The Beauty of Fractals

Images of Complex Dynamical Systems
Author: Heinz-Otto Peitgen,Peter H. Richter
Publisher: Springer Science & Business Media
ISBN: 3642617174
Category: Mathematics
Page: 202
View: 4983

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Now approaching its tenth year, this hugely successful book presents an unusual attempt to publicise the field of Complex Dynamics. The text was originally conceived as a supplemented catalogue to the exhibition "Frontiers of Chaos", seen in Europe and the United States, and describes the context and meaning of these fascinating images. A total of 184 illustrations - including 88 full-colour pictures of Julia sets - are suggestive of a coffee-table book. However, the invited contributions which round off the book lend the text the required formality. Benoit Mandelbrot gives a very personal account, in his idiosyncratic self-centred style, of his discovery of the fractals named after him and Adrien Douady explains the solved and unsolved problems relating to this amusingly complex set.

The Science of Fractal Images


Author: Heinz-Otto Peitgen,Dietmar Saupe
Publisher: Springer Science & Business Media
ISBN: 146123784X
Category: Mathematics
Page: 312
View: 5044

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This book is based on notes for the course Fractals:lntroduction, Basics and Perspectives given by MichaelF. Barnsley, RobertL. Devaney, Heinz-Otto Peit gen, Dietmar Saupe and Richard F. Voss. The course was chaired by Heinz-Otto Peitgen and was part of the SIGGRAPH '87 (Anaheim, California) course pro gram. Though the five chapters of this book have emerged from those courses we have tried to make this book a coherent and uniformly styled presentation as much as possible. It is the first book which discusses fractals solely from the point of view of computer graphics. Though fundamental concepts and algo rithms are not introduced and discussed in mathematical rigor we have made a serious attempt to justify and motivate wherever it appeared to be desirable. Ba sic algorithms are typically presented in pseudo-code or a description so close to code that a reader who is familiar with elementary computer graphics should find no problem to get started. Mandelbrot's fractal geometry provides both a description and a mathemat ical model for many of the seemingly complex forms and patterns in nature and the sciences. Fractals have blossomed enormously in the past few years and have helped reconnect pure mathematics research with both natural sciences and computing. Computer graphics has played an essential role both in its de velopment and rapidly growing popularity. Conversely, fractal geometry now plays an important role in the rendering, modelling and animation of natural phenomena and fantastic shapes in computer graphics.

Chaos and Fractals

The Mathematics Behind the Computer Graphics
Author: Robert L. Devaney,Linda Keen,Kathleen T. Alligood
Publisher: American Mathematical Soc.
ISBN: 0821801376
Category: Mathematics
Page: 148
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This volume contains the proceedings of a highly successful AMS Short Course on Chaos and Fractals, held during the AMS Centennial Celebration in Providence, Rhode Island in August 1988. Chaos and fractals have been the subject of great interest in recent years and have proven to be useful in a variety of areas of mathematics and the sciences. The purpose of the short course was to provide a solid introduction to the mathematics underlying the notions of chaos and fractals. The papers in this book range over such topics as dynamical systems theory, Julia sets, the Mandelbrot set, attractors, the Smale horseshoe, calculus on fractals, and applications to data compression. The authors represented here are some of the top experts in this field. Aimed at beginning graduate students, college and university mathematics instructors, and non-mathematics researchers, this book provides readable expositions of several exciting topics of contemporary research.

Chaos and Fractals

New Frontiers of Science
Author: Heinz-Otto Peitgen,Hartmut Jürgens,Dietmar Saupe
Publisher: Springer Science & Business Media
ISBN: 1475747403
Category: Mathematics
Page: 999
View: 6453

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For almost ten years chaos and fractals have been enveloping many areas of mathematics and the natural sciences in their power, creativity and expanse. Reaching far beyond the traditional bounds of mathematics and science to the realms of popular culture, they have captured the attention and enthusiasm of a worldwide audience. The fourteen chapters of the book cover the central ideas and concepts, as well as many related topics including, the Mandelbrot Set, Julia Sets, Cellular Automata, L-Systems, Percolation and Strange Attractors, and each closes with the computer code for a central experiment. In the two appendices, Yuval Fisher discusses the details and ideas of fractal image compression, while Carl J.G. Evertsz and Benoit Mandelbrot introduce the foundations and implications of multifractals.

An Introduction To Chaotic Dynamical Systems


Author: Robert Devaney
Publisher: Westview Press
ISBN: 0786722673
Category: Science
Page: 416
View: 787

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The study of nonlinear dynamical systems has exploded in the past 25 years, and Robert L. Devaney has made these advanced research developments accessible to undergraduate and graduate mathematics students as well as researchers in other disciplines with the introduction of this widely praised book. In this second edition of his best-selling text, Devaney includes new material on the orbit diagram fro maps of the interval and the Mandelbrot set, as well as striking color photos illustrating both Julia and Mandelbrot sets. This book assumes no prior acquaintance with advanced mathematical topics such as measure theory, topology, and differential geometry. Assuming only a knowledge of calculus, Devaney introduces many of the basic concepts of modern dynamical systems theory and leads the reader to the point of current research in several areas.

