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Chaos and Fractals: An Elementary Introduction

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David P. Feldman
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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The subject of chaos and fractals, like cryptography, can be taught to undergraduates at different levels of mathematical sophistication. On the one hand, a course for mathematics majors can easily be created that uses topological notions such as metric spaces; Iowa State University, for example, offers an upper-level course on the topic. (The last time the course was offered, the text was Chaos: An Introduction to Dynamical Systems by Allgood, Sauer and Yorke.) But it is also possible, as this book compellingly demonstrates, to teach this material to students with no particular college-level mathematics background at all.

This text arose from the author’s experiences teaching a course at the College of the Atlantic for students “who are not math or science majors and who have not necessarily taken calculus.” The prerequisites for reading the text are quite modest: high school algebra should get a reader through most or all of it, and even this material is reviewed in a fifteen-page Appendix at the end of the text. Yet, despite these minimal prerequisites, this book is not a simplistic, dumbed-down account; although the book starts slowly, it never insults the reader’s intelligence, and covers a lot of interesting material.

Although a strong mathematical background is not assumed, mathematics is not avoided. The beginnings of differential calculus are developed in chapter 28, in a nice intuitive way that conveys the basics of the subject without getting bogged down in technicalities. (I suspect that, the next time I teach freshman calculus, I may wind up stealing some of this material for an introductory overview of the course.). For other examples, concepts of probability theory are discussed in chapters 19 and 20, chapter 21 provides a nice account of some of the basic material about cardinal numbers of sets, and chapter 23 is a short overview of the geometry and algebra of complex numbers. (For these reasons, I think the author’s suggestion, in the preface, that this book is suitable as a text for high school students is probably a bit optimistic. Although the book literally doesn’t use much that the average high school senior has not seen, it does require a certain level of maturity, and willingness to think hard, on the part of its readers that the average high school student may not have.)

The text is divided into six main parts (each one except the last consisting of multiple chapters), plus a set of Appendices collectively denoted Part VII. Part I is mostly introductory. The concept of a function is defined from scratch, and applied to the study of iterated functions. This in turn leads to the introduction of the idea of a dynamical system, and a discussion of population models. The final chapter of Part I is a nice digression on the nature of science and some of the controversy surrounding the question of what the assumption of a deterministic universe has to say about free will.

Part II introduces chaos by means of a discussion of sensitive dependence on initial data, and the so-called “butterfly effect”. The logistic equation is a recurring example throughout these chapters, which also introduce the notion of bifurcation diagrams and some statistical concepts.

Fractals are the subject of Part III of the text; they are introduced first by means of examples and then given a more precise definition. The concept of fractal dimension is discussed at some length. Two important examples of fractals, the Julia sets and Mandelbrot set, are the subject of Part IV. (To quote the author: “You are probably familiar with them; these are the psychedelic fractal images that have found their way onto posters, calendars, web pages, book covers, and so on.”) As previously mentioned, a short chapter on complex numbers is included for the benefit of people without much background in them.

In Part V, the author looks at types of dynamical systems other than the discretely-changing single-variable functions that were the focus of study to this point. This necessitates the introduction to the ideas of calculus that was previously mentioned, and which is applied to study such things as systems and phase planes, the Volterra prey-predator system, and the Lorenz equations. There is also a chapter on cellular automata, another kind of dynamical system.

The main textual material ends with Part VI, comprising one single, very short, chapter entitled “Conclusion”, in which the author summarizes what has come before and also offers his opinions as to the significance of the theory for the future. The author poses, and briefly discusses, the questions “Is chaos a scientific revolution on par with the development of quantum mechanics, relativity and calculus? What impact has chaos had, and what impact will it have in the future?”

This is followed by a set of three appendices: a review of high school algebra, a discussion of histograms and distributions, and a guide to further reading. This last chapter deserves specific mention because it not only lists books but also discusses them, a feature I wish more textbooks would emulate. Here, for example, we find a page-long discussion of Gleick’s Chaos: Making a New Science, and some of the controversy it engendered. Another noteworthy, and very attractive, feature of this chapter is that not only does it list journal articles, it contains a discussion of the peer-review process and offers suggestions to students as to how to search for articles.

There is a great deal to like about this book, starting with the author’s writing style, which I found particularly clear and enjoyable. The frequent use of the first-person “I” gives the text a pleasantly informal, chatty quality. Consider, for example, this passage from chapter 23: “I think that the word ‘imaginary’ carries some metaphysical or philosophical baggage that is not helpful. All numbers — complex and real — are imaginary. They are constructs and idealizations of ideas in people’s heads.” The author’s willingness to express opinions (e.g, “Mandelbrot responded to Krantz’s essay in a predictably prickly fashion”) also helps alleviate the dryness that sometimes attends mathematics texts. In addition to this overall conversational tone, the author makes frequent use of marginal notes to help clarify or explain points.

The book is also liberally sprinkled with photographs and pictures. Unfortunately, however, there are no color plates of the kind of beautiful fractal photographs that one typically finds in a book of this nature. This seems like an opportunity missed, but is hardly a critical defect, particularly since this is intended as a textbook, not a coffee table book.

Consistent with its intended use as a textbook, there are exercises at the end of each chapter, something that is often missing in other accounts of fractals and chaos at this level (such as Ian Stewart’s Does God Play Dice or Chaos Under Control by Peak and Frame), most of which are apparently intended as popular science expositions rather than actual texts. Some of these exercises, marked with an asterisk, attempt to lead the student to discover interesting ideas that may later be referred to in the text; others, marked with a music “sharp sign”, call for more mathematical sophistication and are therefore intended for more advanced students. A lengthy solutions manual (178 pages) is available from the publisher.

All in all, this is a very valuable book. I mentioned earlier that its mathematical prerequisites were minimal, but that doesn’t mean an undergraduate math major wouldn’t also benefit from reading it, particularly if he or she were planning to go on to take more sophisticated courses in the subject; this would be a great place to get a preliminary overview of the more difficult material to follow. This is an excellent book, and is highly recommended.

Mark Hunacek ( teaches mathematics at Iowa State University.

I. Introducing Discrete Dynamical Systems
0. Opening Remarks
1. Functions
2. Iterating Functions
3. Qualitative Dynamics
4. Time Series Plots
5. Graphical Iteration
6. Iterating Linear Functions
7. Population Models
8. Newton, Laplace, and Determinism
II. Chaos
9. Chaos and the Logistic Equation
10. The Buttery Effect
11. The Bifurcation Diagram
12. Universality
13. Statistical Stability of Chaos
14. Determinism, Randomness, and Nonlinearity
III. Fractals
15. Introducing Fractals
16. Dimensions
17. Random Fractals
18. The Box-Counting Dimension
19. When do Averages exist?
20. Power Laws and Long Tails
20. Introducing Julia Sets
21. Infinities, Big and Small
IV. Julia Sets and The Mandelbrot Set
22. Introducing Julia Sets
23. Complex Numbers
24. Julia Sets for f(z) = z2 + c
25. The Mandelbrot Set
V. Higher-Dimensional Systems
26. Two-Dimensional Discrete Dynamical Systems
27. Cellular Automata
28. Introduction to Differential Equations
29. One-Dimensional Differential Equations
30. Two-Dimensional Differential Equations
31. Chaotic Differential Equations and Strange Attractors
VI. Conclusion
32. Conclusion
VII. Appendices
A. Review of Selected Topics from Algebra
B. Histograms and Distributions
C. Suggestions for Further Reading