Analysis and simulation of chaotic systems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY ; Berlin ; Heidelberg ; Barcelona ; Hong Kong ; London
Springer
2000
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Applied mathematical sciences
94 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 303 - 309 |
| Beschreibung: | XX, 315 S. graph. Darst. 24 cm |
| ISBN: | 0387989439 |
Internformat
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| 245 | 1 | 0 | |a Analysis and simulation of chaotic systems |c Frank C. Hoppensteadt |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York, NY ; Berlin ; Heidelberg ; Barcelona ; Hong Kong ; London |b Springer |c 2000 | |
| 300 | |a XX, 315 S. |b graph. Darst. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Applied mathematical sciences |v 94 | |
| 500 | |a Literaturverz. S. 303 - 309 | ||
| 650 | 7 | |a Analyse (wiskunde) |2 gtt | |
| 650 | 7 | |a Chaos |2 gtt | |
| 650 | 7 | |a Simulatie |2 gtt | |
| 650 | 4 | |a Chaotic behavior in systems | |
| 650 | 0 | 7 | |a Dynamisches System |0 (DE-588)4013396-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Chaotisches System |0 (DE-588)4316104-2 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1804127843104325632 |
|---|---|
| adam_text | Contents
Acknowledgments v
Introduction xiii
1 Linear Systems 1
1.1 Examples of Linear Oscillators 1
1.1.1 Voltage Controlled Oscillators 2
1.1.2 Filters 3
1.1.3 Pendulum with Variable Support Point 4
1.2 Time Invariant Linear Systems 5
1.2.1 Functions of Matrices 6
1.2.2 exp(At) 7
1.2.3 Laplace Transforms of Linear Systems 9
1.3 Forced Linear Systems with Constant Coefficients .... 10
1.4 Linear Systems with Periodic Coefficients 12
1.4.1 Hill s Equation 14
1.4.2 Mathieu s Equation 15
1.5 Fourier Methods 18
1.5.1 Almost Periodic Functions 18
1.5.2 Linear Systems with Periodic Forcing 21
1.5.3 Linear Systems with Quasiperiodic Forcing .... 22
1.6 Linear Systems with Variable Coefficients: Variation of
Constants Formula 23
1.7 Exercises 24
viii Contents
2 Dynamical Systems 27
2.1 Systems of Two Equations 28
2.1.1 Linear Systems 28
2.1.2 Poincare and Bendixson s Theory 29
2.1.3 x + f(x)x + g(x) = 0 32
2.2 Angular Phase Equations 35
2.2.1 A Simple Clock: A Phase Equation on T1 .... 37
2.2.2 A Toroidal Clock: Denjoy s Theory 38
2.2.3 Systems of N (Angular) Phase Equations .... 40
2.2.4 Equations on a Cylinder: PLL 40
2.3 Conservative Systems 42
2.3.1 Lagrangian Mechanics 42
2.3.2 Plotting Phase Portraits Using Potential Energy . 43
2.3.3 Oscillation Period of x + Ux(x) =0 46
2.3.4 Active Transmission Line 47
2.3.5 Phase Amplitude (Angle Action) Coordinates . . 49
2.3.6 Conservative Systems with N Degrees of Freedom 52
2.3.7 Hamilton Jacobi Theory 53
2.3.8 Liouville s Theorem 56
2.4 Dissipative Systems 57
2.4.1 van der Pol s Equation 57
2.4.2 Phase Locked Loop 57
2.4.3 Gradient Systems and the Cusp Catastrophe ... 62
2.5 Stroboscopic Methods 65
2.5.1 Chaotic Interval Mappings 66
2.5.2 Circle Mappings 71
2.5.3 Annulus Mappings 74
2.5.4 Hadamard s Mappings of the Plane 75
2.6 Oscillations of Equations with a Time Delay 78
2.6.1 Linear Spline Approximations 80
2.6.2 Special Periodic Solutions 81
2.7 Exercises 83
3 Stability Methods for Nonlinear Systems 91
3.1 Desirable Stability Properties of Nonlinear Systems ... 92
3.2 Linear Stability Theorem 94
3.2.1 Gronwall s Inequality 95
3.2.2 Proof of the Linear Stability Theorem 96
3.2.3 Stable and Unstable Manifolds 97
3.3 Liapunov s Stability Theory 99
3.3.1 Liapunov s Functions 99
3.3.2 UAS of Time Invariant Systems 100
3.3.3 Gradient Systems 101
3.3.4 Linear Time Varying Systems 102
3.3.5 Stable Invariant Sets 103
Contents ix
3.4 Stability Under Persistent Disturbances 106
3.5 Orbital Stability of Free Oscillations 108
3.5.1 Definitions of Orbital Stability 109
3.5.2 Examples of Orbital Stability 110
3.5.3 Orbital Stability Under Persistent Disturbances . Ill
3.5.4 Poincare s Return Mapping Ill
3.6 Angular Phase Stability 114
3.6.1 Rotation Vector Method 114
3.6.2 Huygen s Problem 116
3.7 Exercises 118
4 Bifurcation and Topological Methods 121
4.1 Implicit Function Theorems 121
4.1.1 Fredholm s Alternative for Linear Problems . . . 122
4.1.2 Nonlinear Problems: The Invertible Case 126
