Chaotic modelling and simulation: analysis of chaotic models, attractors and forms
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Boca Raton[u.a.]
CRC Press
2009
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index. - Formerly CIP |
| Physical Description: | 349 S. Ill., graf. Darst. |
| ISBN: | 142007900X 9781420079005 |
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| adam_text | IMAGE 1
CHAOTIC MODELLING
AND SIMULATION ANALYSIS OF CHAOTIC MODELS, ATTRACTORS AND FORMS
CHRISTOS H. SKIADAS CHARILAOS SKIADAS
@
CRC PRESS TAYLOR & FRANCIS CROUP BOCA RATON LONDON NEW YORK
CRC PRESS IS AN IMPRINT OF THE TAYLOR & FRANCIS GROUP, AN INFORMA
BUSINESS
A C H A P M AN & HALL BOOK
IMAGE 2
CONTENTS
1 INTRODUCTION 1
1.1 CHAOS IN DIFFERENTIAL EQUATIONS SYSTEMS 1
1.2 CHAOS IN DIFFERENCE EQUATION SYSTEMS 4
1.2.1 THE LOGISTIC MAP 5
1.2.2 DELAY MODEIS 6
1.2.3 THE HENON MODEL 6
1.3 MORE COMPLEX STRUCTURES 8
1.3.1 THREE-DIMENSIONAL AND HIGHER-DIMENSIONAL MODEIS . . .. 8 1.3.2
CONSERVATIVE SYSTEMS 8
1.3.3 ROTATIONS 10
1.3.4 SHAPE AND FORM 12
1.4 CHAOS AND THE UNIVERSE 14
1.4.1 CHAOS IN THE SOLAR SYSTEM 14
1.4.2 CHAOS IN GALAXIES 19
1.4.3 GALACTIC-TYPE POTENTIALS AND THE HENON-HEILES SYSTEM . .. 23 1.4.4
THE CONTOPOULOS SYSTEM 25
1.5 ODDS AND ENDS, AND MILESTONES 27
2 MODELS AND MODELLING 29
2.1 INTRODUCTION 29
2.2 MODEL CONSTRUCTION 30
2.2.1 GROWTH/DECAY MODEIS 30
2.3 MODELLING TECHNIQUES 32
2.3.1 SERIES APPROXIMATION 33
2.3.2 EMPIRICAL MODEL FORMULATION 36
2.3.3 THE CALCULUS OF VARIATIONS APPROACH 38
2.3.4 THE PROBABILISTIC-STOCHASTIC APPROACH 39
2.3.5 DELAY GROWTH FUNCTIONS 40
2.4 CHAOTIC ANALYSIS AND SIMULATION 41
2.5 DETERMINISTIC, STOCHASTIC AND CHAOTIC MODELS 42
3 THE LOGISTIC MODEL 47
3.1 THE LOGISTIC MAP 47
3.1.1 GEOMETRIE ANALYSIS OF THE LOGISTIC 47
3.1.2 ALGEBRAIC ANALYSIS OF THE LOGISTIC 51
3.2 THE BIFURCATION DIAGRAM 58
3.3 OTHER MODELS WITH SIMILAR BEHAVIOUR 61
IMAGE 3
3.4 MODELS WITH DIFFERENT CHAOTIC BEHAVIOUR 62
3.5 THE GRM1 CHAOTIC MODEL 64
3.5.1 GRM1 AND INNOVATION DIFFUSION MODELLING 64
3.5.2 THE GENERALIZED RATIONAL MODEL 66
3.5.3 PARAMETER ESTIMATION FOR THE GRM1 MODEL 68
3.5.4 ILLUSTRATIONS 69
3.6 FURTHER DISCUSSION 71
4 THE DELAY LOGISTIC MODEL 79
4.1 INTRODUCTION 79
4.2 DELAY DIFFERENCE MODELS 79
4.2.1 SIMPLE DELAY OSCILLATION SCHEME 79
4.2.2 THE DELAY LOGISTIC MODEL 81
4.3 TIME DELAY DIFFERENTIAL EQUATIONS 82
4.4 A MORE COMPLICATED DELAY MODEL 84
4.5 A DELAY DIFFERENTIAL LOGISTIC ANALOGUE 86
4.6 OTHER DELAY LOGISTIC MODELS 86
4.7 MODEL BEHAVIOUR FOR LARGE DELAYS 89
4.8 ANOTHER DELAY LOGISTIC MODEL 91
5 THE HENON MODEL 99
5.1 GLOBAL PERIOD DOUBLING BIFURCATIONS IN THE HENON MAP 99
5.1.1 PERIOD DOUBLING BIFURCATIONS WHEN B = -1 100
5.1.2 PERIOD DOUBLING BIFURCATIONS WHEN B - 1 101
5.2 THE COSINE-HENON MODEL 101
