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Chaos Theory Tamed,Garnett P Williams,US Geological Survey Ret. JOSEPH HENRY PRESS,Washington D C 1997,JOSEPH HENRY PRESS. 2101 Constitution Avenue NW Washington DC 20418, The JOSEPH HENRY PRESS an imprint of the NATIONAL ACADEMY PRESS was created by the National. Academy of Sciences and its affiliated institutions with the goal of making books on science technology and. health more widely available to professionals and the public Joseph Henry was one of the founders of the. National Academy of Sciences and a leader of early American science. Any opinions findings conclusions or recommendations expressed in this volume are those of the author and. do not necessarily reflect the views of the National Academy of Sciences or its affiliated institutions. Library of Congress Catalog Card Number 97 73862,International Standard Book Number 0 309 06351 5. Additional copies of this book are available from,JOSEPH HENRY PRESS NATIONAL ACADEMY PRESS.

2101 Constitution Avenue NW,Washington DC 20418,1 800 624 6242 phone. 1 202 334 3313 phone in Washington DC,1 202 334 2451 fax. Visit our Web site to read and order books on line http www nap edu. Copyright 1997 Garnett P Williams All rights reserved. Published by arrangement with Taylor Francis Ltd,Reprinted 1999. Printed in Great Britain, Virtually every branch of the sciences engineering economics and related fields now discusses or refers to. chaos James Gleick s 1987 book Chaos making a new science and a 1988 one hour television program on. chaos aroused many people s curiosity and interest There are now quite a few books on the subject Anyone. writing yet another book on any topic inevitably goes through the routine of justifying it My justification. consists of two reasons, Most books on chaos while praiseworthy in many respects use a high level of math Those books have.

been written by specialists for other specialists even though the authors often label them introductory. Amato 1992 refers to a cultural chasm between the small group of mathematically inclined initiates. who have been touting chaos theory on the one hand and most scientists and I might add everybody. else on the other There are relatively few books for those who lack a strong mathematics and physics. background and who might wish to explore chaos in a particular field More about this later in the Preface. Most books in my opinion don t provide understandable derivations or explanations of many key. concepts such as Kolmogorov Sinai entropy dimensions Fourier analysis Lyapunov exponents and. others At present the best way to get such explanations is either to find a personal guru or to put in gobs of. frustrating work studying the brief condensed advanced treatments given in technical articles. Chaos is a mathematical subject and therefore isn t for everybody However to understand the fundamental. concepts you don t need a background of anything more than introductory courses in algebra trigonometry. geometry and statistics That s as much as you ll need for this book More advanced work on the other hand. does require integral calculus partial differential equations computer programming and similar topics. In this book I assume no prior knowledge of chaos on your part Although chaos covers a broad range of. topics I try to discuss only the most important ones I present them hierarchically Introductory background. perspective takes up the first two chapters Then come seven chapters consisting of selected important material. an auxiliary toolkit from various fields Those chapters provide what I think is a good and necessary. foundation one that can be arduous and time consuming to get from other sources Basic and simple chaos. related concepts follow They in turn are prerequisites for the slightly more advanced concepts that make up. the later chapters That progression means in turn that some chapters are on a very simple level others on a. more advanced level In general I try to present a plain vanilla treatment with emphasis on the idealized case. of low dimensional noise free chaos That case is indispensable for an introduction Some real world data in. contrast often require sophisticated and as yet developing methods of analysis I don t discuss those techniques. The absence of high level math of course doesn t mean that the reading is light entertainment Although there s. no way to avoid some specialized terminology I define such terms in the text as well as in a Glossary Besides. learning and using a new vocabulary a new language is fun and exciting It opens up a new world. My goal then is to present a basic semitechnical introduction to chaos The intended audience consists of. chaos nonspecialists who want a foothold on the fundamentals of chaos theory regardless of their academic. level Such nonspecialists may not be comfortable with the more formal mathematical approaches that some. books follow Moreover many readers myself included often find a formal writing style more difficult to. understand With this wider and less mathematically inclined readership in mind I have deliberately kept the. writing informal we ll instead of we will I d instead of I would etc Traditionalists who are used to a. formal style may be uneasy with this Nonetheless I hope it will help reduce the perceived distance between. subject and reader, I m a geologist hydrologist by training I believe that coming from a peripheral field helps me to see the subject. differently It also helps me to understand and I hope answer the types of questions a nonspecialist has. Finally I hope it will help me to avoid using excessive amounts of specialized jargon. In a nutshell this is an elementary approach designed to save you time and trouble in acquiring many of the. fundamentals of chaos theory It s the book that I wish had been available when I started looking into chaos I. hope it ll be of help to you, In regard to units of measurement I have tried to compromise between what I m used to and what I suspect. most readers are used to I ve used metric units kilometers centimeters etc for length because that s what I. have always used in the scientific field I ve used Imperial units pounds Fahrenheit etc in most other cases. I sincerely appreciate the benefit of useful conversations with and or help from A V Vecchia Brent Troutman. Andrew Fraser Ben Mesander Michael Mundt Jon Nese William Schaffer Randy Parker Michael Karlinger. Leonard Smith Kaj Williams Surja Sharma Robert Devaney Franklin Horowitz and John Moody For. critically reading parts of the manuscript I thank James Doerer Jon Nese Ron Charpentier Brent Troutman A. V Vecchia Michael Karlinger Chris Barton Andrew Fraser Troy Shinbrot Daniel Kaplan Steve Pruess. David Furbish Liz Bradley Bill Briggs Dean Prichard Neil Gershenfeld Bob Devaney Anastasios Tsonis. and Mitchell Feigenbaum Their constructive comments helped reduce errors and bring about a much more. readable and understandable product I also thank Anthony Sanchez Kerstin Williams and Sebastian. Kuzminsky for their invaluable help on the figures. I especially want to thank Roger Jones for his steadfast support perseverance hard work friendly advice and. invaluable expertise as editor He has made the whole process a rewarding experience for me Other authors. should be so fortunate,A constant,A constant or scalar. a A constant b a component dimension in deriving the Hausdorff Besicovich dimension. Intercept of line R ca H KSm and taken as a rough indicator of the accuracy of the measurements. Component dimension in deriving Hausdorff Besicovich dimension elsewhere a derivative. Base of natural logarithms with a value equal to 2 718. A function,Harmonic number, A global counter often representing the ith bin of the group of bins into which we divide values of x. a A counter often representing the jth bin of the group of bins into which we divide values of y b the. imaginary number 1 0 5,Control parameter,k value at time t or observation n.

