Bifurcations: sights, sounds, and mathematics
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Tokyo u.a.
Springer-Verl.
1993
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XL, 468 S. Ill., graph. Darst. |
| ISBN: | 4431701206 3540701206 0387701206 |
Internformat
MARC
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| 245 | 1 | 0 | |a Bifurcations |b sights, sounds, and mathematics |c T. Matsumoto ... |
| 264 | 1 | |a Tokyo u.a. |b Springer-Verl. |c 1993 | |
| 300 | |a XL, 468 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Chaos (théorie des systèmes) |2 ram | |
| 650 | 7 | |a bifurcation |2 inriac | |
| 650 | 7 | |a chaos |2 inriac | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Bifurcation theory | |
| 650 | 4 | |a Chaotic behavior in systems | |
| 650 | 4 | |a Electronic circuits |x Mathematical models | |
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| 689 | 1 | |5 DE-604 | |
| 700 | 1 | |a Matsumoto, Takashi |e Sonstige |4 oth | |
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Datensatz im Suchindex
| _version_ | 1804122664615280640 |
|---|---|
| adam_text | Table of Contents
Preface V
Color Plates XVII
1 Bifurcations Observed from Electronic Circuits 1
1.1 Introduction 1
1.2 The Double Scroll Circuit 2
1.2.1 Circuit and its Dynamics 2
1.2.2 Implementation 4
1.2.3 Experiments 7
A Hopf Bifurcation 7
B Period Doubling Bifurcations of the Periodic Orbit 7
C Chaotic Attractor (Rossler s Spiral type) 7
D Saddle Node Bifurcations of the Periodic Orbit and
Periodic Window 8
E Interior Crisis (The Double Scroll) 8
F Near Heteroclinicity 8
G Boundary Crisis 8
H Sounds 9
1.2.4 Confirmations 10
A Hopf Bifurcation 10
B Period Doubling Bifurcation 11
C Chaotic Attractor (Rossler s Spiral type) 11
D Saddle Node Bifurcation and Periodic Window . . 15
E Interior Crisis (The Double Scroll) 16
F Near Heteroclinicity 16
G Boundary Crisis 16
1.2.5 Summary 17
1.3 Structure of the Double Scroll 20
1.3.1 Geometric Structure 20
1.3.2 Lyapunov Exponents and Lyapunov Dimension 29
A Lyapunov Exponents 29
X Table of Contents
B Computations 32
C Explicit Formula 34
D Lyapunov Dimension 35
E Time Waveforms and Power Spectra 35
1.4 The Double Scroll Circuit is Chaotic in the Sense of Shil nikov . . 35
1.4.1 Statement 35
1.4.2 The Class C 37
1.4.3 Equivalence and Conjugacy Classes of £ 44
1.4.4 Subset CVs 50
A Half Return Map tto 51
B Half Return Map tti 56
C The Map Z 59
D Poincare Map 7r 60
1.4.5 Completion of the Proof 62
1.5 Homoclinic Linkage 74
1.5.1 Introduction 74
1.5.2 Bifurcation Equations 74
A Normal Form 74
B Return Time Coordinates 76
C Periodic Orbits 78
D Bifurcation Conditions for Periodic Orbits .... 78
E Homoclinic Orbits Passing Through O 80
F Homoclinic Orbits Passing Through P+ 80
G Heteroclinic Orbits 81
1.5.3 Global Bifurcations 82
A Homoclinic/Heteroclinic Bifurcation Sets 83
B Homoclinic Linkage 90
C Global Bifurcations of Periodic Windows 96
1.6 The Torus Breakdown Circuit 102
1.6.1 Introduction 102
1.6.2 Observations of Torus Breakdown 102
A The Circuit and its Dynamics 102
B Experiments 104
C Period Adding Sequence 109
D Sounds 110
1.6.3 Analysis 110
A Divergence Zero Boundary Ill
B Trajectories on the Torus 112
C The Folded Torus and the Double Scroll 114
1.7 The Hyperchaotic Circuit 115
1.7.1 Introduction 115
1.7.2 Experiment 115
A Observation 115
B Sounds 116
1.7.3 Confirmation 116
Table of Contents XI
1.8 The Neon Bulb Circuit 120
1.8.1 Introduction 120
1.8.2 Experiment 121
A Observation 121
