The symmetry of chaos:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford [u.a.]
Oxford Univ. Press
2007
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XX, 545 S. Ill., graph. Darst. |
| ISBN: | 9780195310658 |
Internformat
MARC
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| 008 | 080425s2007 ad|| |||| 00||| eng d | ||
| 020 | |a 9780195310658 |9 978-0-19-531065-8 | ||
| 035 | |a (OCoLC)494380346 | ||
| 035 | |a (DE-599)BVBBV023278997 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-20 |a DE-355 |a DE-11 |a DE-19 |a DE-1102 | ||
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| 084 | |a UG 3900 |0 (DE-625)145629: |2 rvk | ||
| 100 | 1 | |a Gilmore, Robert |d 1941- |e Verfasser |0 (DE-588)122549058 |4 aut | |
| 245 | 1 | 0 | |a The symmetry of chaos |c Robert Gilmore and Christophe Letellier |
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2007 | |
| 300 | |a XX, 545 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 Symétrie |2 ram | |
| 650 | 4 | |a Chaotic behavior in systems | |
| 650 | 4 | |a Symmetry (Mathematics) | |
| 650 | 0 | 7 | |a Symmetrie |0 (DE-588)4058724-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Nichtlineares dynamisches System |0 (DE-588)4126142-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Chaos |0 (DE-588)4191419-3 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Nichtlineares dynamisches System |0 (DE-588)4126142-2 |D s |
| 689 | 0 | 1 | |a Symmetrie |0 (DE-588)4058724-1 |D s |
| 689 | 0 | 2 | |a Chaos |0 (DE-588)4191419-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Letellier, Christophe |e Verfasser |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016463831&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016463831 | ||
Datensatz im Suchindex
| _version_ | 1804137596788408320 |
|---|---|
| adam_text | Contents
PARTI
Examples and Simple Applications
1
Introduction
3
1.1
Swift Survey of Symmetries
3
1.1.1
Lasers
4
1.1.2
SunspotData
4
1.1.3
Paleomagnetic Data
4
1.1.4
Fluid Experiments
5
1.1.5
Electronic Circuits
6
1.2
Deeper Probe of Symmetries
6
1.2.1
Navier-Stokes PDEs
-» Lorenz
ODEs
7
1.2.2
Maxwell-Bloch Equations
—>
Laser Equations
10
1.3
Mathematical Motivation
12
1.3.1
Classification Theory
12
1.3.2
Cover-Image Relations
13
1.4
Overview of the Book
13
2
Simple Symmetries
17
2.1
Equations with Two-Fold Symmetry
17
2.1.1
Rotation Symmetry
18
χ
Contents
2.1.2 Inversion
Symmetry
20
2.1.3
Reflection Symmetry
21
2.2
Existence and Uniqueness of Solutions
22
2.3
Symmetry and the Fundamental Theorem
23
2.4
Fixed Point Distributions
24
2.5
Classification of Strange Attractors
28
2.5.1
Background
28
2.5.2
Topological Classification of Strange Attractors
29
2.5.3
Extracting Branched Manifolds from Data
32
2.5.4
Linking Number Tables
34
2.5.5
Symmetry
37
2.5.6
Application
37
2.6
Symbolic Dynamics
38
2.7
Periodic Orbits
39
2.8
Topological Entropy
40
2.9
Return Maps
41
3
Image Dynamical Systems
45
3.1
Diffeomorphisms: Global and Local
45
3.2 2 —> 1
Local Diffeomorphisms
48
3.3
Image Equations
50
3.3.1 Lorenz
Equations: TZz
(π)
51
3.3.2
Burke and Shaw Equations: TZZ
(π)
51
3.3.3
General Case: Uz
(π)
52
3.3.4
General Case: V
53
3.3.5
General Case:
σζ
55
3.4
Fixed Point Distributions
56
3.5
Branched Manifolds and Their Images
58
3.6
Symbolic Dynamics
60
3.7
Periodic Orbits
61
3.8
Poincaré
Sections and First-Return Maps
63
3.9
Tips for Integration
65
4
Covers
67
4.1
Local Diffeomorphisms
68
4.2
Singular Sets
69
4.3
Lifts to Rotation Invariant Systems: Topological Indices
72
4.4
Branched Manifolds
72
Contents xi
4.5
Periodic
Orbits 77
4.6
Poincaré
Sections and First-Return Maps
78
4.7
Fractal Dimensions and Lyapunov Exponents
80
4.8
Continuations
80
4.8.1
Topological Continuation
80
4.8.2
Group Continuation
81
4.9
Horseshoe versus Reverse Horseshoe
82
4.10
Lifts of the
Smale
Horseshoe
84
4.11
Tips for Integration
84
5
Peeling Bifurcations
87
5.1
Structural Stability
88
5.2
The
Perestroika
from
(1,1)
to
(0,1) 89
5.2.1
Branched Manifolds
89
5.2.2
Transition Matrices
91
5.2.3
Return Maps
91
5.2.4
Periodic Orbits
92
5.3
The
Perestroika
from
(0,1 )
to
(0,0) 93
5.3.1
Branched Manifolds
93
5.3.2
Transition Matrices
95
5.3.3
Return Maps
96
5.3.4
Periodic Orbits
96
5.4
Structurally Unstable Strange Attractors
97
5.5
The Peeling Bifurcation
100
5.6
Application: Sunspot Covers
101
6
Three-Fold and Four-Fold Covers
105
6.1
Image Dynamical Systems
] 05
6.1.1 Rössler
System
106
6.1.2
Proto-Lorenz System
106
6.1.3
Proto-Burke and Shaw Equations
107
6.2
η
-Fold Covers and Complex Variables
107
