Applied symbolic dynamics and chaos:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore u.a.
World Scientific
1998
|
| Series: | Directions in Chaos
7 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008321116&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XV, 443 S. graph. Darst. |
| ISBN: | 9810235127 |
Staff View
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Record in the Search Index
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| adam_text | DIRECTIONS IN CHAOS VOL. 7 APPLIED SYMBOLIC DYNAMICS AND CHAOS BAI-LIN
HAO WEI-MOU ZHENG THE INSTITUTE OF THEORETICAL PHYSICS ACADEMIA SINICA,
CHINA VFOE WORLD SCIENTIFIC WL SINAAOORE * NEWJERSEV L SINGAPORE * NEW
JERSEY * LONDON * HONG KONG CONTENTS PREFACE XIII 1 INTRODUCTION 1 1.1
DYNAMICAL SYSTEMS 2 1.1.1 PHASE SPACE AND ORBITS 2 1.1.2 PARAMETERS AND
BIFURCATION OF DYNAMICAL BEHAVIOR 2 1.1.3 EXAMPLES OF DYNAMICAL SYSTEMS
3 1.2 SYMBOLIC DYNAMICS AS COARSE-GRAINED DESCRIPTION OF DYNAMICS . . .
5 1.2.1 FINE-GRAINED AND COARSE-GRAINED DESCRIPTIONS 6 1.2.2 SYMBOLIC
DYNAMICS AS THE SIMPLEST DYNAMICS 6 1.3 ABSTRACT VERSUS APPLIED SYMBOLIC
DYNAMICS 8 1.3.1 ABSTRACT SYMBOLIC DYNAMICS 8 1.3.2 APPLIED SYMBOLIC
DYNAMICS 9 1.4 LITERATURE ON SYMBOLIC DYNAMICS 11 2 SYMBOLIC DYNAMICS OF
UNIMODAL MAPS 13 2.1 SYMBOLIC SEQUENCES IN UNIMODAL MAPS 15 2.1.1
NUMERICAL ORBIT AND SYMBOLIC SEQUENCE 15 2.1.2 SYMBOLIC SEQUENCE AND
FUNCTIONAL COMPOSITION 21 2.1.3 THE WORD-LIFTING TECHNIQUE 22 2.2 THE
QUADRATIC MAP 24 2.2.1 AN OVER-SIMPLIFIED POPULATION MODEL 24 2.2.2
BIFURCATION DIAGRAM OF THE QUADRATIC MAP 26 2.2.3 DARK LINES IN THE
BIFURCATION DIAGRAM 30 2.3 ORDERING OF SYMBOLIC SEQUENCES AND THE
ADMISSIBILITY CONDITION ... 37 2.3.1 PROPERTY OF MONOTONE FUNCTIONS 38
2.3.2 THE ORDERING RULE 38 2.3.3 DYNAMICAL INVARIANT RANGE AND KNEADING
SEQUENCE 42 2.3.4 THE ADMISSIBILITY CONDITION 43 2.4 THE PERIODIC WINDOW
THEOREM 45 2.4.1 PERIODIC WINDOW THEOREM 46 2.4.2 CONSTRUCTION OF MEDIAN
WORDS 49 2.4.3 THE MSS TABLE OF KNEADING SEQUENCES 50 2.4.4 NOMENCLATURE
OF UNSTABLE PERIODIC ORBITS 52 VII VIII CONTENTS 2.5 COMPOSITION RULES
53 2.5.1 THE *-COMPOSITION 53 2.5.2 GENERALIZED COMPOSITION RULE 57
2.5.3 PROOF OF THE GENERALIZED COMPOSITION RULE 60 2.5.4 APPLICATIONS OF
THE GENERALIZED COMPOSITION RULE 62 2.5.5 FURTHER REMARKS ON COMPOSITION
RULES 65 2.6 COARSE-GRAINED CHAOS 67 2.6.1 CHAOS IN THE SURJECTIVE
UNIMODAL MAP 68 2.6.2 CHAOS IN PX 00 MAPS 74 2.7 TOPOLOGICAL ENTROPY 82
2.8 PIECEWISE LINEAR MAPS AND METRIE REPRESENTATION OF SYMBOLIC
SEQUENCES 83 2.8.1 THE TENT MAP AND SHIFT MAP 84 2.8.2 THE A-EXPANSION
OF REAL NUMBERS 85 2.8.3 CHARACTERISTIC FUNCTION OF THE KNEADING