Fractals Everywhere

New Edition
Author: Michael F. Barnsley
Publisher: Courier Corporation
ISBN: 0486320340
Category: Mathematics
Page: 560
View: 5726

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Up-to-date text focuses on how fractal geometry can be used to model real objects in the physical world, with an emphasis on fractal applications. Includes solutions, hints, and a bonus CD.

A First Course in Discrete Dynamical Systems


Author: Richard A. Holmgren
Publisher: Springer Science & Business Media
ISBN: 1441987320
Category: Mathematics
Page: 223
View: 5238

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Given the ease with which computers can do iteration it is now possible for almost anyone to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. Mathematica programs that illustrate the dynamics are included in an appendix.

Invitation to Dynamical Systems


Author: Edward R. Scheinerman
Publisher: Courier Corporation
ISBN: 0486275329
Category: Mathematics
Page: 408
View: 2058

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This text is designed for those who wish to study mathematics beyond linear algebra but are unready for abstract material. Rather than a theorem-proof-corollary exposition, it stresses geometry, intuition, and dynamical systems. 1996 edition.

Chaos and Fractals

A Computer Graphical Journey
Author: C.A. Pickover
Publisher: Elsevier
ISBN: 9780080528861
Category: Computers
Page: 452
View: 1134

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These days computer-generated fractal patterns are everywhere, from squiggly designs on computer art posters to illustrations in the most serious of physics journals. Interest continues to grow among scientists and, rather surprisingly, artists and designers. This book provides visual demonstrations of complicated and beautiful structures that can arise in systems, based on simple rules. It also presents papers on seemingly paradoxical combinations of randomness and structure in systems of mathematical, physical, biological, electrical, chemical, and artistic interest. Topics include: iteration, cellular automata, bifurcation maps, fractals, dynamical systems, patterns of nature created through simple rules, and aesthetic graphics drawn from the universe of mathematics and art. Chaos and Fractals is divided into six parts: Geometry and Nature; Attractors; Cellular Automata, Gaskets, and Koch Curves; Mandelbrot, Julia and Other Complex Maps; Iterated Function Systems; and Computer Art. Additionally, information on the latest practical applications of fractals and on the use of fractals in commercial products such as the antennas and reaction vessels is presented. In short, fractals are increasingly finding application in practical products where computer graphics and simulations are integral to the design process. Each of the six sections has an introduction by the editor including the latest research, references, and updates in the field. This book is enhanced with numerous color illustrations, a comprehensive index, and the many computer program examples encourage reader involvement.

The Computational Beauty of Nature

Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation
Author: Gary William Flake
Publisher: MIT Press
ISBN: 9780262561273
Category: Computers
Page: 493
View: 9138

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Gary William Flake develops in depth the simple idea that recurrent rules can produce rich and complicated behaviors.

Iteration of Rational Functions

Complex Analytic Dynamical Systems
Author: Alan F. Beardon
Publisher: Springer Science & Business Media
ISBN: 9780387951515
Category: Mathematics
Page: 280
View: 4070

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This book makes available a comprehensive, detailed, and organized treatment of the foundations of the theory of iteration of rational functions of a complex variable. The material covered extends from the original memoirs of Fatou and Julia to the recent and important results and methods of Sullivan and Shishikura. Many of the details of the proofs have not occurred in print before. The theory of of dynamical systems and chaos has recently undergone a rapid growth in popularity, in part due to the spectacular computer graphics of Julia sets, fractals, and the Mandelbrot set. This text focuses on the specialized area of complex analytic dynamics, a subject that dates back to 1916 and is currently a very active area in mathematics.

SuperFractals


Author: Michael Fielding Barnsley
Publisher: Cambridge University Press
ISBN: 0521844932
Category: Computers
Page: 453
View: 7396

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SuperFractals, first published in 2006, describes mathematics and algorithms for the first time in book form, with breathtaking colour pictures.

Dynamics Of Complex Systems


Author: Yaneer Bar-yam
Publisher: Westview Press
ISBN: 9780813341217
Category: Science
Page: 864
View: 8429

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The study of complex systems in a unified framework has become recognized in recent years as a new scientific discipline, the ultimate in the interdisciplinary fields. Breaking down the barriers between physics, chemistry, and biology and the so-called soft sciences of psychology, sociology, economics and anthropology, this text explores the universal physical and mathematical principles that govern the emergence of complex systems from simple components. Dynamics of Complex Systems is the first text describing the modern unified study of complex systems. It is designed for upper-undergraduate/beginning graduate level students, and covers a broad range of applications in a broad array of disciplines. A central goal of this text is to develop models and modeling techniques that are useful when applied to all complex systems. This is done by adopting both analytic tools, including statistical mechanics and stochastic dynamics, and computer simulation techniques, such as cellular automata and Monte Carlo. In four sets of paired, self-contained chapters, Yaneer Bar-Yam discusses complex systems in the context of neural networks, protein folding, living organisms, and finally, human civilization itself. He explores fundamental questions about the structure, dynamics, evolution, development and quantitative complexity that apply to all complex systems. In the first chapter, mathematical foundations such as iterative maps and chaos, probability theory and random walks, thermodynamics, information and computation theory, fractals and scaling, are reviewed to enable the text to be read by students and researchers with a variety of backgrounds.

