Chaos and fractals: new frontiers of science
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2004
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 864 S. Ill., graph. Darst. |
| ISBN: | 0387202293 9780387202297 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents
Foreword 1
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 — On Don t Trust Your Computer.............. 37
1.5 Chaos Wipes Out Every Computer............................. 49
2 Classical Fractals and Self Similarity 61
2.1 The Cantor Set ....................................... 65
2.2 The Sierpinski Gasket and Carpet ............................. 76
2.3 The Pascal Triangle..................................... 80
2.4 The Koch Curve....................................... 87
2.5 Space Filling Curves.................................... 92
2.6 Fractals and the Problem of Dimension........................... 104
2.7 The Universality of the Sierpinski Carpet ......................... 110
2.8 Julia Sets .......................................... 120
2.9 Pythagorean Trees...................................... 124
3 Limits and Self Similarity 129
3.1 Similarity and Scaling.................................... 132
3.2 Geometric Series and the Koch Curve ........................... 141
3.3 Corner the New from Several Sides: Pi and the Square Root of Two ........... 147
3.4 Fractals as Solutions of Equations ............................. 162
4 Length, Area and Dimension: Measuring Complexity and Scaling Properties 173
4.1 Finite and Infinite Length of Spirals ............................ 175
4.2 Measuring Fractal Curves and Power Laws ........................ 182
4.3 Fractal Dimension...................................... 192
4.4 The Box Counting Dimension............................... 202
4.5 Borderline Fractals: Devil s Staircase and Peano Curve.................. 210
xii Table of Contents
5 Encoding Images by Simple Transformations 215
5.1 The Multiple Reduction Copy Machine Metaphor..................... 217
5.2 Composing Simple Transformations............................ 220
5.3 Relatives of the Sierpinski Gasket ............................. 230
5.4 Classical Fractals by IFSs.................................. 238
5.5 Image Encoding by IFSs.................................. 244
5.6 Foundation of IFS: The Contraction Mapping Principle.................. 248
5.7 Choosing the Right Metric................................. 258
5.8 Composing Self Similar Images .............................. 262
5.9 Breaking Self Similarity and Self Affinity: Networking with MRCMs.......... 267
6 The Chaos Game: How Randomness Creates Deterministic Shapes 277
6.1 The Fortune Wheel Reduction Copy Machine....................... 280
6.2 Addresses: Analysis of the Chaos Game.......................... 287
6.3 Tuning the Fortune Wheel ................................. 300
6.4 Random Number Generator Pitfall............................. 311
6.5 Adaptive Cut Methods ................................... 319
7 Recursive Structures: Growing Fractals and Plants 329
7.1 L Systems: A Language for Modeling Growth....................... 333
7.2 Growing Classical Fractals with MRCMs ......................... 340
7.3 Turtle Graphics: Graphical Interpretation of L Systems.................. 351
7.4 Growing Classical Fractals with L Systems ........................ 355
7.5 Growing Fractals with Networked MRCMs........................ 367
7.6 L System Trees and Bushes................................. 372
8 Pascal s Triangle: Cellular Automata and Attractors 377
8.1 Cellular Automata...................................... 382
8.2 Binomial Coefficients and Divisibility........................... 393
8.3 IFS: From Local Divisibility to Global Geometry..................... 404
8.4 HIFS and Divisibility by Prime Powers .......................... 412
8.5 Catalytic Converters, or How Many Cells Are Black?................... 420
9 Irregular Shapes: Randomness in Fractal Constructions 423
9.1 Randomizing Deterministic Fractals............................ 425
9.2 Percolation: Fractals and Fires in Random Forests..................... 429
9.3 Random Fractals in a Laboratory Experiment....................... 440
9.4 Simulation of Brownian Motion .............................. 446
9.5 Scaling Laws and Fractional Brownian Motion ...................... 456
9.6 Fractal Landscapes..................................... 462
10 Deterministic Chaos: Sensitivity, Mixing, and Periodic Points 467
