The topology of chaos: Alice in stretch and squeezeland
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
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Weinheim
Wiley-VCH
2011
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| Edition: | 2. rev. and enl. ed. |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XXIV, 593 S. Ill., graph. Darst. 240 mm x 170 mm |
| ISBN: | 9783527410675 3527410678 |
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MARC
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| 245 | 1 | 0 | |a The topology of chaos |b Alice in stretch and squeezeland |c Robert Gilmore and Marc Lefranc |
| 250 | |a 2. rev. and enl. ed. | ||
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| 300 | |a XXIV, 593 S. |b Ill., graph. Darst. |c 240 mm x 170 mm | ||
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IMAGE 1
CONTENTS
PREFACE TO SECOND EDITION XVII
PREFACE TO THE FIRST EDITION XIX
1 INTRODUCTION 1
1.1 BRIEF REVIEW OF USEFUL CONCEPTS 2 1.2 LASER WITH MODULATED LOSSES 4
1.3 OBJECTIVES OF A NEW ANALYSIS PROCEDURE 11
1.4 PREVIEW OF RESULTS 32
1.5 ORGANIZATION OF THIS WORK 14
2 DISCRETE DYNAMICAL SYSTEMS: MAPS 19 2.1 INTRODUCTION 19
2.2 LOGISTIC MAP 20
2.3 BIFURCATION DIAGRAMS 22
2.4 ELEMENTARY BIFURCATIONS IN THE LOGISTIC MAP 25 2.4.1 SADDLE-NODE
BIFURCATION 25 2.4.2 PERIOD-DOUBLING BIFURCATION 29 2.5 MAP CONJUGACY 32
2.5.1 CHANGES OF COORDINATES 32 2.5.2 INVARIANTS OF CONJUGACY 33 2.6
FULLY DEVELOPED CHAOS IN THE LOGISTIC MAP 34
2.6.1 ITERATES OF THE TENT MAP 35 2.6.2 LYAPUNOV EXPONENTS 36 2.6.3
SENSITIVITY TO INITIAL CONDITIONS AND MIXING 37 2.6.4 CHAOS AND DENSITY
OF (UNSTABLE) PERIODIC ORBITS 38
2.6.4.1 NUMBER OF PERIODIC ORBITS OF THE TENT MAP 38 2.6.4.2
EXPANSIVENESS IMPLIES INFINITELY MANY PERIODIC ORBITS 39 2.6.5 SYMBOLIC
CODING OF TRAJECTORIES: FIRST APPROACH 40
2.7 ONE-DIMENSIONAL SYMBOLIC DYNAMICS 42 2.7.1 PARTITIONS 42
2.7.2 SYMBOLIC DYNAMICS OF EXPANSIVE MAPS 44 2.7.3 GRAMMAR OF CHAOS:
FIRST APPROACH 48 2.7.3.1 INTERVAL ARITHMETICS AND INVARIANT INTERVAL 48
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1011200597
DIGITALISIERT DURCH
IMAGE 2
VI I CONTENTS
2.7.3.2 EXISTENCE OF FORBIDDEN SEQUENCES 49
2J A KNEADING THEORY 51 2.7.4.1 ORDERING OF ITINERARIES 52 2.7'.4.2
ADMISSIBLE SEQUENCES 54 2.7.5 BIFURCATION DIAGRAM OF THE LOGISTIC MAP
REVISITED 55
2.7.5.1 SADDLE-NODE BIFURCATIONS 55 2.7.5.2 PERIOD-DOUBLING BIFURCATIONS
56 2.7.5.3 UNIVERSAL SEQUENCE 57 2.7.5.4 SELF-SIMILAR STRUCTURE OF THE