6.2.1
Covers of the
Rössler
Equations
108
6.2.2
Covers of the Proto-Lorenz Equations
109
6.2.3
Covers of the Proto-Burke and Shaw System
110
6.3
Covers with
Сз
Symmetry
110
6.3.1
The Group
110
6.3.2
Invariant Polynomials
111
xii Contents
6.3.3
The Jacobian
111
6.3.4
Covering Equations
112
6.3.5
Topołogical
Index
112
6.3.6
Covering Branched Manifolds
112
6.3.7
Symmetry-Adapted Labeling
114
6.3.8
Transition Matrices
115
6.3.9
Periodic Orbits
115
6.3.10
Linking Numbers
118
6.3.11
Poincaré
Sections and First-Return Maps
121
5.4
Covers with C4 Symmetry
121
6.4.1
The Group
121
6.4.2
Invariant and Covariant Polynomials
122
6.4.3
The Jacobian
123
6.4.4
Covering Equations
123
6.4.5
Topological Index
124
6.4.6
Covering Branched Manifolds
124
6.4.7
Symmetry-Adapted Labeling
124
6.4.8
Transition Matrices
125
6.4.9
Periodic Orbits
125
6.4,10
Linking Numbers
128
6.4.11
Poincaré
Sections and First-Return Maps
128
6.5
Cover Comparisons
130
6.5.1 Rössler
System
130
6.5.2
Proto-Lorenz System
133
6.5.3
Proto-Burke and Shaw System
136
6.6
Covers with V4 Symmetry
141
6.6.1
The Group
141
6.6.2
Invariant Polynomials
141
6.6.3
Invariant Coordinates
141
6.6.4
The Jacobian
142
6.6.5
Covering Equations
142
6.6.6
Topological Index
143
6.6.7
Transition Matrix
144
6.6.8
Branched Manifold
144
6.6.9
Another Cover
144
6.6.10
Comparison of Attractors
145
6.7
Noncommutativity of Lifts
146
6.8
Matrix Index for Lifts of Periodic Orbits
153
Contents xiii
7
M
ultichannel Intermittency
157
7.1
Review of Intermittency
157
7.2
Intermittency and Saddle-Node Bifurcations
158
7.3
Intermittency in Equivariant Dynamical Systems
160
7.4
Two-Channel Intermittency in the
Lorenz
Attractor
161
7.5
Intermittency in Covers of the
Rössler
System
163
7.5.1
Two-Fold Cover
164
7.5.2
Three-Fold Cover
167
7.5.3
Four-Fold Cover
169
8
Driven Two-Dimensional Dynamical Systems
171
8.1
Structure of Dynamical Systems
171
8.1.1
Reducible
172
8.1.2
Fully Reducible
173
8.1.3
Irreducible
173
8.2
Entrainment and Synchronization
173
8.2.1
Entrainment
174
8.2.2
Synchronization
174
8.3
Driving Systems
175
8.3.1 Rössler
System
175
8.3.2 Lorenz
System
175
8.3.3
Harmonic Oscillator
178
8.4
Undriven Nonlinear Oscillators
179
8.4.1
Linear Oscillators
179
8.4.2
Nonlinear Oscillators
179
8.4.3
Fixed Points
179
8.4.4
Stability of Fixed Points
179
8.4.5
Global Stability Conditions
180
8.4.6
Symmetry
182
8.4.7
Origins of These Nonlinear Oscillators
182
8.5
The van
der Pol
Oscillator
185
8.5.1 Rössler
Drive
186
8.5.2 Lorenz
Drive
188
8.5.3
Harmonic Drive
190
8.6
The Duffing Oscillator
197
8.6.1 Rössler
Drive
198
8.6.2 Lorenz
Drive
200
8.6.3
Harmonic Drive
204
xiv Contents
8.7
The Takens-Bogdanov Oscillator
206
8.7.1 Rössler
Drive
208
8.7.2 Lorenz
Drive
210
8.7.3
Harmonic Drive
214
8.8
Modding Out the Symmetry
216
8.9
CN Symmetries
220
8.10
Covers and Images in the Torus
221
8.11
Quantizing Chaos
222
8.11.1
The Equivalence Principle
224
8.11.2
Rotating Transformations
224
8.11.3
Dynamical Measures
225
8.11.4
Universal Image
227
8.11.5
Harmonic Maps
228
8.11.6
Subharmonic Lifts and Quantum Numbers
228
8.11.7
Application to Autonomous Systems
229
9
Larger Symmetries
233
9.1
Complex Dynamical Systems
233
9.1.1
Projection to Five Complex Dimensions
234
9.1.2
Symmetries
234
9.1.3
Dynamics
235
9.1.4
Dynamics in Polar Coordinates
236
9.1.5
Symmetry Reduction
238
9.1.6
Dimensional Reduction
240
9.2
Continuous Rotations
241
9.2.1
Zeghlache-Mandel System
243
9.2.2
Symmetries
243
9.2.3
Fixed Points
244
9.2.4
Dynamics
245
9.2.5
Dynamics in Polar Coordinates
248
9.2.6
Reduction of Dimension by Symmetry
248
9.2.7
Dynamical Reduction
249
9.3
Thomas s System
252
9.3.1
Fixed Points
252
9.3.2
Symmetry
254
9.3.3
Bifurcation Studies
255
9.3.4
Periodic Orbits
256
9.3.5
Period-Doubling Cascades
256
Contents xv
9.4
Symmetry Breaking and Restoration
257
9.4.1
Modeling
257
9.4.2
Entrainment
258
9.4.3
Mutual Entrainment
259
PART II
Mathematical Foundations
10
Group Theory Basics
265
10.1
Dynamical Systems
265
10.2
Change of Basis
267
10.3
Symmetry under Linear Transformations
269
10.4
Groups of Linear Transformations
271
10.5
Properties of Equivanant Dynamical Systems
273
10.6
Partition of Phase Space
275
10.7
Representations of Groups
278
10.7.1
Definition of Representation
278
10.7.2
Equivalent Representations
278
10.7.3
Faithful Representations
279
10.7.4
Reducible Representations
279
10.7.5
Irreducible Representations
280
10.7.6
Arbitrary Representations
281
10.7.7
Real Representations
282
10.7.8
Important Representations
283
10.8
How Many Ways Can a Group Appear?