5.3 AN EXAMPLE OF BIFURCATION AND PERIOD DOUBLING 103
5.4 A DIFFERENTIAL EQUATION ANALOGUE 103
5.5 VARIANTS OF THE HENON DELAY DIFFERENCE EQUATION 104
5.5.1 THE THIRD-ORDER DELAY MODEL 104
5.5.2 SECOND-ORDER DELAY MODEIS 105
5.5.3 FIRST-ORDER DELAY VARIANTS 107
5.5.4 EXPONENTIAL VARIANTS 108
5.6 VARIANTS OF THE HENON SYSTEM EQUATIONS 109
5.7 THE HOLMES AND SINE DELAY MODELS 110
5.7.1 THE HOLMES MODEL 110
5.7.2 THE SINE DELAY MODEL 111
6 THREE-DIMENSIONAL AND HIGHER-DIMENSIONAL MODELS 117
6.1 EQUILIBRIUM POINTS AND CHARACTERISTIC MATRICES 117
6.2 THE LOTKA-VOLTERRA MODEL 118
6.3 THE ARNEODO MODEL 119
6.4 AN AUTOCATALYTIC ATTRACTOR 121
6.5 A FOUR-DIMENSIONAL AUTOCATALYTIC ATTRACTOR 122
6.6 THE ROESSLER MODEL 123
6.6.1 A VARIANT OF THE ROESSLER MODEL 124
IMAGE 4
6.6.2 INTRODUCING ROTATION INTO THEJLOESSLER MODEL 127
6.7 THE LORENZ MODEL 129
6.7.1 THE MODIFIED LORENZ MODEL 131
7 NON-CHAOTIC SYSTEMS 135
7.1 CONSERVATIVE SYSTEMS 135
7.1.1 THE SIMPLEST CONSERVATIVE SYSTEM 137
7.1.2 EQUILIBRIUM POINTS IN HAMILTONIAN SYSTEMS 138
7.2 LINEAR SYSTEMS 139
7.2.1 TRANSFORMATIONS ON LINEAR SYSTEMS 140
7.2.2 QUALITATIVE BEHAVIOUR AT EQUILIBRIUM POINTS 142
7.3 EGG-SHAPED FORMS 144
7.3.1 A SIMPLE EGG-SHAPED FORM 144
7.3.2 A DOUBLE EGG-SHAPED FORM 145
7.3.3 A DOUBLE EGG-SHAPED FORM WITH AN ENVELOPE 146
7.4 SYMMETRIE FORMS 148
7.5 MORE COMPLEX FORMS 150
7.6 HIGHER-ORDER FORMS , 153
8 ROTATIONS 157
8.1 INTRODUCTION 157
8.2 A SIMPLE ROTATION-TRANSLATION SYSTEM OF DIFFERENTIAL EQUATIONS . 158
8.3 A DISCRETE ROTATION-TRANSLATION MODEL 163
8.4 A GENERAL ROTATION-TRANSLATION MODEL 169
8.5 ROTATING PARTICLES INSIDE THE EGG-SHAPED FORM 171
8.6 ROTATIONS FOLLOWING AN INVERSE SQUARE LAW 173
9 SHAPE AND FORM 179
9.1 INTRODUCTION 179
9.1.1 SYMMETRY AND PLANE ISOMETRIES 180
9.2 ISOMETRIES IN MODELLING 184
9.2.1 TWO-DIMENSIONAL ROTATION 184
9.3 REFLECTION AND TRANSLATION 186
9.3.1 SPACE CONTRACTION 186
9.4 APPLICATION IN THE IKEDA ATTRACTOR 187
9.5 CHAOTIC ATTRACTORS AND ROTATION-REFLECTION 188
9.6 EXPERIMENTING WITH ROTATION AND REFLECTION 191
9.6.1 THE EFFECT OF SPACE CONTRACTION ON ROTATION-TRANSLATION . . . 191
9.6.2 THE EFFECT OF SPACE CONTRACTION AND CHANGE OF REFLECTION ANGLE ON
TRANSLATION-REFLECTION 192
9.6.3 COMPLICATED ROTATION ANGLE FORMS 193
9.6.4 COMPARING ROTATION-REFLECTION 194
9.6.5 A SIMPLE ROTATION-TRANSLATION MODEL 196
9.7 CHAOTIC CIRCULAR FORMS 196
9.8 FURTHER ANALYSIS 200
IMAGE 5
9.8.1 THE SPACE CONTRACTION ROTATION-TRANSLATION CASE
202
10 CHAOTIC ADVECTION 205
10.1 THE SINK PROBLEM 205
10.1.1 CENTRAL SINK 205
10.1.2 THE CONTRACTION PROCESS 207
10.2 NON-CENTRAL SINK 207
10.3 TWO SYMMETRIE SINKS 208
10.3.1 AREF S BLINKING VORTEX SYSTEM 208
10.4 CHAOTIC FORMS WITHOUT SPACE CONTRACTION 213
10.5 OTHER CHAOTIC FORMS 213
10.6 COMPLEX SINUSOIDAL ROTATION ANGLE 216
