Chaos and fractals: new frontiers of science
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Springer
1993
|
| Edition: | Corrected 2. print. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 955 - 969 |
| Physical Description: | XVI, 984 S. zahlr. Ill., graph. Darst. |
| ISBN: | 3540979034 0387979034 |
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Record in the Search Index
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| adam_text | Contents
Preface VII
Authors XI
Foreword 1
Mitchell J. Feigenbaum
Introduction: Causality Principle, Deterministic Laws and Chaos 9
1 The Backbone of Fractals: Feedback and the Iterator 15
1.1 The Principle of Feedback 17
1.2 The Multiple Reduction Copy Machine 23
1.3 Basic Types of Feedback Processes 27
1.4 The Parable of the Parabola — Or: Don t Trust Your Computer 37
1.5 Chaos Wipes Out Every Computer 49
1.6 Program of the Chapter: Graphical Iteration 60
2 Classical Fractals and Self Similarity 63
2.1 The Cantor Set 67
2.2 The Sierpinski Gasket and Carpet 78
2.3 The Pascal Triangle 82
2.4 The Koch Curve 89
2.5 Space Filling Curves 94
2.6 Fractals and the Problem of Dimension 106
2.7 The Universality of the Sierpinski Carpet 112
2.8 Julia Sets 122
2.9 Pythagorean Trees 126
2.10 Program of the Chapter: Sierpinski Gasket by Binary Addresses 132
3 Limits and Self Similarity 135
3.1 Similarity and Scaling 138
3.2 Geometric Series and the Koch Curve 147
3.3 Corner the New from Several Sides: Pi and the Square Root of Two 153
3.4 Fractals as Solution of Equations 168
3.5 Program of the Chapter: The Koch Curve 179
xiv Table of Contents
4 Length, Area and Dimension: Measuring Complexity and Scaling Properties 183
4.1 Finite and Infinite Length of Spirals 185
4.2 Measuring Fractal Curves and Power Laws 192
4.3 Fractal Dimension 202
4.4 The Box Counting Dimension 212
4.5 Borderline Fractals: Devil s Staircase and Peano Curve 220
4.6 Program of the Chapter: The Cantor Set and Devil s Staircase 226
5 Encoding Images by Simple Transformations 229
5.1 The Multiple Reduction Copy Machine Metaphor 231
5.2 Composing Simple Transformations 234
5.3 Relatives of the Sierpinski Gasket 244
5.4 Classical Fractals by IFSs 252
5.5 Image Encoding by IFSs 258
5.6 Foundation of IFS: The Contraction Mapping Principle 263
5.7 Choosing the Right Metric 274
5.8 Composing Self Similar Images 278
5.9 Breaking Self Similarity and Self Affinity or, Networking with MRCMs 283
5.10 Program of the Chapter: Iterating the MRCM 293
6 The Chaos Game: How Randomness Creates Deterministic Shapes 297
6.1 The Fortune Wheel Reduction Copy Machine 300
6.2 Addresses: Analysis of the Chaos Game 307
6.3 Tuning the Fortune Wheel 321
6.4 Random Number Generator Pitfall 333
6.5 Adaptive Cut Methods 341
6.6 Program of the Chapter: Chaos Game for the Fern 350
7 Recursive Structures: Growing of Fractals and Plants 353
7.1 L Systems: A Language For Modeling Growth 357
7.2 Growing Classical Fractals with MRCMs 364
7.3 Turtle Graphics: Graphical Interpretation of L Systems 376
7.4 Growing Classical Fractals with L Systems 380
7.5 Growing Fractals with Networked MRCMs 392
7.6 L System Trees and Bushes 397
7.7 Program of the Chapter: L systems 402
8 Pascal s Triangle: Cellular Automata and Attractors 407
8.1 Cellular Automata 412
8.2 Binomial Coefficients and Divisibility 423
8.3 IFS: From Local Divisibility to Global Geometry 434
8.4 HIFS and Divisibility by Prime Powers 442
8.5 Catalytic Converters or how many Cells are Black? 450
8.6 Program of the Chapter: Cellular Automata 454
Table of Contents xv
9 Irregular Shapes: Randomness in Fractal Constructions 457
9.1 Randomizing Deterministic Fractals 459
9.2 Percolation: Fractals and Fires in Random Forests 463
9.3 Random Fractals in a Laboratory Experiment 475
9.4 Simulation of Brownian Motion 481
9.5 Scaling Laws and Fractional Brownian Motion 491
9.6 Fractal Landscapes 497
9.7 Program of the Chapter: Random Midpoint Displacement 503
10 Deterministic Chaos: Sensitivity, Mixing, and Periodic Points 507
10.1 The Signs of Chaos: Sensitivity 509
10.2 The Signs of Chaos: Mixing and Periodic Points 520
10.3 Ergodic Orbits and Histograms 525
10.4 Paradigm of Chaos: The Kneading of Dough 536
10.5 Analysis of Chaos: Sensitivity, Mixing, and Periodic Points 549
10.6 Chaos for the Quadratic Iterator 560
10.7 Mixing and Dense Periodic Points Imply Sensitivity 569
10.8 Numerics of Chaos: Worth the Trouble or Not? 575
10.9 Program of the Chapter: Time Series and Error Development 581
11 Order and Chaos: Period Doubling and its Chaotic Mirror 585
11.1 The First Step From Order to Chaos: Stable Fixed Points 592
11.2 The Next Step From Order to Chaos: The Period Doubling Scenario 603
11.3 The Feigenbaum Point: Entrance to Chaos 619
11.4 From Chaos to Order: a Mirror Image 628