284
10.9
Subgroups and Cosets
288
10.10
Singular Sets
289
10.11
Orbits (Kinetics and Dynamics)
291
11
Invariant Polynomials
293
11.1
Invariant Polynomials
293
11.1.1
What Are They?
293
11.1.2
How to Construct Invariant Polynomials
294
11.1.3
How Many Invariant Polynomials Are There?
296
11.2
Generators and Relations
299
11.2.1
Integrity Basis
299
11.2.2
How to Construct an Integrity Basis
299
11.2.3
Relations
300
xvi Contents
11.2.4
How to Read the Generating Function
303
11.3
Equivariant Polynomials
306
11.3.1
How Many Equivariant Polynomials Are There?
307
11.3.2
How to Construct Equivariant Polynomials
308
11.4
B asis
Sets for Irreducible Representations
310
12
Equivariant Dynamics in RN
313
12.1
Properties of Equivariant Dynamical Systems
313
12.1.1
Fixed Points
313
12.1.2
Flows on SS{g)
314
12.1.3
Structure of Equivariant Dynamical Systems
315
12.2
Injection RN (X)^RK(p)
317
12.2.1
Jacobian of the Transformation
318
12.2.2
Rank
319
12.2.3
The Inversion Map
319
12.3
Structure of Flows in RK(p)
321
12.3.1
Injected Flow is g
-> 1
Image
322
12.3.2
Invariant Manifold
/дт(0)
323
12.3.3
Stratification IN(c)
324
12.3.4
Attracting Manifold
325
12.3.5
Structure of the Invariant Equations
326
12.4
Projection RK(p) | RN(u)
326
12.4.1
Coordinates
и
and Coordinates
ρ
326
12.4.2
Radicals
329
12.4.3
Inversion
и
-»
ρ
330
12.5
Structure of Flows in RN(u)
330
12.5.1
Composition of Jacobians
331
12.5.2
Form of Hows
332
12.5.3
Polynomial Dynamical Systems
333
13
Covering Dynamical Systems
335
13.1
Image Attractors
335
13.1.1
Phase Space
335
13.1.2
Control Parameter Space
336
13.1.3
Image Dynamical Equations
336
13.1.4
Fixed Points
336
13.1.5
Symbolic Dynamics
338
13.1.6
Transition Matrices
338
Contents xvii
13.1.7 Topological
Entropy
340
13.1.8 Higher
Dimensions
340
13.2 Lifts
to
Covers 341
13.2.1
Symmetry
Group
Q
341
13.2.2 Phase Space 342
13.2.3
Representations
342
13.2.4
Local Diffeomorphism
344
13.3
Covering Attractors
345
13.3.1
Phase Space
345
13.3.2
Control Parameter Space
345
13.3.3
Lipschitz Conditions
346
13.3.4
Fixed Points
346
13.3.5
Lyapunov Exponents and Fractal Dimensions
347
13.3.6
Symbolic Dynamics
347
13.3.7
Transition Matrices
348
13.3.8
Topological Entropy
351
13.4
Index
351
13.5
Spectrum of Covers
353
13.5.1
Topological Selection Rules
353
13.5.2
Connectedness
353
13.5.3
Structurally Stable Covers
354
13.5.4
C4-Equivariant
Covers
355
13.5.5
IVEquivariant Covers
356
13.6
Lifts of Orbits
357
13.7
Structurally Unstable Covers
360
14
Symmetries Due to Symmetry
363
14.1 Schur
Symmetries
363
14.1.1
Schur s Lemmas
363
14.1.2
How to Use Schur s Lemmas
367
14.1.3
Application to Equivariant Dynamical Systems
369
14.2
Cauchy-Riemann Symmetries
372
14.2.1
Cauchy-Riemann Conditions for Analytic Functions
372
14.2.2
Examples for Analytic Functions
374
14.2.3
Application to Equivariant Dynamical Systems
375
14.3
Clebsch-Gordan Symmetries
377
14.3.1
Clebsch-Gordan Coupling Matrices
378
14.3.2
Applications to Finite Groups
378
xviii Contents
14.3.3 Application
to Equivariant Dynamical
Systems 380
14.4
Continuations
383
14.4.1
Analytic Continuation
384
14.4.2
