An experimental approach to nonlinear dynamics and chaos:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Redwood City, California
Addison-Wesley
1992
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| Series: | Studies in nonlinearity
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 340, 45, 35 Seiten Diagramme 1 Diskette |
| ISBN: | 0201554410 |
Staff View
MARC
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| 100 | 1 | |a Tufillaro, Nicholas B. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a An experimental approach to nonlinear dynamics and chaos |c Nicholas B. Tufillaro, Tyler Abbott, Jeremiah Reilly |
| 264 | 1 | |a Redwood City, California |b Addison-Wesley |c 1992 | |
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| 700 | 1 | |a Abbott, Tyler |e Verfasser |4 aut | |
| 700 | 1 | |a Reilly, Jeremiah |e Verfasser |4 aut | |
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Record in the Search Index
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| adam_text | Contents
Preface xi
Introduction 1
What is nonlinear dynamics? 1
What is in this book? 6
Some Terminology: Maps, Flows, and Fractals 9
References and Notes 21
1 Bouncing Ball 23
1.1 Introduction 23
1.2 Model 26
1.2.1 Stationary Table 27
1.2.2 Impact Relation for the Oscillating Table 28
1.2.3 The Equations of Motion: Phase and Velocity Maps 29
1.2.4 Parameters 31
1.3 High Bounce Approximation 32
1.4 Qualitative Description of Motions 35
1.4.1 Trapping Region 36
1.4.2 Equilibrium Solutions 38
1.4.3 Sticking Solutions 38
1.4.4 Period One Orbits and Period Doubling 40
1.4.5 Chaotic Motions 43
1.5 Attractors 47
1.6 Bifurcation Diagrams 49
References and Notes 51
Problems 52
2 Quadratic Map 55
2.1 Introduction 55
2.2 Iteration and Differentiation 60
2.3 Graphical Method 62
2.4 Fixed Points 65
2.5 Periodic Orbits 68
v
vi CONTENTS
2.5.1 Graphical Method 70
2.5.2 Period One Orbits 73
2.5.3 Period Two Orbit 74
2.5.4 Stability Diagram 75
2.6 Bifurcation Diagram 75
2.7 Local Bifurcation Theory 79
2.7.1 Saddle node 80
2.7.2 Period Doubling 82
2.7.3 Transcritical 83
2.8 Period Doubling Ad Infinitum 84
2.9 Sarkovskii s Theorem 89
2.10 Sensitive Dependence 91
2.11 Fully Developed Chaos 93
2.11.1 Hyperbolic Invariant Sets 93
2.11.2 Symbolic Dynamics 96
2.11.3 Topological Conjugacy 99
2.12 Symbolic Coordinates 101
2.12.1 What s in a name? Location 102
2.12.2 Alternating Binary Tree 105
2.12.3 Topological Entropy 107
Usage of Mathematica 110
References and Notes 114
Problems 116
3 String 123
3.1 Introduction 123
3.2 Experimental Apparatus 125
3.3 Single Mode Model 128
3.4 Planar Vibrations: Duffing Equation 132
3.4.1 Equilibrium States 133
3.4.2 Unforced Phase Plane 134
3.4.3 Extended Phase Space 138
3.4.4 Global Cross Section 140
3.5 Resonance and Hysteresis 142
3.5.1 Linear Resonance 142
3.5.2 Nonlinear Resonance 144
3.5.3 Response Curve 146
3.5.4 Hysteresis 148
3.5.5 Basins of Attraction 149
3.6 Homoclinic Tangles 151
3.7 Nonplanar Motions 152
3.7.1 Free Whirling 154
3.7.2 Response Curve 156
3.7.3 Torus Attractor 158
3.7.4 Circle Map 159
CONTENTS vii
3.7.5 Torus Doubling 161
3.8 Experimental Techniques 162
3.8.1 Experimental Cross Section 163
3.8.2 Embedding 166
3.8.3 Power Spectrum 169
3.8.4 Attractor Identification 174
3.8.5 Correlation Dimension 177
References and Notes 180
Problems 183
4 Dynamical Systems Theory 187
4.1 Introduction 187
4.2 Flows and Maps 189
4.2.1 Flows 189
4.2.2 Poincare Map 191
4.2.3 Suspension of a Map 193
4.2.4 Creed and Quest 194
4.3 Asymptotic Behavior and Recurrence 194
4.3.1 Invariant Sets 195
4.3.2 Limit Sets: a, w, and Nonwandering 195
4.3.3 Chain Recurrence 197
4.4 Expansions and Contractions 199
4.4.1 Derivative of a Map 199
4.4.2 Jacobian of a Map 200
4.4.3 Divergence of a Vector Field 201
4.4.4 Dissipative and Conservative 202
4.4.5 Equation of First Variation 203
4.5 Fixed Points 204
4.5.1 Stability 204
4.5.2 Linearization 205
4.5.3 Hyperbolic Fixed Points: Saddles, Sources, and Sinks .... 206
4.6 Invariant Manifolds 207
4.6.1 Center Manifold Theorem 208
4.6.2 Homoclinic and Heteroclinic Points 211
4.7 Example: Laser Equations 215
4.7.1 Steady States 216
4.7.2 Eigenvalues of a 2 x 2 Matrix 216
4.7.3 Eigenvectors 217