SEQUENCE 86 2.8.4 MAPPING OF SUBINTERVALS AND THE STEFAN MATRIX 86 2.8.5
MARKOV PARTITIONS AND GENERATING PARTITIONS 93 2.8.6 METRIE
REPRESENTATION OF SYMBOLIC SEQUENCES 96 2.8.7 PIECEWISE LINEAR EXPANDING
MAP 100 3 MAPS WITH MULTIPLE CRITICAL POINTS 103 3.1 GENERAL DISCUSSION
104 3.1.1 THE ORDERING RULE 105 3.1.2 ADMISSIBILITY AND COMPATIBILITY OF
KNEADING SEQUENCES . . . . 106 3.2 THE ANTISYMMETRIC CUBIC MAP 106 3.2.1
SYMBOLIC SEQUENCES AND THEIR ORDERING 109 3.2.2 ADMISSIBILITY CONDITIONS
110 3.2.3 GENERATION OF SUPERSTABLE MEDIAN WORDS 112 3.3 SYMMETRY
BREAKING AND RESTORATION 118 3.3.1 SYMMETRY BREAKING OF SYMMETRIE ORBITS
120 3.3.2 ANALYSIS OF SYMMETRY RESTORATION 122 3.4 THE GAP MAP 125 3.4.1
THE KNEADING PLANE 127 3.4.2 CONTACTS OF EVEN-ODD TYPE 131 3.4.3
SELF-SIMILAR STRUCTURE IN THE KNEADING PLANE 132 3.4.4 CRITERION FOR
TOPOLOGICAL CHAOS 135 3.5 THE LORENZ-LIKE MAP 138 3.5.1 ORDERING RULE
AND ADMISSIBILITY CONDITIONS 139 3.5.2 CONSTRUCTION OF THE KNEADING
PLANE 139 3.5.3 CONTACTS AND INTERSECTIONS 140 3.5.4 FAREY AND DOUBLING
TRANSFORMATIONS 141 3.6 GENERAL CUBIC MAPS 142 3.6.1 SKELETON, BONES AND
JOINTS IN KNEADING PLANE 145 3.6.2 THE CONSTRUCTION OF THE KNEADING
PLANE 147 3.6.3 THE *-COMPOSITION RULES 151 CONTENTS IX 3.6.4 THE
(-,+,-) TYPE CUBIC MAP 152 3.7 THE SINE-SQUARE MAP 156 3.7.1 SYMBOLIC
SEQUENCES AND WORD-LIFTING TECHNIQUE 157 3.7.2 ORDERING RULE AND
ADMISSIBILITY CONDITIONS 159 3.7.3 GENERATION OF KNEADING SEQUENCES 160
3.7.4 JOINTS AND BONES IN THE KNEADING PLANE 161 3.7.5 SKELETON OF
SUPERSTABLE ORBITS AND EXISTENCE OF TOPOLOGICAL CHAOS 165 3.8 THE
LORENZ-SPARROW MAPS 166 3.8.1 ORDERING AND ADMISSIBILITY OF SYMBOLIC
SEQUENCES 167 3.8.2 GENERATION OF COMPATIBLE KNEADING PAIRS 169 3.8.3
GENERATION OF ADMISSIBLE SEQUENCES FOR GIVEN KNEADING PAIR . 170 3.8.4
METRIE REPRESENTATION OF SYMBOLIC SEQUENCES 172 3.8.5 ONE-PARAMETER
LIMITS OF LORENZ-SPARROW MAPS 173 3.9 PIECEWISE LINEAR MAPS 174 3.9.1
PIECEWISE LINEAR MAPS WITH MULTIPLE CRITICAL POINTS 174 3.9.2 KNEADING
DETERMINANTS 175 4 SYMBOLIC DYNAMICS OF CIRCLE MAPS 177 4.1 THE PHYSICS
OF LINEAR AND NONLINEAR OSCILLATORS 178 4.2 CIRCLE MAPS AND THEIR LIFTS
179 4.2.1 THE RIGID ROTATION * BARE CIRCLE MAP 180 4.2.2 THE SINE-CIRCLE
MAP 182 4.2.3 LIFT OF CIRCLE MAPS 183 4.2.4 ROTATION NUMBER AND ROTATION
INTERVAL 184 4.2.5 ARNOLD TONGUES IN THE PARAMETER PLANE 186 4.3
CONTINUED FRACTIONS AND FAREY TREE 187 4.3.1 FAREY TREE: RATIONAL
FRACTION REPRESENTATION 187 4.3.2 FAREY TREE: CONTINUED FRACTION
REPRESENTATION 188 4.3.3 FAREY TREE: FAREY ADDRESSES AND FAREY MATRICES
191 4.3.4 MORE ON CONTINUED FRACTION AND FAREY REPRESENTATIONS . . . .
