The Chaos Hypertextbook The Chaos Hypertext Book Mathematics in the Age of the Computer 0 Prefaces 0.1 What's New 0.2 About This Book 0.3 Links to This Site 1 Mathematical Experiments 1.1 Iterations & Orbits 1.2 Orbits & Bifurcations 1.3 Universality 2 Strange & Complex 2.1 Strange Attractors 2.2 Julia Sets 2.3 Mandelbrot Sets 3 What is Dimension? 3.1 General Dimension 3.2 Topological Dimension 3.3 Fractal Dimension 4 Measuring Chaos 4.1 Harmonic Oscillator 4.2 Logistic Equation 4.3 Lyapunov Exponent 4.4 Lyapunov Space A Appendices A.1 Printed Resources A.2 Software Resources A.3 Internet Resources A.Z Zoooom into the M-Set 0.1 What's New in The Chaos Hypertextbook Prefaces The Chaos Hypertextbook © 1995-99 by Glenn Elert All Rights Reserved -- Fair Use Encouraged prev | index | next 0.1 What's New in The Chaos Hypertextbook 28 January 1999 l M oved to <hypertextbook.com> from <www.columbia.edu/~gae4>. Changed the name to "The Chaos Hypertextbook" from "Chaos, Fractals, Dimension". l M ade a cool new banner showing a Julia Set Cascade of the type explained in Chapter 2.3. Created this page and a page with links to this site. 7 June 1998 l A dded a series of pages showing a zoom into the Mandelbrot set over 15 orders of magnitude. 19 February 1998 l D iscovered the joys of server-side includes (SSI). Now pages have consistent colophons controlled by one file. If that doesn't mean anything to you, then forget about it. 15 December 1997 l C ompletely revised and updated all four chapters in html format. Gave this new website the name "Chaos, Fractals, Dimension". 10 January 1996 l C ompleted the text copy of the fourth chapter. 27 June 1995 l C ompleted the text copy of the third chapter. 12 June 1995 0.1 What's New in The Chaos Hypertextbook l C ompleted the text copy of the second chapter. 3 May 1995 l C ompleted the text copy of the first chapter. 0.2 About The Chaos Hypertextbook Prefaces The Chaos Hypertextbook © 1995-99 by Glenn Elert All Rights Reserved -- Fair Use Encouraged prev | index | next 0.2 About The Chaos Hypertextbook Who? You and I. I wrote this book for anyone with an interest in chaos, fractals, non-linear dynamics, or mathematics in general. It's a moderately heavy piece of work, requiring a bit of mathematical knowledge, but it is definitely not aimed at mathematicians. My background is in physics and I use mathematics extensively in problem solving. Like many educated people, I also enjoy math as a diversion. This is the audience I am writing for. What? Neat Stuff. In the 1980s, strange new mathematical concepts burst forth from academic isolation to seize the attention of the public. Chaos. A fantastic notion. The study of the uncontainable, the unpredictable, the bizarre. Fractals. Curves and surfaces unlike anything ever seen in mathematics before. Surely, these topics are beyond the comprehension of all but the smartest, most educated, and most specialized geniuses. Wrong! Chaos, fractals, and the related topic of dimension are really not that difficult. One can devote an academic lifetime to them, of course, but the basic introduction presented in this book is no more difficult to understand than the straight line and the parabola. When? Right Now. Some of the topics discussed have roots extending back to the close of the Nineteenth Century. The really flashy stuff had to wait until integrated circuits integrated themselves into daily life. To attract the attention of the media-saturated you've got to have color, pattern, detail, and motion at a level 0.2 About The Chaos Hypertextbook beyond line drawings on paper. You need a computer. Actually, you need a lot of computers and they've got to be cheap, fast, and simple to operate so that many people will use them. You need to live at the dawn of the Twenty-first Century. If you're reading this text you have the tool needed to reproduce every image, movie, and data set found in this book. This is mathematics in the age of the computer. Where? Nowhere and Everywhere. This book can never exist on paper. Although copies of