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Chaos and Fractals : New Frontiers of Science - Heinz-Otto Peitgen

Chaos and Fractals

New Frontiers of Science


Published: 3rd February 2004
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The fourteen chapters of this book cover the central ideas and concepts of chaos and fractals as well as many related topics including: the Mandelbrot set, Julia sets, cellular automata, L-systems, percolation and strange attractors. This new edition has been thoroughly revised throughout. The appendices of the original edition were taken out since more recent publications cover this material in more depth. Instead of the focussed computer programs in BASIC, the authors provide 10 interactive JAVA-applets for this second edition.

From the reviews: "Numerous books have appeared in recent years that either explore the beauty of fractal art, describe techniques for its creation, or investigate some aspect of the related field of chaotic behavior. The present work attempts to accomplish all three goals in one huge volume...the authors should be applauded for their ambitions undertaking." Mathematical Reviews "This book ... contains all one ever wanted to know about fractals, and more. Written by-next to Mandelbrot-the greatest popularizer of the concept of fractal geometry ... It contains a wealth of information on nearly every angle of the topic... I enjoyed reading the book for its lucid approach, its attempt at completeness, and especially, for the large number of illustrative figures and pictures." Zentralblatt Mathematik From the reviews of the second edition: "This is the second ... edition of what has been a bestseller since its first publication in 1992. ... All the laudatory comments heard twelve years ago about this fascinating book remain entirely valid. No one has succeeded in better presenting ... . the presentation has not aged at all - the comprehensiveness of the underlying mathematics and the illustrative power of the figures has never been surpassed. Twelve years after its first edition this book remains a must buy." (Andre Hautot, Physicalia, Vol. 57 (3), 2005) "The book is written for everyone who wants to learn details of chaos theory and fractal geometry, also for readers who have not much knowledge of technical mathematics. In the fourteen chapters the central ideas and concepts of chaos and fractals are developed ... ." (F. Haslinger, Monatshefte fur Mathematik, Vol. 144 (4), 2005) "It is one of the best introductions to chaos and fractals around. ... Unlike some other books on fractals, it can be read by non-specialists ... . The book is beautifully produced and well illustrated so it is a pleasure to read." (Hugh Williams, The Mathematical Gazette, Vol. 90 (5l9), 2006) "The first edition of this vast introduction to chaos and fractals appeared in 1992. This new edition is virtually identical to the original except for some material ... . the book is ... a wonderful tour of a fascinating area of mathematics, and now the reader can take this tour while carrying around a slimmer (but still hefty) volume. ... The authors have a friendly conversational style ... . This is a great book ... ." (Raymond N. Greenwell, MathDL, May, 2005) "Chaos and Fractals: New Frontiers of Science is an amazing introduction to the ideas of fractal geometry and chaotic dynamics ... . The authors have done a tremendous job in explaining quite difficult concepts in an elegant and simple way ... . I enjoyed this book tremendously - the authors have put in a tremendous amount of work in making a vast and interesting subject accessible ... . I wholeheartedly recommend this book to anyone with even a passing interest in the subject matter." (Dr. S. Virmani, Contemporary Physics, Vol. 46 (6), 2005) "There appeared many books in the 1980's and early 1990's that ... required only a limited mathematical background to understand. They made the fractals, chaos and the Mandelbrot and Julia sets quite popular ... . The ... book that is under review here is one of these popular books. ... The book will remain what it has been so far: an outstanding book that contains all you ever wanted to know about fractals and chaos accessible to all levels of mathematically skilled." (Bulletin of the Belgian Mathematical Society, Vol. 12 (3), 2005)