Topological Continuation
386
14.4.3
Group Continuation
388
PART III
Symmetry without Groups: Topology
15
Symmetry without Groups: Topological Symmetry
393
15.1
Covers and Images
393
15.2
Bounding Tori
394
15.2.1
Strange Attractors in R3
395
15.2.2
Blow-Ups of Branched Manifolds
395
15.2.3
The Boundary
396
15.2.4
Euler
Characteristic
397
15.2.5
Poincaré-Hopf
Index Theorem
397
15.2.6
Surface Singularities
398
15.3
Canonical Form
399
15.4
Properties of the Canonical Form
402
15.4.1
Disk Boundary
403
15.4.2
Interior Holes of Two Types
403
15.4.3
Homotopy Group
405
15.4.4
Further Properties
407
15.5
How to Classify Dressed Tori
408
15.5.1
Singular Holes
408
15.5.2
Uniflow Holes
411
15.5.3
Branch Lines
413
15.5.4
Allowable Orbits
414
15.6
Transition Matrices
415
15.6.1
The Cyclic Matrix
416
15.6.2
The Structure Matrix
417
15.6.3
Encoding the Structure Matrix
417
15.7
Enumeration to Genus
9 418
15.7.1
Genus
1 418
15.7.2
Genus
> 1 419
15.7.3
Genus
3 419
15.7.4
Genus
4 419
Contents xix
15.7.5 Genus 5 419
15.7.6 Genus 6 420
15.7.7 Genus 7 420
15.7.8 Genus 8 420
15.7.9 Genus 9 421
15.7.10
Entropy
421
15.8
Compatible Branched Manifolds
426
15.8.1
Return Maps
426
15.8.2
Selection Rules
429
15.8.3
Perestroikas of Branched Manifolds
430
15.9
How to Extract from Experimental Data
432
15.10
Perestroikas of Canonical Tori
437
15.10.1
Exterior Flow Tubes
439
15.10.2
Interior Flow Tubes
442
15.11
Topologically Equivariant Covers
444
16
All the Covers of the Horseshoe
449
16.1
Review: Some of the Covers of the Horseshoe
449
16.2
Universal: Covering Groups and Image Attractors
453
16.2.1
Cartan s Theorem
453
16.2.2
Groups and Diffeomorphisms
455
16.3
All the Covers of the Horseshoe
456
16.3.1
Structurally Unstable Covers
456
16.3.2
Structurally Stable Covers
458
16.4
Intrinsic Embeddings
460
16.4.1
Embeddings of Branched Manifolds
460
16.4.2
Embeddings of Flows
461
16.5
Extrinsic Embeddings
462
16.5.1
Embeddings of Branched Manifolds
462
16.5.2
Embeddings of Flows
463
16.6
Once a Horseshoe, Always a Horseshoe
469
Appendix A A Potpourri of Equivariant Systems
471
A.I Three-Dimensional Systems
471
A.
1.1 Lorenz
System
471
A.
1.2
Thermal Convection Loop
473
A.
1.3
Rikitake System
475
A.
1.4 Homopolar
Dynamo
477
xx Contents
Α.
1.5 Still
Another
Lorenz-Like Attractor 480
Α.
1.6
Chen and Ueta System
480
Α.
1.7
Burke and Shaw System
481
A.
1.8
Leipnik and Newton System
483
A.
1.9
Simple Models for Pulsating Stars
484
A.
1.10
Minimal Jerk System
486
A.
1.11
Kremliovsky System
488
A.
1.12
An Equivariant
Rössler
System
489
A.
1.13
Duan-Wang-Huang System
489
A.
1.14
Matsumoto-Chua System
490
A.
1.15
Multispiral Attractors
493
A.
1.16
Thomas Systems
495
A.
1.17
Liu and Chen System
496
A.
1.18 Lü,
Chen, and Cheng System
499
A.2 Higher Dimensional Systems
501
A.2.1 4D Chaotic System
501
A.2.2 5D Laser Model by Zeghlache and
Mandel 501
A.2.3 6D Chaotic Model for Solar Activity
503
A.2.4 9D Model for
a Rayleigh-Bénard
Convection
505
A.2.5 10D Model for Wave-Wave Interaction in a Plasma
507
A.3 Nonautonomous Systems
508
A.3.
1
van
der Pol
System
508
A.
3.2
Duffing System
509
A.4 Other Cases
511
A.4.