193 4.3.5 FAREY TREE: SYMBOLIC REPRESENTATION 197 4.4 FAREY
TRANSFORMATIONS AND WELL-ORDERED ORBITS 200 4.4.1 WELL-ORDERED SYMBOLIC
SEQUENCES 201 4.4.2 FAREY TRANSFORMATIONS AS COMPOSITION RULES 201 4.3.3
EXTREME PROPERTY OF WELL-ORDERED PERIODIC SEQUENCES . . . . 202 4.4.4
GENERATION OF R AND L * 205 III DX II1111 4.5 CIRCLE MAP WITH
NON-MONOTONE LIFT 207 4.5.1 SYMBOLIC SEQUENCES AND THEIR CONTINUOUS
TRANSFORMATIONS . . 207 4.5.2 ORDERING RULE AND ADMISSIBILITY CONDITION
208 4.5.3 EXISTENCE OF WELL-ORDERED SYMBOLIC SEQUENCES 209 4.5.4 THE
FAREY TRANSFORMATIONS 210 4.5.5 EXISTENCE OF SYMBOLIC SEQUENCE WITHOUT
ROTATION NUMBER . . 211 4.6 KNEADING PLANE OF CIRCLE MAPS 212 X CONTENTS
4.6.1 ARNOLD TONGUE WITH ROTATION NUMBER 1/2 212 4.6.2 DOUBLY
SUPERSTABLE KNEADING SEQUENCES: JOINTS AND BONES . . 213 4.6.3
GENERATION OF KNEADING SEQUENCES K G AND K S 215 4.6.4 CONSTRUCTION OF
THE KNEADING PLANE 217 4.7 PIECEWISE LINEAR CIRCLE MAPS AND TOPOLOGICAL
ENTROPY 218 4.7.1 THE SAWTOOTH CIRCLE MAP 218 4.7.2 CIRCLE MAP WITH
GIVEN KNEADING SEQUENCES 219 4.7.3 KNEADING DETERMINANT AND TOPOLOGICAL
ENTROPY 221 4.7.4 CONSTRUCTION OF A MAP FROM A GIVEN KNEADING SEQUENCE .
. . 222 4.7.5 ROTATION INTERVAL AND WELL-ORDERED PERIODIC SEQUENCES . .
. . 222 5 SYMBOLIC DYNAMICS OF TWO-DIMENSIONAL MAPS 225 5.1 GENERAL
DISCUSSION 227 5.1.1 BI-INFINITE SYMBOLIC SEQUENCES 227 5.1.2
DECOMPOSITION OF THE PHASE PLANE . 229 5.1.3 TANGENCIES AND
ADMISSIBILITY CONDITIONS 230 5.1.4 ADMISSIBILITY CONDITIONS IN SYMBOLIC
PLANE 231 5.2 INVARIANT MANIFOLDS AND DYNAMICAL FOLIATIONS OF PHASE
PLANE 233 5.2.1 STABLE AND UNSTABLE INVARIANT MANIFOLDS 233 5.2.2
DYNAMICAL FOLIATIONS OF THE PHASE PLANE 236 5.2.3 SUMMARY AND DISCUSSION
238 5.3 THE TEL MAP 240 5.3.1 FORWARD AND BACKWARD SYMBOLIC SEQUENCES
241 5.3.2 DYNAMICAL FOLIATIONS OF PHASE SPACE AND THEIR ORDERING . . .
