Author: David P. Feldman
Publisher: Oxford University Press
Release Date: 2012-08-09
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.
Author: Heinz-Otto Peitgen
Publisher: Springer Science & Business Media
Release Date: 2013-06-29
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.
Author: Robert L. Devaney
Publisher: American Mathematical Soc.
Release Date: 1989-01-01
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.
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.
This fascinating book explores the connections between chaos theory, physics, biology, and mathematics. Its award-winning computer graphics, optical illusions, and games illustrate the concept of self-similarity, a typical property of fractals. The author - hailed by Publishers Weekly as a modern Lewis Carroll - conveys memorable insights in the form of puns and puzzles. 1992 edition.
Author: Paul S. Addison
Publisher: Inst of Physics Pub Incorporated
Release Date: 1997
The aim of this textbook is to provide the reader with an elementary introduction to fractal geometry and chaotic dynamics: subjects which have attracted immense interest throughout the scientific and engineering disciplines. The book may be used in part or as a whole to form an introductory course in either or both subject areas. A prominent feature of the book is the use of many illustrations to convey the concepts required for comprehension of the subject. Plenty of problems are provided to test understanding. In addition, advanced mathematics is avoided in order to provide a concise treatment and speed the reader through the subject areas. The book can be used as a text for undergraduate courses or for self-study.
This book provides a broad coverage and has acessible style of exposition. Emphasis is on physical concepts and useful results, rather than rigorous mathematical proofs. Completeing this volume is free and user-friendly software.
Author: Willi-Hans Steeb
Publisher: World Scientific Publishing Company
Release Date: 2016-03-02
This book presents a collection of problems for nonlinear dynamics, chaos theory and fractals. Besides the solved problems, supplementary problems are also added. Each chapter contains an introduction with suitable definitions and explanations to tackle the problems. The material is self-contained, and the topics range in difficulty from elementary to advanced. While students can learn important principles and strategies required for problem solving, lecturers will also find this text useful, either as a supplement or text, since concepts and techniques are developed in the problems.
BACKGROUND Sir Isaac Newton hrought to the world the idea of modeling the motion of physical systems with equations. It was necessary to invent calculus along the way, since fundamental equations of motion involve velocities and accelerations, of position. His greatest single success was his discovery that which are derivatives the motion of the planets and moons of the solar system resulted from a single fundamental source: the gravitational attraction of the hodies. He demonstrated that the ohserved motion of the planets could he explained hy assuming that there is a gravitational attraction he tween any two ohjects, a force that is proportional to the product of masses and inversely proportional to the square of the distance between them. The circular, elliptical, and parabolic orhits of astronomy were v INTRODUCTION no longer fundamental determinants of motion, but were approximations of laws specified with differential equations. His methods are now used in modeling motion and change in all areas of science. Subsequent generations of scientists extended the method of using differ ential equations to describe how physical systems evolve. But the method had a limitation. While the differential equations were sufficient to determine the behavior-in the sense that solutions of the equations did exist-it was frequently difficult to figure out what that behavior would be. It was often impossible to write down solutions in relatively simple algebraic expressions using a finite number of terms. Series solutions involving infinite sums often would not converge beyond some finite time.
Author: Heinz-Otto Peitgen
Publisher: Springer Science & Business Media
Release Date: 2012-12-06
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.
Author: Stephen H. Kellert
Publisher: University of Chicago Press
Release Date: 1994-10-28
Chaos theory has captured scientific and popular attention. What began as the discovery of randomness in simple physical systems has become a widespread fascination with "chaotic" models of everything from business cycles to brainwaves to heart attacks. But what exactly does this explosion of new research into chaotic phenomena mean for our understanding of the world? In this timely book, Stephen Kellert takes the first sustained look at the broad intellectual and philosophical questions raised by recent advances in chaos theory—its implications for science as a source of knowledge and for the very meaning of that knowledge itself.
This textbook, now in its second edition, provides a broad introduction to both continuous and discrete dynamical systems, the theory of which is motivated by examples from a wide range of disciplines. It emphasizes applications and simulation utilizing MATLAB®, Simulink®, the Image Processing Toolbox® and the Symbolic Math toolbox®, including MuPAD. Features new to the second edition include · sections on series solutions of ordinary differential equations, perturbation methods, normal forms, Gröbner bases, and chaos synchronization; · chapters on image processing and binary oscillator computing; · hundreds of new illustrations, examples, and exercises with solutions; and · over eighty up-to-date MATLAB program files and Simulink model files available online. These files were voted MATLAB Central Pick of the Week in July 2013. The hands-on approach of Dynamical Systems with Applications using MATLAB, Second Edition, has minimal prerequisites, only requiring familiarity with ordinary differential equations. It will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a broad range of disciplines such as population dynamics, biology, chemistry, computing, economics, nonlinear optics, neural networks, and physics. Praise for the first edition Summing up, it can be said that this text allows the reader to have an easy and quick start to the huge field of dynamical systems theory. MATLAB/SIMULINK facilitate this approach under the aspect of learning by doing. —OR News/Operations Research Spectrum The MATLAB programs are kept as simple as possible and the author's experience has shown that this method of teaching using MATLAB works well with computer laboratory classes of small sizes.... I recommend ‘Dynamical Systems with Applications using MATLAB’ as a good handbook for a diverse readership: graduates and professionals in mathematics, physics, science and engineering. —Mathematica
Author: Paul Manneville
Publisher: World Scientific
Release Date: 2010
This book (2nd edition) is a self-contained introduction to a wide body of knowledge on nonlinear dynamics and chaos. Manneville emphasises the understanding of basic concepts and the nontrivial character of nonlinear response, contrasting it with the intuitively simple linear response. He explains the theoretical framework using pedagogical examples from fluid dynamics, though prior knowledge of this field is not required. Heuristic arguments and worked examples replace most esoteric technicalities. Only basic understanding of mathematics and physics is required, at the level of what is currently known after one or two years of undergraduate training: elementary calculus, basic notions of linear algebra and ordinary differential calculus, and a few fundamental physical equations (specific complements are provided when necessary). Methods presented are of fully general use, which opens up ample windows on topics of contemporary interest. These include complex dynamical processes such as patterning, chaos control, mixing, and even the Earth's climate. Numerical simulations are proposed as a means to obtain deeper understanding of the intricacies induced by nonlinearities in our everyday environment, with hints on adapted modelling strategies and their implementation.
Author: K. J. Falconer
Publisher: Oxford University Press
Release Date: 2013-09-26
An essential discussion of the popular science and mathematics behind fractals reveals how fractal shapes can be found everywhere in nature from clouds to coastlines, explaining how basic concepts in fractal geometry produced a revolution in mathematical understandings of patterns in the 20th century. Original.