BIFURCATION DIAGRAM 58 2.8 SHIFT DYNAMICAL SYSTEMS, MARKOV PARTITIONS,
AND ENTROPY 59 2.8.1 SHIFTS OF FINITE TYPE AND TOPOLOGICAL MARKOV CHAINS
59 2.8.2 PERIODIC ORBITS AND TOPOLOGICAL ENTROPY OF A MARKOV CHAIN 61
2.8.3 MARKOV PARTITIONS 63 2.8.4 APPROXIMATION BY MARKOV CHAINS 65 2.8.5
ZETA FUNCTION 65 2.8.6 DEALING WITH GRAMMARS 66 2.8.6.1 SIMPLE GRAMMARS
67
2.8.6.2 COMPLICATED GRAMMARS 69 2.9 FINGERPRINTS OF PERIODIC ORBITS AND
ORBIT FORCING 70 2.9.1 PERMUTATION OF PERIODIC POINTS AS A TOPOLOGICAL
INVARIANT 70 2.9.2 TOPOLOGICAL ENTROPY OF A PERIODIC ORBIT 72 2.9.3
PERIOD 3 IMPLIES CHAOS AND SARKOVSKII'S THEOREM 74 2.9'.4 PERIOD 3 DOES
NOT ALWAYS IMPLY CHAOS:
ROLE OF PHASE-SPACE TOPOLOGY 75 2.9.5 PERMUTATIONS AND ORBIT FORCING 75
2.10 TWO-DIMENSIONAL DYNAMICS: SMALE'S HORSESHOE 77 2.10.1 HORSESHOE MAP
77 2.10.2 SYMBOLIC DYNAMICS OF THE INVARIANT SET 78 2.10.3 DYNAMICAL
PROPERTIES 81 2.10.4 VARIATIONS ON THE HORSESHOE MAP: BAKER MAPS 82 2.11
HENON MAP 85
2.11.1 A ONCE-FOLDING MAP 85 2.11.2 SYMBOLIC DYNAMICS OF THE HENON MAP:
CODING 87 2.11.3 SYMBOLIC DYNAMICS OF THE HENON MAP: GRAMMAR 93 2.12
CIRCLE MAPS 96
2.12.1 A NEW GLOBAL TOPOLOGY 96 2.12.2 FREQUENCY LOCKING AND ARNOLD
TONGUES 96 2.12.3 CHAOTIC CIRCLE MAPS AS LIMITS OF ANNULUS MAPS 100 2.13
ANNULUS MAPS 100
2.14 SUMMARY 104
3 CONTINUOUS DYNAMICAL SYSTEMS: FLOWS 105 3.1 DEFINITION OF DYNAMICAL
SYSTEMS 105 3.2 EXISTENCE AND UNIQUENESS THEOREM 106 3.3 EXAMPLES OF
DYNAMICAL SYSTEMS 107
IMAGE 3
CONTENTS I VII
3.3.1 DUFFING EQUATION 307
3.3.2 VAN DER POL EQUATION 109 3.3.3 LORENZ EQUATIONS 111 3.3.4 ROESSLER
EQUATIONS 113 3.3.5 EXAMPLES OF NONDYNAMICAL SYSTEMS 114 3.3.5.1
EQUATION WITH NON-LIPSCHITZ FORCING TERMS 115 3.3.5.2 DELAY DIFFERENTIAL
EQUATIONS 335 3.3.5.3 STOCHASTIC DIFFERENTIAL EQUATIONS 336 3.3.6
ADDITIONAL OBSERVATIONS 117 3.4 CHANGE OF VARIABLES 320
3.4.1 DIFFEOMORPHISMS 120 3.4.2 EXAMPLES 323 3.4.3 STRUCTURE THEORY 124
3.5 FIXED POINTS 125
3.5.1 DEPENDENCE ON TOPOLOGY OF PHASE SPACE 125 3.5.2 HOW TO FIND FIXED
POINTS IN R N 126 3.5.3 BIFURCATIONS OF FIXED POINTS 127 3.5.4 STABILITY
OF FIXED POINTS 130 3.6 PERIODIC ORBITS 131
3.6.1 LOCATING PERIODIC ORBITS IN R"" 1 X S 1 131 3.6.2 BIFURCATIONS OF
FIXED POINTS 132 3.6.3 STABILITY OF FIXED POINTS 133 3.7 FLOWS NEAR
NONSINGULAR POINTS 334 3.8 VOLUME EXPANSION AND CONTRACTION 336 3.9
STRETCHING AND SQUEEZING 137 3.10 THE FUNDAMENTAL IDEA 138 3.11 SUMMARY
139