11.5 Intermittency and Crises: The Backdoors to Chaos 640
11.6 Program of the Chapter: Final State Diagram 651
12 Strange Attractors: The Locus of Chaos 655
12.1 A Discrete Dynamical System in Two Dimensions: Henon s Attractor 659
12.2 Continuous Dynamical Systems: Differential Equations 678
12.3 The Rossler Attractor 686
12.4 The Lorenz Attractor 697
12.5 Quantitative Characterization of Strange Chaotic Attractors: Ljapunov Exponents . . . 709
12.6 Quantitative Characterization of Strange Chaotic Attractors: Dimensions 721
12.7 The Reconstruction of Strange Attractors 745
12.8 Fractal Basin Boundaries 757
12.9 Program of the Chapter: Rossler Attractor 766
13 Julia Sets: Fractal Basin Boundaries 769
13.1 Julia Sets as Basin Boundaries 771
13.2 Complex Numbers — A Short Introduction 776
13.3 Complex Square Roots and Quadratic Equations 784
13.4 Prisoners versus Escapees 789
¦ 13.5 Equipotentials and Field Lines for Julia Sets 800
13.6 Binary Decomposition, Field Lines and Dynamics 812
xvi Table of Contents
13.7 Chaos Game and Self Similarity for Julia Sets 820
13.8 The Critical Point and Julia Sets as Cantor Sets 826
13.9 Quaternion Julia Sets 837
13.10 Program of the Chapter: Julia Sets 839
14 The Mandelbrot Set: Ordering the Julia Sets 841
14.1 From the Structural Dichotomy to the Binary Decomposition 843
14.2 The Mandelbrot Set — A Road Map for Julia Sets 855
14.3 The Mandelbrot Set as a Table of Content 878
14.4 Program of the Chapter: The Mandelbrot Set 896
A A Discussion of Fractal Image Compression 903
Yuval Fisher
A.I Self Similarity in Images 906
A.2 A Special MRCM 908
A.3 Encoding Images 912
A.4 Ways to Partition Images 914
A.5 Implementation Notes 917
B Multifractal Measures 921
Carl J. G. Evertsz and Benoit B. Mandelbrot
B.I Introduction 922
B.2 The Binomial and Multinomial Measures 927
B.3 Methods for Estimating the Function /(a) from Data 938
B.4 Probabilistic Roots of Multifractals. Role of f(a) in Large Deviation Theory .... 944
B.5 Some Applications, and Advanced Multifractals 952
Bibliography 955
Index 971
|
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| dewey-full | 003.857 514.74 |
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| edition | Corrected 2. print. |
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| physical | XVI, 984 S. zahlr. Ill., graph. Darst. |
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| spelling | Peitgen, Heinz-Otto 1945- Verfasser (DE-588)120260387 aut Chaos and fractals new frontiers of science Heinz-Otto Peitgen ; Hartmut Jürgens ; Dietmar Saupe Corrected 2. print. New York, NY [u.a.] Springer 1993 XVI, 984 S. zahlr. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 955 - 969 Chaos (théorie des systèmes) ram Fractales ram Chaotic behavior in systems Fractals Fraktal (DE-588)4123220-3 gnd rswk-swf Chaos (DE-588)4191419-3 gnd rswk-swf Chaostheorie (DE-588)4009754-7 gnd rswk-swf Chaostheorie (DE-588)4009754-7 s DE-604 Fraktal (DE-588)4123220-3 s Chaos (DE-588)4191419-3 s 1\p DE-604 Jürgens, Hartmut 1955- Verfasser (DE-588)120260441 aut Saupe, Dietmar 1954- Verfasser (DE-588)12026045X aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005405007&sequence=000002&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 | Peitgen, Heinz-Otto 1945- Jürgens, Hartmut 1955- Saupe, Dietmar 1954- Chaos and fractals new frontiers of science Chaos (théorie des systèmes) ram Fractales ram Chaotic behavior in systems Fractals Fraktal (DE-588)4123220-3 gnd Chaos (DE-588)4191419-3 gnd Chaostheorie (DE-588)4009754-7 gnd |
| subject_GND | (DE-588)4123220-3 (DE-588)4191419-3 (DE-588)4009754-7 |
| title | Chaos and fractals new frontiers of science |
| title_auth | Chaos and fractals new frontiers of science |
| title_exact_search | Chaos and fractals new frontiers of science |
| title_full | Chaos and fractals new frontiers of science Heinz-Otto Peitgen ; Hartmut Jürgens ; Dietmar Saupe |
| title_fullStr | Chaos and fractals new frontiers of science Heinz-Otto Peitgen ; Hartmut Jürgens ; Dietmar Saupe |
| title_full_unstemmed | Chaos and fractals new frontiers of science Heinz-Otto Peitgen ; Hartmut Jürgens ; Dietmar Saupe |
| title_short | Chaos and fractals |
| title_sort | chaos and fractals new frontiers of science |
| title_sub | new frontiers of science |
| topic | Chaos (théorie des systèmes) ram Fractales ram Chaotic behavior in systems Fractals Fraktal (DE-588)4123220-3 gnd Chaos (DE-588)4191419-3 gnd Chaostheorie (DE-588)4009754-7 gnd |
| topic_facet | Chaos (théorie des systèmes) Fractales Chaotic behavior in systems Fractals Fraktal Chaos Chaostheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005405007&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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