10.7 A SPECIAL ROTATION-TRANSLATION MODEL 219
10.8 OTHER ROTATION-TRANSLATION MODELS 219
10.8.1 ELLIPTIC ROTATION-TRANSLATION 219
10.8.2 ROTATION-TRANSLATION WITH SPECIAL ROTATION ANGLE 220
11 CHAOS IN GALAXIES AND RELATED SIMULATIONS 223
11.1 INTRODUCTION 223
11.2 CHAOS IN THE SOLAR SYSTEM 225
11.3 GALAXY MODELS AND MODELLING 228
11.3.1 A SPECIAL ROTATION-TRANSLATION IMAGE 234
11.4 ROTATION-REFLECTION 235
11.5 RELATIVITY IN ROTATION-TRANSLATION SYSTEMS 236
11.6 OTHER RELATIVISTIC FORMS 241
11.7 GALACTIC CLUSTERS 247
11.8 RELATIVISTIC REFLECTION-TRANSLATION . . 248
11.9 ROTATING DISKS OF PARTICLES 248
11.9.1 A CIRCULAR ROTATING DISK 248
11.9.2 THE ROTATING ELLIPSOID 250
11.10 ROTATING PARTICLES UNDER DISTANT ATTRACTING MASSES 253
11.10.1 ONE ATTRACTING MASS 253
11.10.2 THE AREA OF THE CHAOTIC REGION IN GALAXY SIMULATIONS . . . . 255
11.10.3 THE SPEED OF PARTICLES 257
11.11 TWO EQUAL ATTRACTING MASSES IN OPPOSITE DIRECTIONS 258
11.11.1 SYMMETRIE UNEQUAL ATTRACTING MASSES 260
11.12 TWO ATTRACTING EQUAL NONSYMMETRIC MASSES 263
12 GALACTIC-TYPE POTENTIALS AND THE HENON-HEILES SYSTEM 267
12.1 INTRODUCTION 267
12.2 THE HENON-HEILES SYSTEM 268
12.3 DISCRETE ANALOGUES TO THE HENON-HEILES SYSTEM 270
12.4 PATHS OF PARTICLES IN THE HENON-HEILES SYSTEM 272
12.5 OTHER FORMS FOR THE HAMILTONIAN 273
12.6 THE SIMPLEST FORM FOR THE HAMILTONIAN 275
IMAGE 6
12.7 GRAVITATIONAL ATTRACTION 275
12.8 A LOGARITHMIC POTENTIAL 279
12.9 HAMILTONIANS WITH A GALACTIC TYPE POTENTIAL: THE CONTOPOULOS SYSTEM
279
12.10 ANOTHER SIMPLE HAMILTONIAN SYSTEM 281
13 ODDSANDENDS 285
13.1 FORCED NONLINEAR OSCILLATORS 285
13.2 THE EFFECT OF NOISE IN THREE-DIMENSIONAL MODELS 285
13.3 THE LOTKA-VOLTERRA THEORY FOR THE GROWTH OF TWO CONFLICTING
POPULATIONS 288
13.4 THEPENDULUM 290
13.5 A SPECIAL SECOND-ORDER DIFFERENTIAL EQUATION 292
13.6 OTHER PATTERNS AND CHAOTIC FORMS 292
14 MILESTONES 297
REFERENCES 303
INDEX , 345
|
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| author | Skiadas, Christos H. Skiadas, Charilaos |
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| discipline | Informatik Mathematik |
| format | Book |
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| id | DE-604.BV035338548 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:31:37Z |
| institution | BVB |
| isbn | 142007900X 9781420079005 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017142875 |
| oclc_num | 244695399 |
| open_access_boolean | |
| owner | DE-473 DE-BY-UBG DE-11 |
| owner_facet | DE-473 DE-BY-UBG DE-11 |
| physical | 349 S. Ill., graf. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | CRC Press |
| record_format | marc |