4.1.3 Nonlinear Problems: The Noninvertible Case . . . 128
4.2 Solving Some Bifurcation Equations 129
4.2.1 q = : Newton s Polygons 130
4.3 Examples of Bifurcations 132
4.3.1 Exchange of Stabilities 132
4.3.2 Andronov Hopf Bifurcation 133
4.3.3 Saddle Node on Limit Cycle Bifurcation 134
4.3.4 Cusp Bifurcation Revisited 134
4.3.5 Canonical Models and Bifurcations 135
4.4 Fixed Point Theorems 136
4.4.1 Contraction Mapping Principle 136
4.4.2 Wazewski s Method 138
4.4.3 Sperner s Method 141
4.4.4 Measure Preserving Mappings 142
4.5 Exercises 142
5 Regular Perturbation Methods 145
5.1 Perturbation Expansions 147
5.1.1 Gauge Functions: The Story of o, O 147
5.1.2 Taylor s Formula 148
5.1.3 Pade s Approximations 148
5.1.4 Laplace s Methods 150
5.2 Regular Perturbations of Initial Value Problems 152
5.2.1 Regular Perturbation Theorem 152
5.2.2 Proof of the Regular Perturbation Theorem ... 153
5.2.3 Example of the Regular Perturbation Theorem . 155
5.2.4 Regular Perturbations for 0 t oo 155
5.3 Modified Perturbation Methods for Static States 157
5.3.1 Nondegenerate Static State Problems Revisited . 158
5.3.2 Modified Perturbation Theorem 158
x Contents
5.3.3 Example: q = 1 160
5.4 Exercises 161
6 Iterations and Perturbations 163
6.1 Resonance 164
6.1.1 Formal Perturbation Expansion of Forced Oscilla¬
tions 166
6.1.2 Nonresonant Forcing 167
6.1.3 Resonant Forcing 170
6.1.4 Modified Perturbation Method for Forced Oscilla¬
tions 172
6.1.5 Justification of the Modified Perturbation Method 173
6.2 Duffing s Equation 174
6.2.1 Modified Perturbation Method 175
6.2.2 Duffing s Iterative Method 176
6.2.3 Poincare Linstedt Method 177
6.2.4 Frequency Response Surface 178
6.2.5 Subharmonic Responses of Duffing s Equation . . 179
6.2.6 Damped Duffing s Equation 181
6.2.7 Duffing s Equation with Subresonant Forcing . . 182
6.2.8 Computer Simulation of Duffing s Equation . . . 184
6.3 Boundaries of Basins of Attraction 186
6.3.1 Newton s Method and Chaos 187
6.3.2 Computer Examples 188
6.3.3 Fractal Measures 190
6.3.4 Simulation of Fractal Curves 191
6.4 Exercises 194
7 Methods of Averaging 195
7.1 Averaging Nonlinear Systems 199
7.1.1 The Nonlinear Averaging Theorem 200
7.1.2 Averaging Theorem for Mean Stable Systems . . 202
7.1.3 A Two Time Scale Method for the Full Problem . 203
7.2 Highly Oscillatory Linear Systems 204
7.2.1 dx/dt = eB{t)x 205
7.2.2 Linear Feedback System 206
7.2.3 Averaging and Laplace s Method 207
7.3 Averaging Rapidly Oscillating Difference Equations . . . 207
7.3.1 Linear Difference Schemes 210
7.4 Almost Harmonic Systems 214
7.4.1 Phase Amplitude Coordinates 215
7.4.2 Free Oscillations 216
7.4.3 Conservative Systems 219
7.5 Angular Phase Equations 223
7.5.1 Rotation Vector Method 224
Contents xi
7.5.2 Rotation Numbers and Period Doubling Bifurca¬
tions 227
7.5.3 Euler s Forward Method for Numerical Simulation 227
7.5.4 Computer Simulation of Rotation Vectors .... 229
7.5.5 Near Identity Flows on S1 x S1 231
7.5.6 KAM Theory 233
7.6 Homogenization 234
7.7 Computational Aspects of Averaging 235
7.7.1 Direct Calculation of Averages 236
7.7.2 Extrapolation 237
7.8 Averaging Systems with Random Noise 238
7.8.1 Axioms of Probability Theory 238
7.8.2 Random Perturbations 241
7.8.3 Example of a Randomly Perturbed System .... 242
7.9 Exercises 243
8 Quasistatic State Approximations 249
8.1 Some Geometrical Examples of Singular Perturbation
Problems 254
8.2 Quasistatic State Analysis of a Linear Problem 257
8.2.1 Quasistatic Problem 258
8.2.2 Initial Transient Problem 261
8.2.3 Composite Solution 263
8.2.4 Volterra Integral Operators with Kernels Near 6 . 264
8.3 Quasistatic State Approximation for Nonlinear Initial Value
Problems 264
8.3.1 Quasistatic Manifolds 265
8.3.2 Matched Asymptotic Expansions 268
8.3.3 Construction of QSSA 270
8.3.4 The Case T = oo 271
8.4 Singular Perturbations of Oscillations 273
8.4.1 Quasistatic Oscillations 274
8.4.2 Nearly Discontinuous Oscillations 279
8.5 Boundary Value Problems 281
8.6 Nonlinear Stability Analysis near Bifurcations 284
8.6.1 Bifurcating Static States 284
8.6.2 Nonlinear Stability Analysis of Nonlinear Oscilla¬
tions 287
8.7 Explosion Mode Analysis of Rapid Chemical Reactions . 289
8.8 Computational Schemes Based on QSSA 292