10.1 The Signs of Chaos: Sensitivity .............................. 469
10.2 The Signs of Chaos: Mixing and Periodic Points...................... 480
10.3 Ergodic Orbits and Histograms............................... 485
10.4 Metaphor of Chaos: The Kneading of Dough ....................... 496
Table of Contents xiii
10.5 Analysis of Chaos: Sensitivity, Mixing, and Periodic Points................509
10.6 Chaos for the Quadratic Iterator ..............................520
10.7 Mixing and Dense Periodic Points Imply Sensitivity....................529
10.8 Numerics of Chaos: Worth the Trouble or Not?......................535
11 Order and Chaos: Period Doubling and Its Chaotic Mirror 541
11.1 The First Step from Order to Chaos: Stable Fixed Points .................548
11.2 The Next Step from Order to Chaos: The Period Doubling Scenario...........559
11.3 The Feigenbaum Point: Entrance to Chaos.........................575
11.4 From Chaos to Order: A Mirror Image...........................583
11.5 Intermittency and Crises: The Backdoors to Chaos ....................595
12 Strange Attractors: The Locus of Chaos 605
12.1 A Discrete Dynamical System in Two Dimensions: Henon s Attractor.......... 609
12.2 Continuous Dynamical Systems: Differential Equations.................. 628
12.3 The Rossler Attractor.................................... 636
12.4 The Lorenz Attractor.................................... 647
12.5 Quantitative Characterization of Strange Chaotic Attractors: Ljapunov Exponents .... 659
12.6 Quantitative Characterization of Strange Chaotic Attractors: Dimensions......... 671
12.7 The Reconstruction of Strange Attractors ......................... 694
12.8 Fractal Basin Boundaries.................................. 706
13 Julia Sets: Fractal Basin Boundaries 715
13.1 Julia Sets as Basin Boundaries...............................717
13.2 Complex Numbers — A Short Introduction........................722
13.3 Complex Square Roots and Quadratic Equations......................729
13.4 Prisoners versus Escapees..................................733
13.5 Equipotentials and Field Lines for Julia Sets........................744
13.6 Binary Decomposition, Field Lines and Dynamics.....................756
13.7 Chaos Game and Self Similarity for Julia Sets.......................764
13.8 The Critical Point and Julia Sets as Cantor Sets ......................769
13.9 Quaternion Julia Sets....................................780
14 The Mandelbrot Set: Ordering the Julia Sets 783
14.1 From the Structural Dichotomy to the Binary Decomposition............... 785
14.2 The Mandelbrot Set — A Road Map for Julia Sets .................... 797
14.3 The Mandelbrot Set as a Table of Content......................... 820
Bibliography 839
Index 853
|
| any_adam_object | 1 |
| author | Peitgen, Heinz-Otto 1945- Jürgens, Hartmut 1955- Saupe, Dietmar 1954- |
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| author_sort | Peitgen, Heinz-Otto 1945- |
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| discipline | Physik Informatik Mathematik |
| edition | 2. ed. |
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| indexdate | 2025-02-20T06:58:45Z |
| institution | BVB |
| isbn | 0387202293 9780387202297 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010601682 |
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| physical | XIII, 864 S. Ill., graph. Darst. |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Springer |
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| spellingShingle | Peitgen, Heinz-Otto 1945- Jürgens, Hartmut 1955- Saupe, Dietmar 1954- Chaos and fractals new frontiers of science Caos sbt Chaostheorie Fraktal Frattali sbt Chaotic behavior in systems Fractals Fraktal (DE-588)4123220-3 gnd Chaostheorie (DE-588)4009754-7 gnd Chaos (DE-588)4191419-3 gnd |
| subject_GND | (DE-588)4123220-3 (DE-588)4009754-7 (DE-588)4191419-3 |
| 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 | Caos sbt Chaostheorie Fraktal Frattali sbt Chaotic behavior in systems Fractals Fraktal (DE-588)4123220-3 gnd Chaostheorie (DE-588)4009754-7 gnd Chaos (DE-588)4191419-3 gnd |
| topic_facet | Caos Chaostheorie Fraktal Frattali Chaotic behavior in systems Fractals Chaos |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010601682&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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Indexes
THWS Schweinfurt Zentralbibliothek Lesesaal
| Call Number: |
2000 SK 350 P379 C4 |
|---|---|
| Copy 1 | ausleihbar Available Place a Hold |