4 TOPOLOGICAL INVARIANTS 141 4.1 STRETCHING AND SQUEEZING MECHANISMS 343
4.2 LINKING NUMBERS 145
4.2.1 DEFINITIONS 346 4.2.2 REIDEMEISTER MOVES 347 4.2.3 BRAIDS 348
4.2.4 EXAMPLES 353
4.2.5 LINKING NUMBERS FOR A HORSESHOE 353 4.2.6 LINKING NUMBERS FOR THE
LORENZ ATTRACTOR 154 4.2.7 LINKING NUMBERS FOR THE PERIOD-DOUBLING
CASCADE 354 4.2.8 LOCAL TORSION 155 4.2.9 WRITHE AND TWIST 356 4.2.10
ADDITIONAL PROPERTIES 358 4.3 RELATIVE ROTATION RATES 159 4.3.1
DEFINITION 360
4.3.2 COMPUTING RELATIVE ROTATION RATES 160 4.3.3 HORSESHOE MECHANISM
163
IMAGE 4
VIII I CONTENTS
4.3.4 ADDITIONAL PROPERTIES 168
4.4 RELATION BETWEEN LINKING NUMBERS AND RELATIVE ROTATION RATES 169 4.5
ADDITIONAL USES OF TOPOLOGICAL INVARIANTS 170 4.5.1 BIFURCATION
ORGANIZATION 270 4.5.2 TORUS ORBITS 171 4.5.3 ADDITIONAL REMARKS 171 4.6
SUMMARY 174
5 BRANCHED MANIFOLDS 175 5.1 CLOSED LOOPS 175 5.2 WHAT DOES THIS HAVE TO
DO WITH DYNAMICAL SYSTEMS? 378 5.3 GENERAL PROPERTIES OF BRANCHED
MANIFOLDS 178 5.4 BIRMAN-WILLIAMS THEOREM 181 5.4.1 BIRMAN-WILLIAMS
PROJECTION 182 5.4.2 STATEMENT OF THE THEOREM 183 5.5 RELAXATION OF
RESTRICTIONS 384 5.5.1 STRONGLY CONTRACTING RESTRICTION 184 5.5.2
HYPERBOLIC RESTRICTION 185 5.6 EXAMPLES OF BRANCHED MANIFOLDS 186 5.6.1
SMALE-ROESSLER SYSTEM 186 5.6.2 LORENZ SYSTEM 188 5.6.3 DUFFING SYSTEM
389 5.6.4 VAN DER POL SYSTEM 192 5.7 UNIQUENESS AND NONUNIQUENESS 294
5.7.1 LOCAL MOVES 195 5.7.2 GLOBAL MOVES 197 5.8 STANDARD FORM 200 5.9
TOPOLOGICAL INVARIANTS 201 5.9.1 KNEADING THEORY 202 5.9.2 LINKING
NUMBERS 205 5.9.3 RELATIVE ROTATION RATES 207 5.10 ADDITIONAL PROPERTIES
207
5.10.1 PERIOD AS LINKING NUMBER 208 5.10.2 EBK-IIKE EXPRESSION FOR
PERIODS 208 5.10.3 POINCARE SECTION 209 5.10.4 BLOW-UP OF BRANCHED
MANIFOLDS 210
5.10.5 BRANCHED-MANIFOLD SINGULARITIES 211 5.10.6 CONSTRUCTING A
BRANCHED MANIFOLD FROM A MAP 212 5.10.7 TOPOLOGICAL ENTROPY 233 5.11
SUBTEMPLATES 236
5.11.1 TWO ALTERNATIVES 236 5.11.2 A CHOICE 218 5.11.3 TOPOLOGICAL
ENTROPY 229
5.11.4 SUBTEMPLATES OF THE SMALE HORSESHOE 221
IMAGE 5
CONTENTS I IX
5.11.5 SUBTEMPLATES INVOLVING TONGUES 222
5.12 SUMMARY 224
6 TOPOLOGICAL ANALYSIS PROGRAM 227 6.1 BRIEF SUMMARY OF THE TOPOLOGICAL
ANALYSIS PROGRAM 227 6.2 OVERVIEW OF THE TOPOLOGICAL ANALYSIS PROGRAM
228 6.2.1 FIND PERIODIC ORBITS 228 6.2.2 EMBED IN R 3 229
6.2.3 COMPUTE TOPOLOGICAL INVARIANTS 230 6.2.4 IDENTIFY TEMPLATE 230
6.2.5 VERIFY TEMPLATE 233 6.2.6 MODEL DYNAMICS 232 6.2.7 VALIDATE MODEL
233 6.3 DATA 234