242 5.3.3 FORBIDDEN AND ALLOWED ZONES IN SYMBOLIC PLANE 249 5.3.4 THE
ADMISSIBILITY CONDITIONS 252 5.3.5 SUMMARY 255 5.4 THE LOZI MAP 256
5.4.1 FORWARD AND BACKWARD SYMBOLIC SEQUENCES 258 5.4.2 DYNAMICAL
FOLIATIONS OF THE PHASE SPACE 259 5.4.3 ORDERING OF THE FORWARD AND
BACKWARD FOLIATIONS . . . . . . . 267 5.4.4 ALLOWED AND FORBIDDEN ZONES
IN SYMBOLIC PLANE 269 5.4.5 DISCUSSION OF THE ADMISSIBILITY CONDITION
273 5.5 THE HENON MAP 275 5.5.1 FIXED POINTS AND THEIR STABILITY 276
5.5.2 DETERMINATION OF PARTITION LINES IN PHASE PLANE 278 5.5.3
HENON-TYPE SYMBOLIC DYNAMICS 282 5.5.4 SYMBOLIC ANALYSIS AT TYPICAL
PARAMETER VALUES 283 5.6 THE DISSIPATIVE STANDARD MAP 287 5.6.1
DYNAMICAL FOLIATIONS OF THE PHASE PLANE 287 5.6.2 ORDERING OF SYMBOLIC
SEQUENCES 289 5.6.3 SYMBOLIC PLANE AND ADMISSIBILITY OF SYMBOLIC
SEQUENCES . . . 291 5.7 THE STADIUM BILLIARD PROBLEM 294 5.7.1 A CODING
BASED ON LIFTING 295 CONTENTS XI 5.7.2 RELATION TO OTHER CODINGS 298
5.7.3 THE HALF-STADIUM 300 5.7.4 SUMMARY 301 6 APPLICATION TO ORDINARY
DIFFERENTIAL EQUATIONS 303 6.1 GENERAL DISCUSSION 305 6.1.1 THREE TYPES
OF ODES 305 6.1.2 ON NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS 306
6.1.3 NUMERICAL CALCULATION OF THE POINCARE MAPS 308 6.2 THE
PERIODICALLY FORCED BRUSSELATOR 312 6.2.1 THE BRUSSELATOR VIEWED FROM
THE STANDARD MAP 314 6.2.2 TRANSITION FROM ANNULAR TO INTERVAL DYNAMICS
317 6.2.3 SYMBOLIC ANALYSIS OF INTERVAL DYNAMICS 321 6.3 THE LORENZ
EQUATIONS 326 6.3.1 SUMMARY OF KNOWN PROPERTIES 328 6.3.2 CONSTRUCTION
OF POINCARE AND RETURN MAPS 330 6.3.3 ONE-DIMENSIONAL SYMBOLIC DYNAMICS
ANALYSIS 334 6.3.4 SYMBOLIC DYNAMICS OF THE 2D POINCARE MAPS 337 6.3.5
STABLE PERIODIC ORBITS 345 6.3.6 CONCLUDING REMARKS 352 6.4 SUMMARY OF
OTHER ODE SYSTEMS 352 6.4.1 THE DRIVEN TWO-WELL DUFFING EQUATION 353
6.4.2 THE NMR-LASER MODEL 354 7 COUNTING THE NUMBER OF PERIODIC ORBITS
355 7.1 PERIODIC VERSUS CHAOTIC REGIMES 355 7.1.1 STABLE VERSUS UNSTABLE
PERIODS IN 1D MAPS . 356 7.1.2 NOTATIONS AND SUMMARY OF RESULTS 358
7.1.3 A FEW NUMBER THEORY NOTATIONS AND FUNCTIONS 361 7.2 NUMBER OF
PERIODIC ORBITS IN A CLASS OF ONE-PARAMETER MAPS . . . . 362 7.2.1
NUMBER OF ADMISSIBLE WORDS IN SYMBOLIC DYNAMICS 362 7.2.2 NUMBER OF
TANGENT AND PERIOD-DOUBLING BIFURCATIONS 363 7.2.3 RECURSION FORMULA FOR
THE TOTAL NUMBER OF PERIODS 365 7.2.4 SYMMETRY TYPES OF PERIODIC
SEQUENCES 366 7.2.5 EXPLICIT SOLUTIONS TO THE RECURRENCE RELATIONS 370
7.2.6 FINITE LAMBDA AUTO-EXPANSION OF REAL NUMBERS 371 7.3 OTHER ASPECTS
OF THE COUNTING PROBLEM 373 7.3.1 THE NUMBER OF ROOTS OF THE DARK LINE
EQUATION 373 7.3.2 NUMBER OF SADDLE NODES IN FORMING SMALE HORSESHOE 373
7.3.3 NUMBER OF SOLUTIONS OF RENORMALIZATION GROUP EQUATIONS . . 374 7.4
COUNTING FORMULAE FOR GENERAL CONTINUOUS MAPS 375 7.5 NUMBER OF PERIODS
IN MAPS WITH DISCONTINUITY 377 7.5.1 NUMBER OF PERIODS IN THE GAP MAP
377 7.5.2 NUMBER OF PERIODS IN THE LORENZ-LIKE MAP 379 XII CONTENTS 7.6
