There have been many books written about chaos and dynamical systems. Most of them have been at the advanced level, for graduate students and professionals in the field. Other books have been written for the layperson and, while descriptive, these less technical books lack the mathematics to really explain what chaos is all about. But now Richard Kautz has written an excellent text that is suitable for undergraduates and provides the mathematical detail necessary to give a thorough introduction to chaos. Chaos is a superb book that presents chaos and chaotic systems so that readers will understand the concepts and understand what is so fascinating about this phenomenon.
The book begins with a short introduction where we meet the Tilt-A-Whirl, an amusement park ride that, because of its unique design, takes a passenger through an undulating curve in a spinning car. The rider experiences unpredictable motion. This motion is, of course, chaotic; to the riders it looks as if what the car will do and how the passenger will be turned or pushed is simply unpredictable and, therefore, fun. Thus the reader meets chaos at an amusement park. Later, at the end of the book, Kautz details just how this amazing ride was designed and why it behaves chaotically. This is a good real-world example and the details are enough to help you both understand the ride and the mathematics that describe the ride's motion.
But chaos is more than a ride. Kautz gives the reader a gentle, yet thorough, introduction to the mathematics so that reader can simulate chaotic systems on his own. There are, for example, equations of motion for projectiles and planets. The equations are calculus-free — not differential equations, but rather finite difference equations. These equations give the reader a clear way to understand the phenomenon without knowing calculus. In most simulations, of course, difference equations are what one uses in research and so the reader can directly simulate the systems as any researcher might do.
The book introduces readers to equations of motion for projectiles, celestial mechanics, pendulum motion, and white noise. Throughout the text the writing is lucid and conversational. The reader not only sees the mathematics, but he is treated to short biographies of the scientists involved. For example, for celestial mechanics we meet Galileo Galilei, Tycho Brahe, and Johannes Kepler; for projectile motion we meet Isaac Newton, and for music and heat we meet Joseph Fourier. The biographies of the scientists, short though they are, add a human dimension to the text that is sometimes lacking in other books that provide the science alone.
Chaos comes with a CD-ROM so you can run some of the simulations without having to program the equations yourself. That's an added benefit and just one of the terrific features available for the reader.
If you are looking for a good introduction where the mathematics is well explained and the examples are plentiful, then I highly recommend this book.
David S. Mazel welcomes your feedback; he can be contacted at mazeld at gmail dot com.