8.8.1 Direct Calculation of x0{h),y0(h) 293
8.8.2 Extrapolation Method 294
8.9 Exercises 295
Supplementary Exercises 301
xii Contents
References 303
Index , 311
|
| any_adam_object | 1 |
| author | Hoppensteadt, Frank C. 1938- |
| author_GND | (DE-588)11268792X |
| author_facet | Hoppensteadt, Frank C. 1938- |
| author_role | aut |
| author_sort | Hoppensteadt, Frank C. 1938- |
| author_variant | f c h fc fch |
| building | Verbundindex |
| bvnumber | BV013144346 |
| callnumber-first | Q - Science |
| callnumber-label | Q172 |
| callnumber-raw | Q172.5.C45 QA1 |
| callnumber-search | Q172.5.C45 QA1 |
| callnumber-sort | Q 3172.5 C45 |
| callnumber-subject | Q - General Science |
| classification_rvk | SK 520 SK 810 SK 845 ST 340 UG 3900 |
| classification_tum | MAT 587f DAT 780f |
| ctrlnum | (OCoLC)611105884 (DE-599)BVBBV013144346 |
| dewey-full | 003/.857 510 |
| dewey-hundreds | 000 - Computer science, information, general works 500 - Natural sciences and mathematics |
| dewey-ones | 003 - Systems 510 - Mathematics |
| dewey-raw | 003/.857 510 |
| dewey-search | 003/.857 510 |
| dewey-sort | 13 3857 |
| dewey-tens | 000 - Computer science, information, general works 510 - Mathematics |
| discipline | Physik Informatik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV013144346 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:39:47Z |
| institution | BVB |
| isbn | 0387989439 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008954749 |
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| physical | XX, 315 S. graph. Darst. 24 cm |
| publishDate | 2000 |
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| publisher | Springer |
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| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Hoppensteadt, Frank C. 1938- Verfasser (DE-588)11268792X aut Analysis and simulation of chaotic systems Frank C. Hoppensteadt 2. ed. New York, NY ; Berlin ; Heidelberg ; Barcelona ; Hong Kong ; London Springer 2000 XX, 315 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 94 Literaturverz. S. 303 - 309 Analyse (wiskunde) gtt Chaos gtt Simulatie gtt Chaotic behavior in systems Dynamisches System (DE-588)4013396-5 gnd rswk-swf Chaotisches System (DE-588)4316104-2 gnd rswk-swf Chaostheorie (DE-588)4009754-7 gnd rswk-swf 1\p (DE-588)4006432-3 Bibliografie gnd-content 2\p (DE-588)4123623-3 Lehrbuch gnd-content Chaotisches System (DE-588)4316104-2 s Dynamisches System (DE-588)4013396-5 s DE-604 Chaostheorie (DE-588)4009754-7 s Applied mathematical sciences 94 (DE-604)BV000005274 94 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008954749&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 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Hoppensteadt, Frank C. 1938- Analysis and simulation of chaotic systems Applied mathematical sciences Analyse (wiskunde) gtt Chaos gtt Simulatie gtt Chaotic behavior in systems Dynamisches System (DE-588)4013396-5 gnd Chaotisches System (DE-588)4316104-2 gnd Chaostheorie (DE-588)4009754-7 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4316104-2 (DE-588)4009754-7 (DE-588)4006432-3 (DE-588)4123623-3 |
| title | Analysis and simulation of chaotic systems |
| title_auth | Analysis and simulation of chaotic systems |
| title_exact_search | Analysis and simulation of chaotic systems |
| title_full | Analysis and simulation of chaotic systems Frank C. Hoppensteadt |
| title_fullStr | Analysis and simulation of chaotic systems Frank C. Hoppensteadt |
| title_full_unstemmed | Analysis and simulation of chaotic systems Frank C. Hoppensteadt |
| title_short | Analysis and simulation of chaotic systems |
| title_sort | analysis and simulation of chaotic systems |
| topic | Analyse (wiskunde) gtt Chaos gtt Simulatie gtt Chaotic behavior in systems Dynamisches System (DE-588)4013396-5 gnd Chaotisches System (DE-588)4316104-2 gnd Chaostheorie (DE-588)4009754-7 gnd |
| topic_facet | Analyse (wiskunde) Chaos Simulatie Chaotic behavior in systems Dynamisches System Chaotisches System Chaostheorie Bibliografie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008954749&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT hoppensteadtfrankc analysisandsimulationofchaoticsystems |