6.3.1 DATA REQUIREMENTS 235 6.3.2 PROCESSING IN THE TIME DOMAIN 236
6.3.3 PROCESSING IN THE FREQUENCY DOMAIN 238 6.3.3.1 HIGH-FREQUENCY
FILTER 238
6.3.3.2 LOW-FREQUENCY FILTER 238 6.3.3.3 DERIVATIVES AND INTEGRALS 239
6.3.3.4 HUBERT TRANSFORMS 240 6.3.3.5 FOURIER INTERPOLATION 241
6.3.3.6 TRANSFORM AND INTERPOLATION 242 6.4 EMBEDDINGS 243
6.4.1 EMBEDDINGS FOR PERIODICALLY DRIVEN SYSTEMS 244 6.4.2 DIFFERENTIAL
EMBEDDINGS 244 6.4.3 DIFFERENTIAL-INTEGRAL EMBEDDINGS 247 6.4.4
EMBEDDINGS WITH SYMMETRY 248 6.4.5 TIME-DELAY EMBEDDINGS 249 6.4.6
COUPLED-OSCILLATOR EMBEDDINGS 251
6.4.7 SVD PROJECTIONS 252 6.4.8 SVD EMBEDDINGS 254 6.4.9 EMBEDDING
THEOREMS 254 6.5 PERIODIC ORBITS 256
6.5.1 CLOSE RETURNS PLOTS FOR FLOWS 256 6.5.1.1 CLOSE RETURNS HISTOGRAMS
258 6.5.1.2 TESTS FOR CHAOS 258 6.5.2 CLOSE RETURNS IN MAPS 259
6.5.2.1 FIRST RETURN MAP 259 6.5.2.2 JRTH RETURN MAP 260 6.5.3 METRIC
METHODS 263 6.6 COMPUTATION OF TOPOLOGICAL INVARIANTS 262
6.6.1 EMBED ORBITS 262 6.6.2 LINKING NUMBERS AND RELATIVE ROTATION RATES
262 6.6.3 LABEL ORBITS 263
IMAGE 6
X I CONTENTS
6.7 IDENTIFY TEMPLATE 263
6.7.1 PERIOD-1 AND PERIOD-2 ORBITS 263 6.7.2 MISSING ORBITS 264 6.7.3
MORE COMPLICATED BRANCHED MANIFOLDS 264
6.8 VALIDATE TEMPLATE 264 6.8.1 PREDICT ADDITIONAL TOPLOGICAL INVARIANTS
265 6.8.2 COMPARE 265
6.8.3 GLOBAL PROBLEM 265 6.9 MODEL DYNAMICS 265 6.10 VALIDATE MODEL 268
6.10.1 QUALITATIVE VALIDATION 269 6.10.2 QUANTITATIVE VALIDATION 269
6.11 SUMMARY 270
7 FOLDING MECHANISMS: A 2 271 7.1 BELOUSOV-ZHABOTINSKII CHEMICAL
REACTION 272 7.1.1 LOCATION OF PERIODIC ORBITS 273 7.1.2 EMBEDDING
ATTEMPTS 274
7.1.3 TOPOLOGICAL INVARIANTS 278 7.1.4 TEMPLATE 282 7.1.5 DYNAMICAL
PROPERTIES 281 7.1.6 MODELS 283
7.1.7 MODEL VERIFICATION 283 7.2 LASER WITH SATURABLE ABSORBER 285 7.3
STRINGED INSTRUMENT 288 7.3.1 EXPERIMENTAL ARRANGEMENT 288 7.3.2 FLOW
MODELS 290 7.3.3 DYNAMICAL TESTS 291 73 A TOPOLOGICAL ANALYSIS 291 7.4
LASERS WITH LOW-INTENSITY SIGNALS 294 7A.I SVD EMBEDDING 295
7.4.2 TEMPLATE IDENTIFICATION 296 7.4.3 RESULTS OF THE ANALYSIS 297 7.5
THE LASERS IN LILLE 297
7.5.1 CLASS B LASER MODEL 298 7.5.2 CO 2 LASER WITH MODULATED LOSSES 304
7.5.3 ND-DOPEDYAG LASER 308 7.5.4 ND-DOPED FIBER LASER 311 7.5.5
SYNTHESIS OF RESULTS 318 7.6 THE LASER IN ZARAGOZA 322
7.7 NEURON WITH SUBTHRESHOLD OSCILLATIONS 328 7.8 SUMMARY 334
8 TEARING MECHANISMS: AJ 337 8.1 LORENZ EQUATIONS 337
8.1.1 FIXED POINTS 338
IMAGE 7
CONTENTS I XI
8.1.2 STABILITY OF FIXED POINTS 339