| spelling | Skiadas, Christos H. Verfasser (DE-588)170349071 aut Chaotic modelling and simulation analysis of chaotic models, attractors and forms Christos H. Skiadas ; Charilaos Skiadas Boca Raton[u.a.] CRC Press 2009 349 S. Ill., graf. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index. - Formerly CIP Chaotic behavior in systems Mathematical models Mathematisches Modell Chaostheorie (DE-588)4009754-7 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Chaotisches System (DE-588)4316104-2 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s Chaostheorie (DE-588)4009754-7 s DE-604 Chaotisches System (DE-588)4316104-2 s Skiadas, Charilaos Verfasser aut http://www.gbv.de/dms/ilmenau/toc/572459203.PDF lizenzfrei Inhaltsverzeichnis GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017142875&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Skiadas, Christos H. Skiadas, Charilaos Chaotic modelling and simulation analysis of chaotic models, attractors and forms Chaotic behavior in systems Mathematical models Mathematisches Modell Chaostheorie (DE-588)4009754-7 gnd Dynamisches System (DE-588)4013396-5 gnd Chaotisches System (DE-588)4316104-2 gnd |
| subject_GND | (DE-588)4009754-7 (DE-588)4013396-5 (DE-588)4316104-2 |
| title | Chaotic modelling and simulation analysis of chaotic models, attractors and forms |
| title_auth | Chaotic modelling and simulation analysis of chaotic models, attractors and forms |
| title_exact_search | Chaotic modelling and simulation analysis of chaotic models, attractors and forms |
| title_full | Chaotic modelling and simulation analysis of chaotic models, attractors and forms Christos H. Skiadas ; Charilaos Skiadas |
| title_fullStr | Chaotic modelling and simulation analysis of chaotic models, attractors and forms Christos H. Skiadas ; Charilaos Skiadas |
| title_full_unstemmed | Chaotic modelling and simulation analysis of chaotic models, attractors and forms Christos H. Skiadas ; Charilaos Skiadas |
| title_short | Chaotic modelling and simulation |
| title_sort | chaotic modelling and simulation analysis of chaotic models attractors and forms |
| title_sub | analysis of chaotic models, attractors and forms |
| topic | Chaotic behavior in systems Mathematical models Mathematisches Modell Chaostheorie (DE-588)4009754-7 gnd Dynamisches System (DE-588)4013396-5 gnd Chaotisches System (DE-588)4316104-2 gnd |
| topic_facet | Chaotic behavior in systems Mathematical models Mathematisches Modell Chaostheorie Dynamisches System Chaotisches System |
| url | http://www.gbv.de/dms/ilmenau/toc/572459203.PDF http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017142875&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT skiadaschristosh chaoticmodellingandsimulationanalysisofchaoticmodelsattractorsandforms AT skiadascharilaos chaoticmodellingandsimulationanalysisofchaoticmodelsattractorsandforms |
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