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Chaos and Dynamical Systems

David P. Feldman
Princeton University Press
Publication Date: 
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[Reviewed by
David S. Mazel
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David Feldman has written a gentle and loving introduction to dynamical systems. I found his book excellent and well-worth my time to read and think about. If you have not ever read anything about dynamical systems and if a simple differential equation is your friend (or future friend) then this book is for you. The mathematics is clear and well-explained. The examples are as simple as possible while still illustrating the points. Sections that might be difficult are marked and I found them clearly explained in any case.
If, however, you are an expert on dynamical systems then this book will not teach you anything new. Still, I would encourage experts to read this book because Feldman’s writing style is so gentle and clear that even experts will learn about expository writing. And if that’s all you get from his book, you’ll be the richer for it.  Feldman teaches a Massive Open Online Course (MOOC) on this topic with videos that could accompany the book
Now, let’s look inside his remarkable book.  The text begins with iterated functions of one-dimensional equations. The examples include square root functions and quadratics. The equations illustrate the idea of fixed points and periodic orbits. There are excellent plots to show the behavior of the equations. The next topic is differential equations and here, too, Feldman gives an excellent introduction with an example of a cooling (or warming) water bottle. It’s a bit simple but it shows how one can view a differential equation. Even an expert would find this discussion valuable if only to see how one can simplify what is otherwise a possibly difficult topic.
The book then gives the reader what has to be one of the most easily understandable discussions of mathematical modeling. Here we find Laplace’s demon, the all-knowing creature that can predict behaviors with deterministic equations. I have studied and read about chaos for decades yet this was the first time I read of Laplace’s demon in connection to chaos. The demon is a perfect creature to enlist in the discussion of sensitive dependence on initial conditions and non-linear equations.
Feldman makes the case for simplified models by quoting Virginia Marie Peterson, the wife of naturalist Roger Tory Peterson. The passage is about drawings but it applies to models and is extraordinary about how we should view models. I’d like to quote it from the book:
A drawing can do much more than a photograph to emphasize the field marks. A photograph is a record of a fleeting instant; a drawing is a composite of the artist’s experience. The artist can edit out, show field marks to best advantage, and delete unnecessary clutter. He can choose position and stress basic color and pattern unmodified by transitory light and shade. A photograph is subject to the vagaries of color temperature, make of film, time of day, angle of view, skill of the photographer, and just plain luck. The artist has more options and far more control even though he may at times use photographs for reference. This is not a diatribe against photography; Dr. Peterson was an obsessive photographer as well as an artist and fully aware of the differences. Whereas a photograph can have a living immediacy a good drawing is really more instructive.
This quotation provides an excellent context to understand later chapters discussing the logistic equation, for example, and how it simplifies population changes but captures the essence of population fluctuations.  By the way, Feldman peppers his book with other instructive comments similar to Peterson’s quote. I found them not just helpful but insightful as to how we might few our research and place our work in context to nature.
Next, we discover the logistic equation and its sensitive dependence on initial conditions. The equation is:
\( x_{n+1} = r \cdot x_{n}(1-x_{n}) \)
The book explores the behavior of this equation as the gain term \( r \) is varied. This is, of course, standard fare for any book on chaos and Feldman tells us little that is new. But, again, how he tells us about the behavior of the equation is the essence of clear, simplistic, prose. He takes us through its periodic behavior, its aperiodic behavior and its chaotic behavior.  We learn how this sort of equation exhibits deterministic randomness. This discussion is counterintuitive because random seems the polar opposite of deterministic. Yet, we find the
distribution of outputs from the logistics equation is indeed distributed as a random variable.  It’s a discussion most readers should find instructive and puzzling.
The next chapter is a little more technical when we come to bifurcations and hysteresis. Bifurcations are places where the equilibria points can split and hysteresis is where the evolution of a chaotic system demonstrates a path dependence. Both notions are not intuitive but the book gives the reader a good idea of what these mean. The discussion is a bit superficial but it is completely in-line with the expected audience for the book and is well-done.
The chapter on universality of chaos is esoteric and difficult to describe here. But, the following chapter discussing phase space gives you a gentle idea of what phase space is. However, a better and more detailed discussion is David Nolte’s The Tangled Tale of Phase Space, which Feldman references. The final technical chapter about strange attractors is, as the rest of the book, a lovely introduction of these fascinating and endlessly complicated shapes. We are treated to a light and intimate meeting of the Rössler attractor. We see its complexity and are shown how, with the method of delays, we can reconstruct its 3-space shape based on a single time-series of one variable.
In summary, this is an excellent introduction to chaotic systems with well-written prose, clear examples, and a comprehensive list of references for further reading. Chaos and Dynamical Systems is a book for everyone from the layman to the expert. Each will find it useful, informative, and a model of what a popular mathematics book should be.


David S. Mazel is a practicing engineer in Washington, DC. He welcomes your thoughts and feedback. He can be reached at
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