B Sounds 122
1.8.3 Arnold Tongues 122
1.8.4 Rotation Numbers 123
1.9 The .R L Diode Circuit 124
1.9.1 Experiment 1 124
1.9.2 Analysis 1 125
A The Dynamics 125
B Two Dimensional Map Model 129
C The Bifurcation Scenario 129
1.9.3 Experiment 2 133
1.9.4 Analysis 2 133
2 Bifurcations of Continuous Piecewise Liriear Vector Fields 139
2.1 Introduction 139
2.2 Definition and Standard Forms of Continuous Piecewise Linear Maps 140
2.2.1 Definition of Piecewise Linear Maps 140
2.2.2 Standard Forms of CPL Maps with the Boundary Set in
General Position 142
2.2.3 Standard Forms of CPL Functions 146
2.2.4 Examples of CPL functions 157
2.3 Normal Forms of Piecewise Linear Vector Fields 163
2.3.1 Notations 165
2.3.2 Normal Forms of Linear Vector Fields with a Boundary . . 180
2.3.3 Normal Forms of Degenerate Affine Vector Fields with a
Boundary 191
2.3.4 Normal Forms of Two Region Piecewise Linear Vector Fields 210
2.3.5 Normal Forms of Proper Two Region Piecewise Linear Vec¬
tor Fields 237
2.4 Multiregion Systems and Chaotic Attractors 244
2.4.1 Attractors in Three Dimensional Three Region System . . . 244
2.4.2 The Piecewise Linear Lorenz Attractor 249
2.4.3 The Piecewise Linear Duffing Attractor 254
2.5 Bifurcation Equations of Piecewise Linear Vector Fields 257
2.5.1 Normal Forms of Three Dimensional Two Region Systems . 258
2.5.2 The Tangent Map of Poincare Full Return Maps 264
2.5.3 The Return Time Coordinates 267
2.5.4 Bifurcation Equations of Three Dimensional Two Region
Systems 269
A Homoclinic Bifurcations . . . 270
B Heteroclinic Bifurcations 272
2.5.5 Bifurcation Equations of Periodic Orbits 274
XII Tabie of Contents
2.6 Bifurcation Sets 278
2.6.1 Homoclinic/Heteroclinic Bifurcation Sets 279
A Bifurcation Sets for Principal Homoclinic Orbits . 279
B Subsidiary Homoclinic Bifurcation Sets and Hete
roclinic Bifurcation Sets 281
2.6.2 Bifurcation Sets for Periodic Orbits 281
A Saddle Node Bifurcation Sets 286
B Period Doubling Bifurcation Sets 286
C Windows 286
2.6.3 Computing Bifurcation Sets 294
3 Fundamental Concepts in Bifurcations 297
3.1 Introduction 297
3.2 Fundamental Notions for Dynamical Systems 299
3.2.1 Definitions and Examples of Dynamical Systems 299
3.2.2 Orbits and Invariant Sets in Dynamical Systems 305
3.2.3 Linearization at Equilibrium Points and the Theorem of
Hartman Grobman 312
3.2.4 Stable and Unstable Manifolds 318
3.2.5 Topological Equivalence and Structural Stability 323
3.2.6 Bifurcation 326
3.2.7 Framework for the Bifurcation Theory 329
3.3 Local Bifurcations around Equilibrium Points in Vector Fields . • . 330
3.3.1 Center Manifolds 330
3.3.2 Normal Forms 338
3.3.3 Codimension One Bifurcations 343
A Saddle Node Bifurcation 343
B Hopf Bifurcation 346
3.3.4 Bogdanov Takens Bifurcation 351
3.3.5 Symmetry and Bifurcations 355
3.3.6 Other Degenerate Singularities 360
3.4 Dynamics and Bifurcations for Discrete Dynamical Systems .... 362
3.4.1 Discrete Dynamical Systems 362
3.4.2 Basic Theorems and Structural Stability 367
3.4.3 Elementary Bifurcations 368
A Saddle Node Bifurcation 368
B Period Doubling Bifurcation 369
C Hopf Bifurcation 371
3.4.4 One Dimensional Mapping (1) 373
A Elementary Bifurcations for Quadratic Family . . 374
B The Case of /x 2 376
C The Case of /z = 2 378
3.4.5 One Dimensional Mapping (2) 380
3.4.6 Horseshoe 384
A Topological Horseshoe 384
Table of Contents XIII
B Hyperbolicity 386
C Transverse Homoclinic Points and Horseshoes . . 390
3.4.7 Further Developments 393
A One Dimensional Quadratic Family 393