Chaos and Fractals

An Elementary Introduction
Author: David P. Feldman
Publisher: Oxford University Press
ISBN: 0199566437
Category: Mathematics
Page: 408
View: 2328

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For students with a background in elementary algebra, this book provides a vivid introduction to the key phenomena and ideas of chaos and fractals, including the butterfly effect, strange attractors, fractal dimensions, Julia Sets and the Mandelbrot Set, power laws, and cellular automata. The book includes over 200 end-of-chapter exercises.

Introduction to Discrete Dynamical Systems and Chaos


Author: Mario Martelli
Publisher: John Wiley & Sons
ISBN: 1118031121
Category: Mathematics
Page: 344
View: 4636

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A timely, accessible introduction to the mathematics of chaos. The past three decades have seen dramatic developments in the theory of dynamical systems, particularly regarding the exploration of chaotic behavior. Complex patterns of even simple processes arising in biology, chemistry, physics, engineering, economics, and a host of other disciplines have been investigated, explained, and utilized. Introduction to Discrete Dynamical Systems and Chaos makes these exciting and important ideas accessible to students and scientists by assuming, as a background, only the standard undergraduate training in calculus and linear algebra. Chaos is introduced at the outset and is then incorporated as an integral part of the theory of discrete dynamical systems in one or more dimensions. Both phase space and parameter space analysis are developed with ample exercises, more than 100 figures, and important practical examples such as the dynamics of atmospheric changes and neural networks. An appendix provides readers with clear guidelines on how to use Mathematica to explore discrete dynamical systems numerically. Selected programs can also be downloaded from a Wiley ftp site (address in preface). Another appendix lists possible projects that can be assigned for classroom investigation. Based on the author's 1993 book, but boasting at least 60% new, revised, and updated material, the present Introduction to Discrete Dynamical Systems and Chaos is a unique and extremely useful resource for all scientists interested in this active and intensely studied field. An Instructor's Manual presenting detailed solutions to all the problems in the book is available upon request from the Wiley editorial department.

Dynamics in One Complex Variable. (AM-160)

(AM-160) - Third Edition
Author: John Milnor
Publisher: Princeton University Press
ISBN: 9781400835539
Category: Mathematics
Page: 320
View: 3497

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This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. This third edition contains a number of minor additions and improvements: A historical survey has been added, the definition of Lattés map has been made more inclusive, and the écalle-Voronin theory of parabolic points is described. The résidu itératif is studied, and the material on two complex variables has been expanded. Recent results on effective computability have been added, and the references have been expanded and updated. Written in his usual brilliant style, the author makes difficult mathematics look easy. This book is a very accessible source for much of what has been accomplished in the field.

Complex Dynamical Systems


Author: Robert L. Devaney
Publisher: American Mathematical Soc.
ISBN: 9780821867549
Category:
Page: N.A
View: 4465

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In the last fifteen years, the Mandelbrot set has emerged as one of the most recognizable objects in mathematics. While there is no question of its beauty, relatively few people appreciate the fact that the mathematics behind such images is equally beautiful. This book presents lectures delivered during the AMS Short Course entitled ``Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets'', held at the Joint Mathematics Meetings in Cincinnati in January 1994. The lectures cover a wide range of topics, including the classical work of Julia and Fatou on local dynamics of analytic maps as well as recent work on the dynamics of quadratic and cubic polynomials, the geometry of Julia sets, and the structure of various parameter spaces. Among the other topics are recent results on Yoccoz puzzles and tableaux, limiting dynamics near parabolic points, the spider algorithm, extensions of the theory to rational maps, Newton's method, and entire transcendental functions. Much of the book is accessible to anyone with a background in the basics of dynamical systems and complex analysis.

Chaos and Complexity in Psychology

The Theory of Nonlinear Dynamical Systems
Author: Stephen J. Guastello,Matthijs Koopmans,David Pincus
Publisher: Cambridge University Press
ISBN: 1139867261
Category: Psychology
Page: N.A
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While many books have discussed methodological advances in nonlinear dynamical systems theory (NDS), this volume is unique in its focus on NDS's role in the development of psychological theory. After an introductory chapter covering the fundamentals of chaos, complexity and other nonlinear dynamics, subsequent chapters provide in-depth coverage of each of the specific topic areas in psychology. A concluding chapter takes stock of the field as a whole, evaluating important challenges for the immediate future. The chapters are written by experts in the use of NDS in each of their respective areas, including biological, cognitive, developmental, social, organizational and clinical psychology. Each chapter provides an in-depth examination of theoretical foundations and specific applications and a review of relevant methods. This edited collection represents the state of the art in NDS science across the disciplines of psychology.