1
Three Hamiltonian Flows
511
A
.4.2
1-D Delay Differential Equation
515
A.4.3
3D
Discontinuous System
515
References
519
Index
529
|
| adam_txt |
Contents
PARTI
Examples and Simple Applications
1
Introduction
3
1.1
Swift Survey of Symmetries
3
1.1.1
Lasers
4
1.1.2
SunspotData
4
1.1.3
Paleomagnetic Data
4
1.1.4
Fluid Experiments
5
1.1.5
Electronic Circuits
6
1.2
Deeper Probe of Symmetries
6
1.2.1
Navier-Stokes PDEs
-» Lorenz
ODEs
7
1.2.2
Maxwell-Bloch Equations
—>
Laser Equations
10
1.3
Mathematical Motivation
12
1.3.1
Classification Theory
12
1.3.2
Cover-Image Relations
13
1.4
Overview of the Book
13
2
Simple Symmetries
17
2.1
Equations with Two-Fold Symmetry
17
2.1.1
Rotation Symmetry
18
χ
Contents
2.1.2 Inversion
Symmetry
20
2.1.3
Reflection Symmetry
21
2.2
Existence and Uniqueness of Solutions
22
2.3
Symmetry and the Fundamental Theorem
23
2.4
Fixed Point Distributions
24
2.5
Classification of Strange Attractors
28
2.5.1
Background
28
2.5.2
Topological Classification of Strange Attractors
29
2.5.3
Extracting Branched Manifolds from Data
32
2.5.4
Linking Number Tables
34
2.5.5
Symmetry
37
2.5.6
Application
37
2.6
Symbolic Dynamics
38
2.7
Periodic Orbits
39
2.8
Topological Entropy
40
2.9
Return Maps
41
3
Image Dynamical Systems
45
3.1
Diffeomorphisms: Global and Local
45
3.2 2 —> 1
Local Diffeomorphisms
48
3.3
Image Equations
50
3.3.1 Lorenz
Equations: TZz
(π)
51
3.3.2
Burke and Shaw Equations: TZZ
(π)
51
3.3.3
General Case: Uz
(π)
52
3.3.4
General Case: V
53
3.3.5
General Case:
σζ
55
3.4
Fixed Point Distributions
56
3.5
Branched Manifolds and Their Images
58
3.6
Symbolic Dynamics
60
3.7
Periodic Orbits
61
3.8
Poincaré
Sections and First-Return Maps
63
3.9
Tips for Integration
65
4
Covers
67
4.1
Local Diffeomorphisms
68
4.2
Singular Sets
69
4.3
Lifts to Rotation Invariant Systems: Topological Indices
72
4.4
Branched Manifolds
72
Contents xi
4.5
Periodic
Orbits 77
4.6
Poincaré
Sections and First-Return Maps
78
4.7
Fractal Dimensions and Lyapunov Exponents
80
4.8
Continuations
80
4.8.1
Topological Continuation
80
4.8.2
Group Continuation
81
4.9
Horseshoe versus Reverse Horseshoe
82
4.10
Lifts of the
Smale
Horseshoe
84
4.11
Tips for Integration
84
5
Peeling Bifurcations
87
5.1
Structural Stability
88
5.2
The
Perestroika
from
(1,1)
to
(0,1) 89
5.2.1
Branched Manifolds
89
5.2.2
Transition Matrices
91
5.2.3
Return Maps
91
5.2.4
Periodic Orbits
92
5.3
The
Perestroika
from
(0,1 )
to
(0,0) 93
5.3.1
Branched Manifolds
93
5.3.2
Transition Matrices
95
5.3.3
Return Maps
96
5.3.4
Periodic Orbits
96
5.4
Structurally Unstable Strange Attractors
97
5.5
The Peeling Bifurcation
100
5.6
Application: Sunspot Covers
101
6
Three-Fold and Four-Fold Covers
105
6.1
Image Dynamical Systems
] 05
6.1.1 Rössler
System
106
6.1.2
Proto-Lorenz System
106
6.1.3
Proto-Burke and Shaw Equations
107
6.2
η
-Fold Covers and Complex Variables
107
6.2.1
Covers of the
Rössler
Equations
108
6.2.2
Covers of the Proto-Lorenz Equations
109
6.2.3
Covers of the Proto-Burke and Shaw System
110
6.3
Covers with
Сз
Symmetry
110
6.3.1
The Group
110
6.3.2
Invariant Polynomials
111
xii Contents
6.3.3
The Jacobian
111
6.3.4
Covering Equations
112
6.3.5
Topołogical
Index
112
6.3.6
Covering Branched Manifolds
112
6.3.7
Symmetry-Adapted Labeling
114
6.3.8
Transition Matrices
115
6.3.9
Periodic Orbits
115
6.3.10
Linking Numbers
118
6.3.11
Poincaré
Sections and First-Return Maps
121
5.4
Covers with C4 Symmetry
121
6.4.1
The Group
121
6.4.2
Invariant and Covariant Polynomials
122
6.4.3
The Jacobian
123
6.4.4
Covering Equations
123
6.4.5
Topological Index
124
6.4.6
Covering Branched Manifolds
124
6.4.7
Symmetry-Adapted Labeling
124
6.4.8
Transition Matrices
125
6.4.9
Periodic Orbits
125
6.4,10
Linking Numbers
128
6.4.11
Poincaré
Sections and First-Return Maps
128
6.5
Cover Comparisons
130
6.5.1 Rössler
System
130
6.5.2
Proto-Lorenz System
133
6.5.3
Proto-Burke and Shaw System
136
6.6
Covers with V4 Symmetry
141
6.6.1
The Group
141
6.6.2
Invariant Polynomials
141
6.6.3