the linear text have existed on paper in the past and will again in the future, this is really a hypertext document. Move your finger over the linear text, press on a diagram or word and you leave behind a fingerprint. Move your cursor over the hypertext, press on a diagram or word and you're off viewing another page. This book will never exist again as it does now. I intend to update and modify it on an irregular basis (that is, whenever I feel like it). Portions of this book were originally composed with Microsoft Word 5.1 running on a Macintosh LC. After Mosaic sparked the explosion of the World Wide Web in 1994, I knew that I would eventually transfer it to HTML. When the next tidal wave inundates the computer world, chances are this book will be washed away with it. This book does not exist anywhere. There is no entity that contains it. I play with an edition of it that lives in my Power Macintosh, back it up on to Zip disks, download it to a server hidden somewhere on the planet, you access it, and copies of it bounce around the Internet until they land in your cache. This is not a book. Why? Why Not. I began writing Chaos, Fractals, Dimension because I was interested in the topics presented. After I saw how easy it all was to understand, it grew and grew until it covered over one hundred pages of double-spaced text. The augmented, hypertext version as it existed on 15 December 1997 was submitted as the final integrated essay for a Master of Science degree in secondary science education from Teachers College, Columbia University. 0.2 About The Chaos Hypertextbook Expanded Table of Contents Prefaces 1. What's New at The Chaos Hypertextbook 2. About The Chaos Hypertextbook <-- you are here 3. Links to The Chaos Hypertextbook 1. Mathematical Experiments The first chapter introduces the basics of one-dimensional iterated maps. Say what? Take a function y = f(x). Substitute some number into it. Take the answer and run it through the function again. Keep doing this forever. This is called iteration. The numbers generated exhibit three types of behavior: steady-state, periodic, and chaotic. In the 1970's, a whole new branch of mathematics arose from the simple experiments described in this chapter. 1. Iteration & Orbits 2. Orbits & Bifurcations 3. Universality 2. Strange & Complex The second chapter extends the idea of an iterated map into two dimensions, three dimensions, and complex numbers. This leads to the creation of mathematical monsters called fractals. A fractal is a geometric pattern exhibiting an infinite level of repeating, self-similar detail that can't be described with classical geometry. They are quite interesting to look at and have captured a lot of attention. This chapter describes the methods for constructing some of them. 1. Strange Attractors 2. Julia Sets 3. Mandelbrot Sets 3. What is Dimension? The third chapter deals with some of the definitions and applications of the word dimension. A fractal is an object with a fractional dimension. Well, 0.2 About The Chaos Hypertextbook not exactly, but close enough for now. What does this mean? The answer lies in the many definitions of dimension. 1. General Dimension 2. Topological Dimension 3. Fractal Dimension 4. Measuring Chaos The fourth chapter compares linear and non-linear dynamics. The harmonic oscillator is a continuous, first-order, differential equation used to model physical systems. The logistic equation is a discrete, second-order, difference equation used to model animal populations. So similar and yet so alike. The harmonic oscillator is quite well behaved. The paramenters of the system determine what it does. The logistic equation is unruly. It jumps from order to chaos without warning. A parameter that discriminates among these behaviors would enable us to measure chaos. 