4.7.4 Stable Focus 218
4.8 Smale Horseshoe 218
4.8.1 From Tangles to Horseshoes 220
4.8.2 Horseshoe Map 222
4.8.3 Symbolic Dynamics 226
4.8.4 From Horseshoes to Tangles 228
4.9 Hyperbolicity 230
viii CONTENTS
4.10 Lyapunov Characteristic Exponent 231
References and Notes 233
Problems 235
5 Knots and Templates 239
5.1 Introduction 239
5.2 Periodic Orbit Extraction 242
5.2.1 Algorithm 244
5.2.2 Local Torsion 246
5.2.3 Example: Duffing Equation 247
5.3 Knot Theory 249
5.3.1 Crossing Convention 253
5.3.2 Reidemeister Moves 253
5.3.3 Invariants and Linking Numbers 254
5.3.4 Braid Group 256
5.3.5 Framed Braids 259
5.4 Relative Rotation Rates 260
5.5 Templates 265
5.5.1 Motivation and Geometric Description 266
5.5.2 Algebraic Description 272
5.5.3 Location of Knots 277
5.5.4 Calculation of Relative Rotation Rates 281
5.6 Intertwining Matrices 284
5.6.1 Horseshoe 284
5.6.2 Lorenz 284
5.7 Duffing Template 287
References and Notes 289
Problems 291
Appendix A: Bouncing Ball Code 293
Appendix B: Exact Solutions for a Cubic Oscillator 300
Appendix C: Ode Overview 302
Appendix D: Discrete Fourier Transform 305
Appendix E: Henon s Trick 307
Appendix F: Periodic Orbit Extraction Code 309
Appendix G: Relative Rotation Rate Package 314
Appendix H: Historical Comments 323
Appendix I: Projects 327
CONTENTS ix
Commonly Used Notation 333
Index 335
Bouncing Ball User s Guide
Quadratic Map User s Guide
|
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| id | DE-604.BV008062099 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:13:42Z |
| institution | BVB |
| isbn | 0201554410 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005305181 |
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| owner | DE-12 DE-703 DE-20 DE-83 DE-188 DE-384 |
| owner_facet | DE-12 DE-703 DE-20 DE-83 DE-188 DE-384 |
| physical | xiv, 340, 45, 35 Seiten Diagramme 1 Diskette |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| publisher | Addison-Wesley |
| record_format | marc |
| series2 | Studies in nonlinearity |
| spelling | Tufillaro, Nicholas B. Verfasser aut An experimental approach to nonlinear dynamics and chaos Nicholas B. Tufillaro, Tyler Abbott, Jeremiah Reilly Redwood City, California Addison-Wesley 1992 xiv, 340, 45, 35 Seiten Diagramme 1 Diskette txt rdacontent n rdamedia nc rdacarrier Studies in nonlinearity Nichtlineares dynamisches System (DE-588)4126142-2 gnd rswk-swf Chaos (DE-588)4191419-3 gnd rswk-swf Chaotisches System (DE-588)4316104-2 gnd rswk-swf Chaotisches System (DE-588)4316104-2 s Nichtlineares dynamisches System (DE-588)4126142-2 s DE-604 Chaos (DE-588)4191419-3 s Abbott, Tyler Verfasser aut Reilly, Jeremiah Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005305181&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tufillaro, Nicholas B. Abbott, Tyler Reilly, Jeremiah An experimental approach to nonlinear dynamics and chaos Nichtlineares dynamisches System (DE-588)4126142-2 gnd Chaos (DE-588)4191419-3 gnd Chaotisches System (DE-588)4316104-2 gnd |
| subject_GND | (DE-588)4126142-2 (DE-588)4191419-3 (DE-588)4316104-2 |
| title | An experimental approach to nonlinear dynamics and chaos |
| title_auth | An experimental approach to nonlinear dynamics and chaos |
| title_exact_search | An experimental approach to nonlinear dynamics and chaos |
| title_full | An experimental approach to nonlinear dynamics and chaos Nicholas B. Tufillaro, Tyler Abbott, Jeremiah Reilly |
| title_fullStr | An experimental approach to nonlinear dynamics and chaos Nicholas B. Tufillaro, Tyler Abbott, Jeremiah Reilly |
| title_full_unstemmed | An experimental approach to nonlinear dynamics and chaos Nicholas B. Tufillaro, Tyler Abbott, Jeremiah Reilly |
| title_short | An experimental approach to nonlinear dynamics and chaos |
| title_sort | an experimental approach to nonlinear dynamics and chaos |
| topic | Nichtlineares dynamisches System (DE-588)4126142-2 gnd Chaos (DE-588)4191419-3 gnd Chaotisches System (DE-588)4316104-2 gnd |
| topic_facet | Nichtlineares dynamisches System Chaos Chaotisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005305181&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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