B Lozi Map 395
C Henon Map 395
D Homoclinic Tangency 396
3.5 Bifurcations of Homoclinic and Heteroclinic Orbits in Vector Fields 398
3.5.1 Persistence of Homoclinic/Heteroclinic Orbits and the Mel
nikov Integral 398
3.5.2 Shil nikov Theorem 406
3.5.3 Gluing Bifurcations for Heteroclinic Orbits and Exponential
Expansion 411
3.5.4 T points and Gluing Bifurcations with Different Saddle
Indices 417
3.5.5 Homoclinic Doubling Bifurcation 421
A Motivation 421
B Homoclinic Doubling Bifurcation Theorems .... 424
C Proof of the Homoclinic Doubling Bifurcation The¬
orems 427
D Further Development 431
3.5.6 Bifurcation Generating Geometric Lorenz Attractors from
Homoclinic Orbits 432
3.5.7 Local Bifurcations and Global Bifurcations 441
References 445
Index 459
Credits 467
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| id | DE-604.BV008269126 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:17:28Z |
| institution | BVB |
| isbn | 4431701206 3540701206 0387701206 |
| language | English |
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| physical | XL, 468 S. Ill., graph. Darst. |
| publishDate | 1993 |
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| publisher | Springer-Verl. |
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| spelling | Bifurcations sights, sounds, and mathematics T. Matsumoto ... Tokyo u.a. Springer-Verl. 1993 XL, 468 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chaos (théorie des systèmes) ram bifurcation inriac chaos inriac Mathematisches Modell Bifurcation theory Chaotic behavior in systems Electronic circuits Mathematical models Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Chaostheorie (DE-588)4009754-7 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 s DE-604 Chaostheorie (DE-588)4009754-7 s Matsumoto, Takashi Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005462515&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bifurcations sights, sounds, and mathematics Chaos (théorie des systèmes) ram bifurcation inriac chaos inriac Mathematisches Modell Bifurcation theory Chaotic behavior in systems Electronic circuits Mathematical models Verzweigung Mathematik (DE-588)4078889-1 gnd Chaostheorie (DE-588)4009754-7 gnd |
| subject_GND | (DE-588)4078889-1 (DE-588)4009754-7 |
| title | Bifurcations sights, sounds, and mathematics |
| title_auth | Bifurcations sights, sounds, and mathematics |
| title_exact_search | Bifurcations sights, sounds, and mathematics |
| title_full | Bifurcations sights, sounds, and mathematics T. Matsumoto ... |
| title_fullStr | Bifurcations sights, sounds, and mathematics T. Matsumoto ... |
| title_full_unstemmed | Bifurcations sights, sounds, and mathematics T. Matsumoto ... |
| title_short | Bifurcations |
| title_sort | bifurcations sights sounds and mathematics |
| title_sub | sights, sounds, and mathematics |
| topic | Chaos (théorie des systèmes) ram bifurcation inriac chaos inriac Mathematisches Modell Bifurcation theory Chaotic behavior in systems Electronic circuits Mathematical models Verzweigung Mathematik (DE-588)4078889-1 gnd Chaostheorie (DE-588)4009754-7 gnd |
| topic_facet | Chaos (théorie des systèmes) bifurcation chaos Mathematisches Modell Bifurcation theory Chaotic behavior in systems Electronic circuits Mathematical models Verzweigung Mathematik Chaostheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005462515&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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