k value logistic equation at which chaos begins, A chosen lag offset displacement or number of intervals between points or observations. Number position of an iteration observation or period within a sequence e g the nth observation. Scaling ratio,Standard deviation,Variance same as power. Time sometimes measured in actual units and sometimes just in numbers of events no units. An error term,Special variable,A variable,Indicator variable e g time population distance. a The ith value of x ith point or ith bin b all values of x as a group. A trajectory point to which distances from point xi are measured in calculating the correlation dimension. Value of the variable x at the nth observation, Value of the variable x at time or observation t hence x0 x1 x2 etc. Attractor a value of x,Dependent variable or its associated value.

Height of the wave having harmonic number h,Value of dependent variable y at the origin. A variable,Wave amplitude,Correlation sum, A multipurpose or general symbol for dimension including embedding dimension. Capacity a type of dimension,Hausdorff or Hausdorff Besicovich dimension. Information dimension numerically equal to the slope of a straight line on a plot of I arithmetic scale versus. 1 log scale, An observed vector usually not perpendicular to any other observed vectors. Wave frequency, Labeling symbol in definition of correlation dimension.

Entropy sometimes called information entropy,Entropy at time t. Entropy computed as a weighted sum of the entropies of individual phase space compartments. Kolmogorov Sinai K S entropy, Kolmogorov Sinai K S entropy as estimated from incremental redundancies. Self entropy of system X,Joint entropy of systems X and Y. Conditional entropy for system X,Self entropy of system Y. Conditional entropy for system Y, Entropy computed over a particular duration of time t.

Information,Information contributed by compartment i. Total information contributed by all compartments,Information for dynamical system X. Mutual information of coupled systems X and Y,Information of dynamical system Y. Mutual information of coupled systems X and Y, Information needed to describe an attractor or trajectory to within an accuracy. Boltzmann s constant,Length or distance, Estimated length usually by approximations with small straight increments of length.

Wavelength, An estimate of a measure a determination of length area volume etc. A true value of a measure,Total number of data points or observations. Total number of dimensions or variables, Total number of possible bin routes a dynamical system can take during its evolution from an arbitrary starting. time to some later time, Total number of possible or represented states of a system. Number of points contained within a circle sphere or hypersphere of a given radius. Probability, a Probability associated with the ith box sphere value etc b all probabilities of a distribution as a group.

Sequence probability, a Probability of class xi from system X b all probabilities of the various classes of x as a group. Joint probability that system x is in class xi when system Y is in class yj. a Probability of class yj from system Y b all probabilities of the various classes of y as a group. Conditional probability that system Y will be in class yj given that system X is in class xi. Redundancy,Autocorrelation at lag m,Wave period, A unit vector representing any of a set of mutually orthogonal vectors. A vector constructed from an observed vector so as to be orthogonal to similarly constructed vectors of the. A system or ensemble of values of random variable x and its probability distribution. A system or ensemble of values of random variable y and its probability distribution. Fourier cosine coefficient,Fourier sine coefficient. Difference between two computed values of a trajectory for a given iteration number. Orbit difference obtained by extrapolating a straight line back to n 0 on a plot of orbit difference versus. iteration n, Difference between starting values of two trajectories. Characteristic length of scaling device ruler box sphere etc. Largest length of scaling device for which a particular relation holds. a Central or inclusive angle such as the angle subtended during a rotating disk experiment excluding phase. angle or the angle between two vectors b an angular variable or parameter. Lyapunov exponent global not local,Correlation dimension correlation exponent.

Estimated correlation dimension,phase angle,summation symbol. an interval range or difference, incremental redundancy redundancy at a given lag minus redundancy at the previous lag. given or given a value of,BACKGROUND, What is this business called chaos What does it deal with and why do people think it s important Let s. begin with those and similar questions,Introduction. The concept of chaos is one of the most exciting and rapidly expanding research topics of recent decades. Ordinarily chaos is disorder or confusion In the scientific sense chaos does involve some disarray but there s. much more to it than that We ll arrive at a more complete definition in the next chapter. The chaos that we ll study is a particular class of how something changes over time In fact change and time are. the two fundamental subjects that together make up the foundation of chaos The weather Dow Jones industrial. average food prices and the size of insect populations for example all change with time In chaos jargon. learning and using a new vocabulary a new language is fun and exciting It opens up a new world My goal then is to present a basic semitechnical introduction to chaos The intended audience consists of chaos nonspecialists who want a foothold on the fundamentals of chaos theory regardless of their academic level Such nonspecialists may

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