8.1.3 BIFURCATION DIAGRAM 339 8.1.4 TEMPLATES 341 8.1.5 SHIMIZU-MORIOKA
EQUATIONS 343 8.2 OPTICALLY PUMPED MOLECULAR LASER 343
8.2.1 MODELS 344
8.2.2 AMPLITUDES 346 8.2.3 TEMPLATE 346
8.2.4 ORBITS 347
8.2.5 INTENSITIES 350
8.3 FLUID EXPERIMENTS 352
8.3.1 DATA 352
8.3.2 TEMPLATE 353
8.4 WHYA 3 ? 354
8.5 SUMMARY 354
9 UNFOLDINGS 357
9.1 CATASTROPHE THEORY AS A MODEL 357 9.1.1 OVERVIEW 357
9.1.2 EXAMPLE 358
9.1.3 REDUCTION TO A GERM 359 9.1.4 UNFOLDING THE GERM 363 9.1.5 SUMMARY
OF CONCEPTS 362 9.2 UNFOLDING OF BRANCHED MANIFOLDS: BRANCHED MANIFOLDS
AS GERMS 362 9.2.1 UNFOLDING OF FOLDS 362 9.2.2 UNFOLDING OF TEARS 363
9.3 UNFOLDING WITHIN BRANCHED MANIFOLDS: UNFOLDING OF THE HORSESHOE 365
9.3.1 TOPOLOGY OF FORCING: MAPS 365 9.3.2 TOPOLOGY OF FORCING: FLOWS 366
9.3.3 FORCING DIAGRAMS 369
9.3.3.1 ORBITS WITH ZERO ENTROPY 372 9.3.3.2 ORBITS WITH POSITIVE
ENTROPY 372 9.3.3.3 ADDITIONAL COMMENTS 372 9.3.4 BASIS SETS OF ORBITS
374 9.3.5 COEXISTING BASINS 375 9.4 MISSING ORBITS 375
9.5 ROUTES TO CHAOS 377
9.6 ORBIT FORCING AND TOPOLOGICAL ENTROPY: MATHEMATICAL ASPECTS 378
9.6.1 GENERAL OUTLINE 378 9.6.2 BASIC MATHEMATICAL CONCEPTS 379
9.6.2.1 BRAIDS AND BRAID TYPES 379 9.6.2.2 BRAIDS AND SURFACE
HOMEOMORPHISMS 380 9.6.2.3 NIELSEN-THURSTON CLASSIFICATION 381 9.6.2.4
APPLICATION TO PERIODIC ORBITS AND BRAID TYPES 382 9.7 TOPOLOGICAL
MEASURES OF CHAOS IN EXPERIMENTS 383
IMAGE 8
XII I CONTENTS
9.7.1 MIXING IN FLUIDS 383
9.7.2 CHAOS IN AN OPTICAL PARAMETRIC OSCILLATOR 385 9.8 SUMMARY 389
10 SYMMETRY 392
10.1 INFORMATION LOSS AND GAIN 391 10.1.1 INFORMATION LOSS 391 10.1.2
EXCHANGE OF SYMMETRY 392 10.1.3 INFORMATION GAIN 392 10.1.4 SYMMETRIES
OF THE STANDARD SYSTEMS 392 10.2 COVER AND IMAGE RELATIONS 393 10.2.1
GENERAL SETUP 393 10.3 ROTATION SYMMETRY 1: IMAGES 394 10.3.1 IMAGE
EQUATIONS AND FLOWS 394 10.3.2 IMAGE OF BRANCHED MANIFOLDS 396 10.3.3
IMAGE OF PERIODIC ORBITS 398 10.4 ROTATION SYMMETRY 2: COVERS 400 10.4.1
TOPOLOGICAL INDEX 401 10.4.2 COVERS OF BRANCHED MANIFOLDS 402 10.4.3
COVERS OF PERIODIC ORBITS 403 10.5 PEELING: A NEW GLOBAL BIFURCATION 404
10.5.1 ORBIT PERESTROIKA 405 10.5.2 COVERING EQUATIONS 405 10.6
INVERSION SYMMETRY: DRIVEN OSCILLATORS 407 10.7 DUFFING OSCILLATOR 409
10.8 VAN DER POL OSCILLATOR 413 10.9 SUMMARY 438
11 BOUNDING TORI 419
11.1 STRETCHING & FOLDING VS. TEARING & SQUEEZING 420 11.2 INFLATION 421
11.3 BOUNDARY OF INFLATION 422 11.4 INDEX 423
11.5 PROJECTION 424 11.6 NATURE OF SINGULARITIES 426 11.7 TRINIONS 427