Invariant Coordinates
141
6.6.4
The Jacobian
142
6.6.5
Covering Equations
142
6.6.6
Topological Index
143
6.6.7
Transition Matrix
144
6.6.8
Branched Manifold
144
6.6.9
Another Cover
144
6.6.10
Comparison of Attractors
145
6.7
Noncommutativity of Lifts
146
6.8
Matrix Index for Lifts of Periodic Orbits
153
Contents xiii
7
M
ultichannel Intermittency
157
7.1
Review of Intermittency
157
7.2
Intermittency and Saddle-Node Bifurcations
158
7.3
Intermittency in Equivariant Dynamical Systems
160
7.4
Two-Channel Intermittency in the
Lorenz
Attractor
161
7.5
Intermittency in Covers of the
Rössler
System
163
7.5.1
Two-Fold Cover
164
7.5.2
Three-Fold Cover
167
7.5.3
Four-Fold Cover
169
8
Driven Two-Dimensional Dynamical Systems
171
8.1
Structure of Dynamical Systems
171
8.1.1
Reducible
172
8.1.2
Fully Reducible
173
8.1.3
Irreducible
173
8.2
Entrainment and Synchronization
173
8.2.1
Entrainment
174
8.2.2
Synchronization
174
8.3
Driving Systems
175
8.3.1 Rössler
System
175
8.3.2 Lorenz
System
175
8.3.3
Harmonic Oscillator
178
8.4
Undriven Nonlinear Oscillators
179
8.4.1
Linear Oscillators
179
8.4.2
Nonlinear Oscillators
179
8.4.3
Fixed Points
179
8.4.4
Stability of Fixed Points
179
8.4.5
Global Stability Conditions
180
8.4.6
Symmetry
182
8.4.7
Origins of These Nonlinear Oscillators
182
8.5
The van
der Pol
Oscillator
185
8.5.1 Rössler
Drive
186
8.5.2 Lorenz
Drive
188
8.5.3
Harmonic Drive
190
8.6
The Duffing Oscillator
197
8.6.1 Rössler
Drive
198
8.6.2 Lorenz
Drive
200
8.6.3
Harmonic Drive
204
xiv Contents
8.7
The Takens-Bogdanov Oscillator
206
8.7.1 Rössler
Drive
208
8.7.2 Lorenz
Drive
210
8.7.3
Harmonic Drive
214
8.8
Modding Out the Symmetry
216
8.9
CN Symmetries
220
8.10
Covers and Images in the Torus
221
8.11
Quantizing Chaos
222
8.11.1
The Equivalence Principle
224
8.11.2
Rotating Transformations
224
8.11.3
Dynamical Measures
225
8.11.4
Universal Image
227
8.11.5
Harmonic Maps
228
8.11.6
Subharmonic Lifts and Quantum Numbers
228
8.11.7
Application to Autonomous Systems
229
9
Larger Symmetries
233
9.1
Complex Dynamical Systems
233
9.1.1
Projection to Five Complex Dimensions
234
9.1.2
Symmetries
234
9.1.3
Dynamics
235
9.1.4
Dynamics in Polar Coordinates
236
9.1.5
Symmetry Reduction
238
9.1.6
Dimensional Reduction
240
9.2
Continuous Rotations
241
9.2.1
Zeghlache-Mandel System
243
9.2.2
Symmetries
243
9.2.3
Fixed Points
244
9.2.4
Dynamics
245
9.2.5
Dynamics in Polar Coordinates
248
9.2.6
Reduction of Dimension by Symmetry
248
9.2.7
Dynamical Reduction
249
9.3
Thomas's System
252
9.3.1
Fixed Points
252
9.3.2
Symmetry
254
9.3.3
Bifurcation Studies
255
9.3.4
Periodic Orbits
256
9.3.5
Period-Doubling Cascades
256
Contents xv
9.4
Symmetry Breaking and Restoration
257
9.4.1
Modeling
257
9.4.2
Entrainment
258
9.4.3
Mutual Entrainment
259
PART II
Mathematical Foundations
10
Group Theory Basics
265
10.1
Dynamical Systems
265
10.2
Change of Basis
267
10.3
Symmetry under Linear Transformations
269
10.4
Groups of Linear Transformations
271
10.5
Properties of Equivanant Dynamical Systems
273
10.6
Partition of Phase Space
275
10.7
Representations of Groups
278
10.7.1
Definition of Representation
278
10.7.2
Equivalent Representations
278
10.7.3
Faithful Representations
279
10.7.4
Reducible Representations
279
10.7.5
Irreducible Representations
280
10.7.6
Arbitrary Representations
281
10.7.7
Real Representations
282
10.7.8
Important Representations
283
10.8
How Many Ways Can a Group Appear?
284
10.9
Subgroups and Cosets
288
10.10
Singular Sets
289
10.11
Orbits (Kinetics and Dynamics)
291
11
Invariant Polynomials
293
11.1
Invariant Polynomials
293
11.1.1
What Are They?
293
11.1.2
How to Construct Invariant Polynomials
294
11.1.3
How Many Invariant Polynomials Are There?
296
11.2
Generators and Relations
299
11.2.1
Integrity Basis
299
11.2.2
How to Construct an Integrity Basis
299
11.2.3
Relations
300
xvi Contents
11.2.4
How to Read the Generating Function
303
11.3
Equivariant Polynomials
306
11.3.1
How Many Equivariant Polynomials Are There?