Forewordp. 1
Introduction: Causality Principle, Deterministic Laws and Chaosp. 9
The Backbone of Fractals: Feedback and the Iteratorp. 15
The Principle of Feedbackp. 17
The Multiple Reduction Copy Machinep. 23
Basic Types of Feedback Processesp. 27
The Parable of the Parabola--Or: Don't Trust Your Computerp. 37
Chaos Wipes Out Every Computerp. 49
Classical Fractals and Self-Similarityp. 61
The Cantor Setp. 65
The Sierpinski Gasket and Carpetp. 76
The Pascal Trianglep. 80
The Koch Curvep. 87
Space-Filling Curvesp. 92
Fractals and the Problem of Dimensionp. 104
The Universality of the Sierpinski Carpetp. 110
Julia Setsp. 120
Pythagorean Treesp. 124
Limits and Self-Similarityp. 129
Similarity and Scalingp. 132
Geometric Series and the Koch Curvep. 141
Corner the New from Several Sides: Pi and the Square Root of Twop. 147
Fractals as Solutions of Equationsp. 162
Length, Area and Dimension: Measuring Complexity and Scaling Propertiesp. 173
Finite and Infinite Length of Spiralsp. 175
Measuring Fractal Curves and Power Lawsp. 182
Fractal Dimensionp. 192
The Box-Counting Dimensionp. 202
Borderline Fractals: Devil's Staircase and Peano Curvep. 210
Encoding Images by Simple Transformationsp. 215
The Multiple Reduction Copy Machine Metaphorp. 217
Composing Simple Transformationsp. 220
Relatives of the Sierpinski Gasketp. 230
Classical Fractals by IFSsp. 238
Image Encoding by IFSsp. 244
Foundation of IFS: The Contraction Mapping Principlep. 248
Choosing the Right Metricp. 258
Composing Self-Similar Imagesp. 262
Breaking Self-Similarity and Self-Affinity: Networking with MRCMsp. 267
The Chaos Game: How Randomness Creates Deterministic Shapesp. 277
The Fortune Wheel Reduction Copy Machinep. 280
Addresses: Analysis of the Chaos Gamep. 287
Tuning the Fortune Wheelp. 300
Random Number Generator Pitfallp. 311
Adaptive Cut Methodsp. 319
Recursive Structures: Growing Fractals and Plantsp. 329
L-Systems: A Language for Modeling Growthp. 333
Growing Classical Fractals with MRCMsp. 340
Turtle Graphics: Graphical Interpretation of L-Systemsp. 351
Growing Classical Fractals with L-Systemsp. 355
Growing Fractals with Networked MRCMsp. 367
L-System Trees and Bushesp. 372
Pascal's Triangle: Cellular Automata and Attractorsp. 377
Cellular Automatap. 382
Binomial Coefficients and Divisibilityp. 393
IFS: From Local Divisibility to Global Geometryp. 404
HIFS and Divisibility by Prime Powersp. 412
Catalytic Converters, or How Many Cells Are Black?p. 420
Irregular Shapes: Randomness in Fractal Constructionsp. 423
Randomizing Deterministic Fractalsp. 425
Percolation: Fractals and Fires in Random Forestsp. 429
Random Fractals in a Laboratory Experimentp. 440
Simulation of Brownian Motionp. 446
Scaling Laws and Fractional Brownian Motionp. 456
Fractal Landscapesp. 462
Deterministic Chaos: Sensitivity, Mixing, and Periodic Pointsp. 467
The Signs of Chaos: Sensitivityp. 469
The Signs of Chaos: Mixing and Periodic Pointsp. 480
Ergodic Orbits and Histogramsp. 485
Metaphor of Chaos: The Kneading of Doughp. 496
Analysis of Chaos: Sensitivity, Mixing, and Periodic Pointsp. 509
Chaos for the Quadratic Iteratorp. 520
Mixing and Dense Periodic Points Imply Sensitivityp. 529
Numerics of Chaos: Worth the Trouble or Not?p. 535
Order and Chaos: Period-Doubling and Its Chaotic Mirrorp. 541
The First Step from Order to Chaos: Stable Fixed Pointsp. 548
The Next Step from Order to Chaos: The Period-Doubling Scenariop. 559
The Feigenbaum Point: Entrance to Chaosp. 575
From Chaos to Order: A Mirror Imagep. 583
Intermittency and Crises: The Backdoors to Chaosp. 595
Strange Attractors: The Locus of Chaosp. 605
A Discrete Dynamical System in Two Dimensions: Henon's Attractorp. 609
Continuous Dynamical Systems: Differential Equationsp. 628
The Rossler Attractorp. 636
The Lorenz Attractorp. 647
Quantitative Characterization of Strange Chaotic Attractors: Ljapunov Exponentsp. 659
Quantitative Characterization of Strange Chaotic Attractors: Dimensionsp. 671
The Reconstruction of Strange Attractorsp. 694
Fractal Basin Boundariesp. 706
Julia Sets: Fractal Basin Boundariesp. 715
Julia Sets as Basin Boundariesp. 717
Complex Numbers--A Short Introductionp. 722
Complex Square Roots and Quadratic Equationsp. 729
Prisoners versus Escapeesp. 733
Equipotentials and Field Lines for Julia Setsp. 744
Binary Decomposition, Field Lines and Dynamicsp. 756
Chaos Game and Self-Similarity for Julia Setsp. 764
The Critical Point and Julia Sets as Cantor Setsp. 769
Quaternion Julia Setsp. 780
The Mandelbrot Set: Ordering the Julia Setsp. 783
From the Structural Dichotomy to the Binary Decompositionp. 785
The Mandelbrot Set--A Road Map for Julia Setsp. 797
The Mandelbrot Set as a Table of Contentp. 820
Bibliographyp. 839
Indexp. 853
Table of Contents provided by Rittenhouse. All Rights Reserved.

ISBN: 9780387202297
ISBN-10: 0387202293
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 864
Published: 3rd February 2004
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 5.08
Weight (kg): 1.94
Edition Number: 2
Edition Type: Revised