11.8 POINCARE SURFACE OF SECTION 429
11.9 CONSTRUCTION OF CANONICAL FORMS 429 11.10 PERESTROIKAS 432 11.10.1
ENLARGING BRANCHES 433 11.10.2 STARVING BRANCHES 433
11.11 SUMMARY 435
12 REPRESENTATION THEORY FOR STRANGE ATTRACTORS 437 12.1 EMBEDDINGS,
REPRESENTATIONS, EQUIVALENCE 438 12.2 SIMPLEST CLASS OF STRANGE
ATTRACTORS 439
IMAGE 9
CONTENTS I XIII
12.3 REPRESENTATION LABELS 440
12.3.1 PARITY 440 12.3.2 GLOBAL TORSION 441 12.3.3 KNOT TYPE 445 12.4
EQUIVALENCE OF REPRESENTATIONS WITH INCREASING DIMENSION 446 12.4.1
PARITY 447 12.4.2 KNOT TYPE 447 12.4.3 GLOBAL TORSION 448 12.5 GENUS-G
ATTRACTORS 450 12.6 REPRESENTATION LABELS 451 12.6.1 PARITY 451 12.6.2
MULTITORSION INDEX 451 12.6.3 KNOT TYPE 452 12.7 EQUIVALENCE IN
INCREASING DIMENSION 453 12.7.1 PARITY AND KNOT TYPE 453 12.7.2
MULTITORSION INDEX 453 12.8 SUMMARY 455
13 FLOWS IN HIGHER DIMENSIONS 457 13.1 REVIEW OF CLASSIFICATION THEORY
IN 2? 3 457 13.2 GENERAL SETUP 459
13.2.1 SPECTRUM OF LYAPUNOV EXPONENTS 459 13.2.2 DOUBLE PROJECTION 461
13.3 FLOWS IN R 4 462
13.3.1 CYCLIC PHASE SPACES 462 13.3.2 FLOPPINESS AND RIGIDITY 462 13.3.3
SINGULARITIES IN RETURN MAPS 463 13.4 CUSPS IN WEAKLY COUPLED, STRONGLY
DISSIPATIVE CHAOTIC SYSTEMS 466 13.4.1 COUPLED LOGISTIC MAPS 466 13.4.2
COUPLED DIODE RESONATORS 469 13.5 CUSP BIFURCATION DIAGRAMS 470 13.5.1
CUSP RETURN MAPS 472 13.5.2 STRUCTURE IN THE CONTROL PLANE 472 13.5.3
COMPARISON WITH THE FOLD 474 13.6 NONLOCAL SINGULARITIES 475 13.6.1
MULTIPLE CUSPS 475 13.7 GLOBAL BOUNDARY CONDITIONS 477 13.8 FROM BRAIDS
TO TRIANGULATIONS: TOWARD A KINEMATICS IN HIGHER
DIMENSIONS 482
13.8.1 KNOT THEORY IN THREE DIMENSIONS AND BEYOND 481 13.8.2 FROM
NONINTERSECTION TO ORIENTATION PRESERVATION 482 13.8.3 SINGULARITIES IN
HIGHER DIMENSIONS 490 13.9 SUMMARY 490
14 PROGRAM FOR DYNAMICAL SYSTEMS THEORY 493 14.1 REDUCTION OF DIMENSION
494
IMAGE 10
XIV I CONTENTS
14.2 EQUIVALENCE 496
14.3 STRUCTURE THEORY 497 14.3.1 REDUCIBILITY OF DYNAMICAL SYSTEMS 497
14.4 GERMS 498
14.5 UNFOLDING 500 14.6 PATHS 502
14.7 RANK 502
14.7.1 STRETCHING AND SQUEEZING 503 14.8 COMPLEX EXTENSIONS 504 14.9
COXETER-DYNKIN DIAGRAMS 504 14.10 REAL FORMS 506 14.11 LOCAL VS. GLOBAL
CLASSIFICATION 507 14.12 COVER-IMAGE RELATIONS 508 14.13 SYMMETRY
BREAKING AND RESTORATION 508 14.13.1 ENTRAINMENT AND SYNCHRONIZATION 509
14.14 SUMMARY 511
APPENDIX A DETERMINING TEMPLATES FROM TOPOLOGICAL INVARIANTS 513 A.I THE
FUNDAMENTAL PROBLEM 523 A.2 FROM TEMPLATE MATRICES TO TOPOLOGICAL