307
11.3.2
How to Construct Equivariant Polynomials
308
11.4
B asis
Sets for Irreducible Representations
310
12
Equivariant Dynamics in RN
313
12.1
Properties of Equivariant Dynamical Systems
313
12.1.1
Fixed Points
313
12.1.2
Flows on SS{g)
314
12.1.3
Structure of Equivariant Dynamical Systems
315
12.2
Injection RN (X)^RK(p)
317
12.2.1
Jacobian of the Transformation
318
12.2.2
Rank
319
12.2.3
The Inversion Map
319
12.3
Structure of Flows in RK(p)
321
12.3.1
Injected Flow is \g\
-> 1
Image
322
12.3.2
Invariant Manifold
/дт(0)
323
12.3.3
Stratification IN(c)
324
12.3.4
Attracting Manifold
325
12.3.5
Structure of the Invariant Equations
326
12.4
Projection RK(p) | RN(u)
326
12.4.1
Coordinates
и
and Coordinates
ρ
326
12.4.2
"Radicals"
329
12.4.3
Inversion
и
-»
ρ
330
12.5
Structure of Flows in RN(u)
330
12.5.1
Composition of Jacobians
331
12.5.2
Form of Hows
332
12.5.3
Polynomial Dynamical Systems
333
13
Covering Dynamical Systems
335
13.1
Image Attractors
335
13.1.1
Phase Space
335
13.1.2
Control Parameter Space
336
13.1.3
Image Dynamical Equations
336
13.1.4
Fixed Points
336
13.1.5
Symbolic Dynamics
338
13.1.6
Transition Matrices
338
Contents xvii
13.1.7 Topological
Entropy
340
13.1.8 Higher
Dimensions
340
13.2 Lifts
to
Covers 341
13.2.1
Symmetry
Group
Q
341
13.2.2 Phase Space 342
13.2.3
Representations
342
13.2.4
Local Diffeomorphism
344
13.3
Covering Attractors
345
13.3.1
Phase Space
345
13.3.2
Control Parameter Space
345
13.3.3
Lipschitz Conditions
346
13.3.4
Fixed Points
346
13.3.5
Lyapunov Exponents and Fractal Dimensions
347
13.3.6
Symbolic Dynamics
347
13.3.7
Transition Matrices
348
13.3.8
Topological Entropy
351
13.4
Index
351
13.5
Spectrum of Covers
353
13.5.1
Topological Selection Rules
353
13.5.2
Connectedness
353
13.5.3
Structurally Stable Covers
354
13.5.4
C4-Equivariant
Covers
355
13.5.5
IVEquivariant Covers
356
13.6
Lifts of Orbits
357
13.7
Structurally Unstable Covers
360
14
Symmetries Due to Symmetry
363
14.1 Schur
Symmetries
363
14.1.1
Schur's Lemmas
363
14.1.2
How to Use Schur's Lemmas
367
14.1.3
Application to Equivariant Dynamical Systems
369
14.2
Cauchy-Riemann Symmetries
372
14.2.1
Cauchy-Riemann Conditions for Analytic Functions
372
14.2.2
Examples for Analytic Functions
374
14.2.3
Application to Equivariant Dynamical Systems
375
14.3
Clebsch-Gordan Symmetries
377
14.3.1
Clebsch-Gordan Coupling Matrices
378
14.3.2
Applications to Finite Groups
378
xviii Contents
14.3.3 Application
to Equivariant Dynamical
Systems 380
14.4
Continuations
383
14.4.1
Analytic Continuation
384
14.4.2
Topological Continuation
386
14.4.3
Group Continuation
388
PART III
Symmetry without Groups: Topology
15
Symmetry without Groups: "Topological Symmetry"
393
15.1
Covers and Images
393
15.2
Bounding Tori
394
15.2.1
Strange Attractors in R3
395
15.2.2
Blow-Ups of Branched Manifolds
395
15.2.3
The Boundary
396
15.2.4
Euler
Characteristic
397
15.2.5
Poincaré-Hopf
Index Theorem
397
15.2.6
Surface Singularities
398
15.3
Canonical Form
399
15.4
Properties of the Canonical Form
402
15.4.1
Disk Boundary
403
15.4.2
Interior Holes of Two Types
403
15.4.3
Homotopy Group
405
15.4.4
Further Properties
407
15.5
How to Classify Dressed Tori
408
15.5.1
Singular Holes
408
15.5.2
Uniflow Holes
411
15.5.3
Branch Lines
413
15.5.4
Allowable Orbits
414
15.6
Transition Matrices
415
15.6.1
The Cyclic Matrix
416
15.6.2
The Structure Matrix
417
15.6.3
Encoding the Structure Matrix
417
15.7
Enumeration to Genus
9 418
15.7.1
Genus
1 418
15.7.2
Genus
> 1 419
15.7.3
Genus
3 419
15.7.4
Genus
4 419
Contents xix
15.7.5 Genus 5 419
15.7.6 Genus 6 420
15.7.7 Genus 7 420
15.7.8 Genus 8 420
15.7.9 Genus 9 421
15.7.10
Entropy
421
15.8
Compatible Branched Manifolds
426
15.8.1
Return Maps
426
15.8.2
Selection Rules
429
15.8.3
Perestroikas of Branched Manifolds
430
15.9
How to Extract from Experimental Data
432
15.10
Perestroikas of Canonical Tori
437
15.10.1
Exterior Flow Tubes
439
15.10.2
Interior Flow Tubes
442
15.11
"Topologically Equivariant" Covers
444
16
All the Covers of the Horseshoe
449
16.1
Review: Some of the Covers of the Horseshoe
449
16.2
Universal: Covering Groups and Image Attractors
453
16.2.1
Cartan's Theorem
453
16.2.2
Groups and Diffeomorphisms
455
16.3
All the Covers of the Horseshoe
456
16.3.1
Structurally Unstable Covers
456
16.3.2
Structurally Stable Covers
458
16.4
Intrinsic Embeddings
460
16.4.1
Embeddings of Branched Manifolds
460
16.4.2
Embeddings of Flows
461
16.5
Extrinsic Embeddings
462
16.5.1
Embeddings of Branched Manifolds
462
16.5.2
Embeddings of Flows
463
16.6
Once a Horseshoe, Always a Horseshoe
469
Appendix A A Potpourri of Equivariant Systems
471
A.I Three-Dimensional Systems
471
A.
1.1 Lorenz
System
471
A.
1.2
Thermal Convection Loop
473
A.
1.3
Rikitake System
475
A.
1.4 Homopolar
Dynamo
477
xx Contents
Α.
1.5 Still
Another
Lorenz-Like Attractor 480
Α.
1.6
Chen and Ueta System
480
Α.