SUMMARY OF THE COUNTING PROBLEM 381 7.7 CYCLE EXPANSION FOR TOPOLOGICAL
ENTROPY 381 8 SYMBOLIC DYNAMICS AND GRAMMATICAL COMPLEXITY 385 8.1
FORMAL LANGUAGES AND THEIR COMPLEXITY 386 8.1.1 FORMAL LANGUAGE 386
8.1.2 CHOMSKY HIERARCHY OF GRAMMATICAL COMPLEXITY 388 8.1.3 THE L-SYSTEM
389 8.2 REGULAER LANGUAGE AND FINITE AUTOMATON 390 8.2.1 FINITE AUTOMATON
390 8.2.2 REGULAER LANGUAGE 391 8.2.3 STEFAN MATRIX AS TRANSFER FUNCTION
FOR AUTOMATON 391 8.3 BEYOND REGULAER LANGUAGES 395 8.3.1 FEIGENBAUM AND
GENERALIZED FEIGENBAUM LIMITING SETS . . . . 396 8.3.2 EVEN AND ODD
FIBONACCI SEQUENCES 396 8.3.3 ODD MAXIMAL PRIMITIVE PREFIXES AND
KNEADING MAP 398 8.3.4 EVEN MAXIMAL PRIMITIVE PREFIXES AND DISTINCT
EXCLUDED BLOCKS 401 8.4 SUMMARY OF RESULTS 402 9 SYMBOLIC DYNAMICS AND
KNOT THEORY 403 9.1 KNOTS AND LINKS 404 9.2 KNOTS AND LINKS FROM
UNIMODAL MAPS 406 9.3 LINKING NUMBERS 410 9.4 DISCUSSION 411 APPENDIX
413 A.L PROGRAM TO GENERATE ADMISSIBLE SEQUENCES 413 A.2 PROGRAM TO DRAW
DYNAMICAL FOLIATIONS OF A 2D MAP 419 REFERENCES 423 R.L BOOKS 423 R.2
PAPERS 424 SUBJECT INDEX 439
|
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| id | DE-604.BV012276065 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T10:27:28Z |
| institution | BVB |
| isbn | 9810235127 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008321116 |
| oclc_num | 40073248 |
| open_access_boolean | |
| owner | DE-384 DE-29T |
| owner_facet | DE-384 DE-29T |
| physical | XV, 443 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | World Scientific |
| record_format | marc |
| series | Directions in Chaos |
| series2 | Directions in Chaos |
| spellingShingle | Hao, Bai-Lin Zheng, Wei-Mou Applied symbolic dynamics and chaos Directions in Chaos Chaotic behavior in systems Symbolic dynamics Chaostheorie (DE-588)4009754-7 gnd Symbolische Dynamik (DE-588)4482797-0 gnd |
| subject_GND | (DE-588)4009754-7 (DE-588)4482797-0 |
| title | Applied symbolic dynamics and chaos |
| title_auth | Applied symbolic dynamics and chaos |
| title_exact_search | Applied symbolic dynamics and chaos |
| title_full | Applied symbolic dynamics and chaos Bai-Lin Hao ; Wei-Mou Zheng |
| title_fullStr | Applied symbolic dynamics and chaos Bai-Lin Hao ; Wei-Mou Zheng |
| title_full_unstemmed | Applied symbolic dynamics and chaos Bai-Lin Hao ; Wei-Mou Zheng |
| title_short | Applied symbolic dynamics and chaos |
| title_sort | applied symbolic dynamics and chaos |
| topic | Chaotic behavior in systems Symbolic dynamics Chaostheorie (DE-588)4009754-7 gnd Symbolische Dynamik (DE-588)4482797-0 gnd |
| topic_facet | Chaotic behavior in systems Symbolic dynamics Chaostheorie Symbolische Dynamik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008321116&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV003950003 |
| work_keys_str_mv | AT haobailin appliedsymbolicdynamicsandchaos AT zhengweimou appliedsymbolicdynamicsandchaos |