preserving maps, homoclinic tangles and strange
attractors 259
6.7.1 Introductory remarks 259
6.7.2 Periodic orbits and island chains 261
6.7.3 Chaotic orbits and homoclinic tangles 264
6.7.4 Strange attracting sets 267
6.8 Symbolic dynamics 271
6.9 New directions 279
6.9.1 Introductory remarks 279
6.9.2 Iterated function schemes 280
6.9.3 Cellular automata 284
Exercises 288
Bibliography 303
Hints to Exercises 306
Index 326
|
| any_adam_object | 1 |
| author | Arrowsmith, David K. Place, Colin M. |
| author_GND | (DE-588)113968876 |
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| edition | 1. ed. |
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| indexdate | 2024-07-09T16:49:12Z |
| institution | BVB |
| isbn | 0412390701 0412390809 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-004222875 |
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| physical | X, 330 S. zahlr. graph. Darst. |
| publishDate | 1992 |
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| publisher | Chapman & Hall |
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| spelling | Arrowsmith, David K. Verfasser (DE-588)113968876 aut Dynamical systems differential equations, maps and chaotic behaviour D. K. Arrowsmith and C. M. Place 1. ed. London u.a. Chapman & Hall 1992 X, 330 S. zahlr. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Equacoes diferenciais larpcal Sistemas dinamicos larpcal Differentiable dynamical systems Differential equations Abbildung Mathematik (DE-588)4000044-8 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Chaostheorie (DE-588)4009754-7 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s Differentialgleichung (DE-588)4012249-9 s Abbildung Mathematik (DE-588)4000044-8 s Chaostheorie (DE-588)4009754-7 s 1\p DE-604 Place, Colin M. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004222875&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Arrowsmith, David K. Place, Colin M. Dynamical systems differential equations, maps and chaotic behaviour Equacoes diferenciais larpcal Sistemas dinamicos larpcal Differentiable dynamical systems Differential equations Abbildung Mathematik (DE-588)4000044-8 gnd Differentialgleichung (DE-588)4012249-9 gnd Chaostheorie (DE-588)4009754-7 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4000044-8 (DE-588)4012249-9 (DE-588)4009754-7 (DE-588)4013396-5 |
| title | Dynamical systems differential equations, maps and chaotic behaviour |
| title_auth | Dynamical systems differential equations, maps and chaotic behaviour |
| title_exact_search | Dynamical systems differential equations, maps and chaotic behaviour |
| title_full | Dynamical systems differential equations, maps and chaotic behaviour D. K. Arrowsmith and C. M. Place |
| title_fullStr | Dynamical systems differential equations, maps and chaotic behaviour D. K. Arrowsmith and C. M. Place |
| title_full_unstemmed | Dynamical systems differential equations, maps and chaotic behaviour D. K. Arrowsmith and C. M. Place |
| title_short | Dynamical systems |
| title_sort | dynamical systems differential equations maps and chaotic behaviour |
| title_sub | differential equations, maps and chaotic behaviour |
| topic | Equacoes diferenciais larpcal Sistemas dinamicos larpcal Differentiable dynamical systems Differential equations Abbildung Mathematik (DE-588)4000044-8 gnd Differentialgleichung (DE-588)4012249-9 gnd Chaostheorie (DE-588)4009754-7 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Equacoes diferenciais Sistemas dinamicos Differentiable dynamical systems Differential equations Abbildung Mathematik Differentialgleichung Chaostheorie Dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004222875&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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