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Differential Equations, Dynamical Systems, and an Introduction to Chaos

Morris W. Hirsch, Stephen Smale, and Robert L. Devaney
Academic Press
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The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Annalisa Crannell
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The first edition of Hirsch and Smale's Differential Equations, Dynamical Systems, and Linear Algebra has been a standard on mathematical bookshelves for three decades. Now middle-aged, this 30 year old text has gotten a facelift and a new convertible. Well, not really a new convertible, but it did drop the faithful-yet-staid "and Linear Algebra" from the title (but not from the contents), to replaced by the alluring young "and An Introduction to Chaos". Along the way, it picked up a third author, the prolific and entertaining Bob Devaney.

The first hint of the new approach appears subtly on page 1. The first example (dx/dt = ax) is word-for-word the same as the old text, with this exception: in 1974, the authors wrote, "Here a denotes a constant." The new text reads, "Also, a is a parameter; for each value of a we have a different differential equation." Readers familiar with the first edition will find much that is familiar, but with an increased emphasis on the behavior of families of solutions.

Of course the material in the book has been reorganized a bit. Not only has some content shifted during rewriting, but, as noted above, there is a consistent emphasis on understanding the geometry and behavior of solutions to differential equations. The exercises (which were earlier called "problems") have moved from the end of each section to the end of each chapter. In addition, there are three chapters with new material: The Lorenz System, Discrete Dynamical Systems, and Homoclinic Phenomena. Starting about a third of the way through the book, each chapter concludes with an "Exploration" that allows students to pursue an extended, open-ended project.

The "facelift" mentioned above is the addition of many new graphs and improvements in the appearance of the old ones. The book is nice to look at — even fairly easy to skim — because of these. Devaney's quirky humor pokes through occasionally. My favorite example is Figure 4.1 on page 63, which depicts the behavior of families of solutions to X' = AX depending on the trace and determinant of the matrix A. The caption reads, "The trace determinant plane. Any resemblance to any of the authors' faces is purely coincidental." Sure enough, when I glanced back at the figure, I saw a smiley face among the collected graphs. This visual mnemonic is cute enough that it's easy to remember; this picture alone is worth the $80 price tag, I think.

Annalisa Crannell's primary research is in topological dynamical systems, but she is also active in developing curricular materials for courses on "Mathematics and Art" as well as materials for writing across the curriculum. 

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