INVARIANTS 525 A.2.1 CLASSIFICATION OF PERIODIC ORBITS BY SYMBOLIC NAMES
515 A.2.2 ALGEBRAIC DESCRIPTION OF A TEMPLATE 516 A.2.3 LOCAL TORSION
517 A.2 A RELATIVE ROTATION RATES: EXAMPLES 517 A.2.5 RELATIVE ROTATION
RATES: GENERAL CASE 539 A.3 IDENTIFYING TEMPLATES FROM INVARIANTS 523
A.3.1 USING AN INDEPENDENT SYMBOLIC CODING 524 A.3.2 SIMULTANEOUS
DETERMINATION OF SYMBOLIC NAMES AND TEMPLATE 527 A.4 CONSTRUCTING
GENERATING PARTITIONS 531 A.4.1 SYMBOLIC ENCODING AS AN INTERPOLATION
PROCESS 531 A.4.2 GENERATING PARTITIONS FOR EXPERIMENTAL DATA 535 A.4.3
COMPARISON WITH METHODS BASED ON HOMOCLINIC TANGENCIES 536 A.4.4
SYMBOLIC DYNAMICS ON THREE SYMBOLS 538 A.5 SUMMARY 539
APPENDIX B EMBEDDINGS 542 B.I DIFFEOMORPHISMS 542 B.2 MAPPINGS OF DATA
543 B.2.1 TOO LITTLE DATA 543
B.2.2 TOO MUCH DATA 545 B.2.3 JUST THE RIGHT AMOUNT OF DATA 547 B.3
TESTS FOR EMBEDDINGS 547 B.4 TESTS OF EMBEDDING TESTS 549
B.4.1 TRIAL DATA SET 549 B.5 GEOMETRIC TESTS FOR EMBEDDINGS 550 B.5.1
FRACTAL DIMENSION ESTIMATION 550
IMAGE 11
CONTENTS I XV
B.5.2 FALSE NEAR NEIGHBOR ESTIMATES 553
B.6 DYNAMICAL TESTS FOR EMBEDDINGS 554 B.7 TOPOLOGICAL TEST FOR
EMBEDDINGS 555 B.8 POSTMORTEM ON EMBEDDING TESTS 557
B.8.1 GENERALITY 557 B.8.2 COMPUTATIONAL LOAD 557 B.8.3 VARIABILITY 558
B.8.4 STATISTICS 559 B.8.5 PARAMETERS 560
B.8.6 NOISE 560
B.8.7 THE SELF-INTERSECTION PROBLEM 563 B.8.8 RELIABILITY AND
LIMITATIONS 563 B.9 STATIONARITY 562
B.10 BEYOND EMBEDDINGS 563 B.LL SUMMARY 563
APPENDIX C FREQUENTLY ASKED QUESTIONS 565 C.I IS TEMPLATE ANALYSIS VALID
FOR NON-HYPERBOLIC SYSTEMS? 565 C.2 CAN TEMPLATE ANALYSIS BE APPLIED TO
WEAKLY DISSIPATIVE SYSTEMS? 566
C.3 WHAT ABOUT HIGHER-DIMENSIONAL SYSTEMS? 567
REFERENCES 569
INDEX 581 |
| any_adam_object | 1 |
| author | Gilmore, Robert 1941- Lefranc, Marc |
| author_GND | (DE-588)122549058 |
| author_facet | Gilmore, Robert 1941- Lefranc, Marc |
| author_role | aut aut |
| author_sort | Gilmore, Robert 1941- |
| author_variant | r g rg m l ml |
| building | Verbundindex |
| bvnumber | BV039727762 |
| classification_rvk | SK 350 SK 810 UG 3900 |
| ctrlnum | (OCoLC)767796428 (DE-599)DNB1011200597 |
| dewey-full | 515.39 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.39 |
| dewey-search | 515.39 |
| dewey-sort | 3515.39 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| edition | 2. rev. and enl. ed. |
| format | Book |
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| id | DE-604.BV039727762 |