1.7
Burke and Shaw System
481
A.
1.8
Leipnik and Newton System
483
A.
1.9
Simple Models for Pulsating Stars
484
A.
1.10
Minimal Jerk System
486
A.
1.11
Kremliovsky System
488
A.
1.12
An Equivariant
Rössler
System
489
A.
1.13
Duan-Wang-Huang System
489
A.
1.14
Matsumoto-Chua System
490
A.
1.15
Multispiral Attractors
493
A.
1.16
Thomas Systems
495
A.
1.17
Liu and Chen System
496
A.
1.18 Lü,
Chen, and Cheng System
499
A.2 Higher Dimensional Systems
501
A.2.1 4D Chaotic System
501
A.2.2 5D Laser Model by Zeghlache and
Mandel 501
A.2.3 6D Chaotic Model for Solar Activity
503
A.2.4 9D Model for
a Rayleigh-Bénard
Convection
505
A.2.5 10D Model for Wave-Wave Interaction in a Plasma
507
A.3 Nonautonomous Systems
508
A.3.
1
van
der Pol
System
508
A.
3.2
Duffing System
509
A.4 Other Cases
511
A.4.
1
Three Hamiltonian Flows
511
A
.4.2
1-D Delay Differential Equation
515
A.4.3
3D
Discontinuous System
515
References
519
Index
529 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Gilmore, Robert 1941- Letellier, Christophe |
| author_GND | (DE-588)122549058 |
| author_facet | Gilmore, Robert 1941- Letellier, Christophe |
| author_role | aut aut |
| author_sort | Gilmore, Robert 1941- |
| author_variant | r g rg c l cl |
| building | Verbundindex |
| bvnumber | BV023278997 |
| callnumber-first | Q - Science |
| callnumber-label | QA174 |
| callnumber-raw | QA174.7.S96 |
| callnumber-search | QA174.7.S96 |
| callnumber-sort | QA 3174.7 S96 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 350 UG 3900 |
| ctrlnum | (OCoLC)494380346 (DE-599)BVBBV023278997 |
| dewey-full | 003/.857 |
| dewey-hundreds | 000 - Computer science, information, general works |
| dewey-ones | 003 - Systems |
| dewey-raw | 003/.857 |
| dewey-search | 003/.857 |
| dewey-sort | 13 3857 |
| dewey-tens | 000 - Computer science, information, general works |
| discipline | Physik Informatik Mathematik |
| discipline_str_mv | Physik Informatik Mathematik |
| format | Book |
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| id | DE-604.BV023278997 |
| illustrated | Illustrated |
| index_date | 2024-07-02T20:39:01Z |
| indexdate | 2024-07-09T21:14:49Z |
| institution | BVB |
| isbn | 9780195310658 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016463831 |
| oclc_num | 494380346 |
| open_access_boolean | |
| owner | DE-20 DE-355 DE-BY-UBR DE-11 DE-19 DE-BY-UBM DE-1102 |
| owner_facet | DE-20 DE-355 DE-BY-UBR DE-11 DE-19 DE-BY-UBM DE-1102 |
| physical | XX, 545 S. Ill., graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| spelling | Gilmore, Robert 1941- Verfasser (DE-588)122549058 aut The symmetry of chaos Robert Gilmore and Christophe Letellier Oxford [u.a.] Oxford Univ. Press 2007 XX, 545 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chaos (théorie des systèmes) ram Symétrie ram Chaotic behavior in systems Symmetry (Mathematics) Symmetrie (DE-588)4058724-1 gnd rswk-swf Nichtlineares dynamisches System (DE-588)4126142-2 gnd rswk-swf Chaos (DE-588)4191419-3 gnd rswk-swf Nichtlineares dynamisches System (DE-588)4126142-2 s Symmetrie (DE-588)4058724-1 s Chaos (DE-588)4191419-3 s DE-604 Letellier, Christophe Verfasser aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016463831&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gilmore, Robert 1941- Letellier, Christophe The symmetry of chaos Chaos (théorie des systèmes) ram Symétrie ram Chaotic behavior in systems Symmetry (Mathematics) Symmetrie (DE-588)4058724-1 gnd Nichtlineares dynamisches System (DE-588)4126142-2 gnd Chaos (DE-588)4191419-3 gnd |
| subject_GND | (DE-588)4058724-1 (DE-588)4126142-2 (DE-588)4191419-3 |
| title | The symmetry of chaos |
| title_auth | The symmetry of chaos |
| title_exact_search | The symmetry of chaos |
| title_exact_search_txtP | The symmetry of chaos |
| title_full | The symmetry of chaos Robert Gilmore and Christophe Letellier |
| title_fullStr | The symmetry of chaos Robert Gilmore and Christophe Letellier |
| title_full_unstemmed | The symmetry of chaos Robert Gilmore and Christophe Letellier |
| title_short | The symmetry of chaos |
| title_sort | the symmetry of chaos |
| topic | Chaos (théorie des systèmes) ram Symétrie ram Chaotic behavior in systems Symmetry (Mathematics) Symmetrie (DE-588)4058724-1 gnd Nichtlineares dynamisches System (DE-588)4126142-2 gnd Chaos (DE-588)4191419-3 gnd |
| topic_facet | Chaos (théorie des systèmes) Symétrie Chaotic behavior in systems Symmetry (Mathematics) Symmetrie Nichtlineares dynamisches System Chaos |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016463831&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT gilmorerobert thesymmetryofchaos AT letellierchristophe thesymmetryofchaos |