1. Harmonic Oscillator 2. Logistic Equation 3. Lyapunov Exponent 4. Lyapunov Space Appendices 1. Printed Resources 2. Software Resources <-- unusually large page 3. Internet Resources <-- unusually large page 0.3 Links to The Chaos Hypertextbook Prefaces The Chaos Hypertextbook © 1995-99 by Glenn Elert All Rights Reserved -- Fair Use Encouraged prev | index | next 0.3 Links to The Chaos Hypertextbook l A rt and Science, Dr Jean Debord l B ob's Fractal Links, by Bob I assume l B ookmarks, Erkki Kurenniemi, Heureka: The Finnish Science Center l B ookmarks for Tomas B. Klos, University of Groningen l C haos Exercise Connections, J. Saunders l C haos Mathematics, Learning Fractal Geometry, Metaculture l C haos Theory, StudyWeb l C SCI4446/6446 Course Materials for Spring 1999, Liz Bradley, University of Colorado l D eterminitve Chaos Software and related links, Stock Market Timing Using Advanced Mathematics l D ynamical Sytems III (063091), Bob Johnson, University of Durham l E D 265i -- Unit 4 Lecture -- An Anatomy of a Web Site, Kathy Rutkowski, George Washington University l E DU2 : Level 2, E.J. Inglis-Arkell l E nlasces a Otra Interesantes Paginas, José Luis De la Cruz Lázaro l E nlaces y referencias sobre caos y fractales, Gonzalo Alvarez, Instituto de Física Aplicada l E xcite Education: Science & Nature: Mathematics: Fractals & Complexity l F ractal & Fractal, Jiang Ching Kuen, Tsing Hua University, look under "¸H§Îªº*Æ*Ç" 0.3 Links to The Chaos Hypertextbook l F ractal Links, MathsNet, Anglia Multimedia l F ractal Links on Paul N. Lee's Website, look for "various software" l F ractal Top, Garr Lystad l F ractalTrees, Simon Woodside, University of Waterloo l F ractal Web Sites, Fractal Domains, Dennis C. De Mars l F ractals - Links, J.P. Louvet, Université Bordeaux l F rédéric Paccaut, Universite de Bourgogne l F un Science Stuff, Bellarmine College l G eneral Links, Kevin Raulerson l I Jornada de Física y Matemática l Id eas and Activities, Vicki F. & Richard M. Sharp, California State University, Northridge l In teressante links, Educatieve Faculteit Amsterdam l In vestigating The Mathematics of Complexity l Ju an P. Cerezo's Bookmarks, La Universidad Autónoma de Madrid l K aosplock, Lars Rosenberg l K not A Braid of Links, Canadian Mathematical Community (CAMEL) l K uo-Chang Chen's Homepage, University of Minnesota l L 'art et la science, Dr Jean Debord l le c1, Guy Zimmerman, Techno-Z Fachhochschule Salzburg l L ecture for Feb 03, 1999 - Physics 351, Larry Gladney, University of Pennsylvania l L inks, Institut für Physiologie, Freie Universität Berlin l L inks, Math Teacher's Home Page, Toronto District School Board l L inks on Chaos and Links on Chaos, Chaos 21, Inje University l L ookSmart / World - Library - Sciences - Math - Fractals & Complexity l F ractals & Complexity, Looksmart l M ath (question), Hong Kong School Net 0.3 Links to The Chaos Hypertextbook l M @ths: Fractals! Diana Rolf l M andelbrot Exhibition, Jonathan Bowen, Oxford University l M andelbrot Set, Dale Winter, University of Michigan l M ath 335 Chaos, Fractals and Dynamics, Randall Pyke, University of Toronto l M ath 357 Complex Functions, Dr. John Maharry, Franklin College l M ath, Jason Vestuto, University of Maryland l M ath Forum Internet Resource Collection, linked on numerous pages in the index l M ath Sites, South Fayette Township School District l M athematics, Mason Library Information System, Keene State College l M athematics, Todd R. Shaw, University of Utah l M athematics, Science and Engineering, Ralph Carmichael l M athpuzzle.com, Ed Pegg Jr, l M olecular Modeling, Michael C. Tims, University of Maryland l N etscape Search: Science: Math: Chaos l N etWatch (19 February 1999), American Association for the Advancement of Science (AAAS) l N onlinear Dynamics, Mathematics Archives, University of Tennesee, Knoxville l N onlinear Sites, Chaos at Maryland, University of Maryland l N onlinearity and Complexity Home Page, Moses A. Boudourides, Democritus University of Thrace l O ton kirjanmerkit, Otto Hyvärinen l P hysics 4267 - Introductory Nonlinear Dynamics & Chaos, Georgia Institute of Technology l P hysics and astronomy links, Aarhus Universitet l P ouÏitá literatura, Vít Prudil, MasarykUniversity l P rincipia - Internet, Eduardo René Rodríguez Ávila