| illustrated | Illustrated |
| indexdate | 2024-07-21T00:16:57Z |
| institution | BVB |
| isbn | 9783527410675 3527410678 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024575730 |
| oclc_num | 767796428 |
| open_access_boolean | |
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| owner_facet | DE-20 DE-11 DE-19 DE-BY-UBM |
| physical | XXIV, 593 S. Ill., graph. Darst. 240 mm x 170 mm |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Wiley-VCH |
| record_format | marc |
| spelling | Gilmore, Robert 1941- Verfasser (DE-588)122549058 aut The topology of chaos Alice in stretch and squeezeland Robert Gilmore and Marc Lefranc 2. rev. and enl. ed. Weinheim Wiley-VCH 2011 XXIV, 593 S. Ill., graph. Darst. 240 mm x 170 mm txt rdacontent n rdamedia nc rdacarrier Chaostheorie (DE-588)4009754-7 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Chaotisches System (DE-588)4316104-2 gnd rswk-swf Chaotisches System (DE-588)4316104-2 s Topologie (DE-588)4060425-1 s DE-604 Chaostheorie (DE-588)4009754-7 s 1\p DE-604 Lefranc, Marc Verfasser aut X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3718097&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024575730&sequence=000001&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 | Gilmore, Robert 1941- Lefranc, Marc The topology of chaos Alice in stretch and squeezeland Chaostheorie (DE-588)4009754-7 gnd Topologie (DE-588)4060425-1 gnd Chaotisches System (DE-588)4316104-2 gnd |
| subject_GND | (DE-588)4009754-7 (DE-588)4060425-1 (DE-588)4316104-2 |
| title | The topology of chaos Alice in stretch and squeezeland |
| title_auth | The topology of chaos Alice in stretch and squeezeland |
| title_exact_search | The topology of chaos Alice in stretch and squeezeland |
| title_full | The topology of chaos Alice in stretch and squeezeland Robert Gilmore and Marc Lefranc |
| title_fullStr | The topology of chaos Alice in stretch and squeezeland Robert Gilmore and Marc Lefranc |
| title_full_unstemmed | The topology of chaos Alice in stretch and squeezeland Robert Gilmore and Marc Lefranc |
| title_short | The topology of chaos |
| title_sort | the topology of chaos alice in stretch and squeezeland |
| title_sub | Alice in stretch and squeezeland |
| topic | Chaostheorie (DE-588)4009754-7 gnd Topologie (DE-588)4060425-1 gnd Chaotisches System (DE-588)4316104-2 gnd |
| topic_facet | Chaostheorie Topologie Chaotisches System |
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