There is a significant and (I hope) growing genre in the physics literature, that of the informative introductory book. Such a book lands in the valley between two extremes of scientific exposition. On the one hand, a popular book can often introduce a nonexpert reader to a subject, with a focus on the big ideas and the history of a field and the personalities that shaped it. Such a book can, however, be frustrating owing to its lack of detail and specific results about its field. On the other hand, a standard textbook can be exhaustive in detail, but omit the background thinking that gives the subject life and meaning.
Feldman's Chaos and Dynamical Systems deliberately and deftly navigates the gap between these extremes. The book presents heady mathematical and physical ideas, occasionally bordering on philosophy, yet developed in meaningful detail so that the reader knows what is being talked about. The particular genius of Feldman's pedagogy is a well-chosen selection of just a few examples that are explored in detail, so that the big ideas are not arbitrarily other-worldly, but arise in a very natural way.
Feldman begins by defining a dynamical system as “a mathematical system that changes in time according to a well-specified rule.” He illustrates this concept in two versions, iterated functions and ordinary differential equations. The centerpiece of the early part of the book is the logistic function, f(x) = rx(1-x), which he takes care to introduce as a very simple model of population dynamics. Feldman iterates this function to produce the time series x{n + 1} = f(xn), each iteration xn representing the population of a generation. He takes care to present this in detail and to show examples of the iteration in many displayed figures. (A minor quibble: In some of these figures, the reader is asked to distinguish between black and grey markers that are not always obviously distinguishable.) More, he encourages readers to explore this iteration on their own, emphasizing that there is, in principle, nothing mysterious here.
And yet. Repeated iteration of the logistic equation yields a set of values that converge to a certain limit. Or, depending on the value of r, a set of values that alternate between two limits, or four, or perhaps values that fluctuate wildly. Guiding us along the way, and with nothing up his sleeve, Feldman shows us chaos arising within this simple system, enabling him to introduce the butterfly effect and to meditate on the apparent randomness of a well-defined dynamical system. Sensitive dependence on initial conditions puts the lie to Laplace's famous maxim that, given the dynamical laws and the initial conditions of everything, all future motion in the universe could be predicted. Not so, says chaos. There is a kind of “philosophy of the practical” at work in Feldman's exposition. He cares not just about the results, but what to make of the results.
The other thread of Feldman's approach is ordinary differential equations as a dynamical system. Here, he begins with a discussion that should be eye-opening and empowering for students. Having been taught for years to find solutions to differential equations, students in Chapter 2 are instead invited to think about how the solutions might work. Is the time derivative positive or negative, and under what circumstances? What might the long-time behavior be? Feldman adapts the logistic equation into a differential equation, and then adds the idea of a “harvest,” representing, for example, the reduction in the population of fish in lake due to fishing. The equation is able to see a stable population go suddenly unstable under a small change of the harvest rate. This leads to ideas of bifurcations and universal features that seem to apply to many chaotic systems, regardless of their physical origin. Feldman is again teasing big ideas from the simplicity of his starting point. At this point, it would have been great to see data from some particular population of animals (or any relevant system) where some of the striking ideas play out, but Feldman does not take this opportunity to do so.
The greatest strength of this book is the logic and clarity of the discussion. Each topic, each idea, is developed and explained. At each step, we know where Feldman's discussion is going to go, and why, and how he gets there. Because everything makes sense along the way, we are able to appreciate the final big idea, that of strange attractors, in the final two chapters. He presents a chaotic differential equation, whose solutions are fairly regular and trace nearly predictable orbits in phase space (“attractors”), while yet chaotically fluctuating over part of the orbit on each pass (“strange!”). Feldman points out that the existence of strange attractors implies that a complex, nonlinear system, such as the climate, can have an overall predictable trend, whereas fluctuations like the day-to-day weather can still appear unpredictable. For this lesson alone, the book should be required reading in science for non-scientist classes, or perhaps for congressional staff.
A secondary strength is that Feldman places the specific results in a broader context. He spends a delightful Chapter 3 discussing the role of mathematical models. Again, students are typically taught to develop everything in detail from first principles in physics classes, but the world encourages us to simplify things in a meaningful way, to find and model those things that really matter. Feldman also, throughout the book, raises issues such as catastrophe theory or emergence, without the space or time to really develop them. But here he steps in with a large set of pointers to additional reading, with some discussion of what one can expect to get from each reference. He is a useful guide through the subject, even as he parts ways with us to let us continue our own journey.
In the preface of his magisterial work The Variational Principles of Mechanics, Cornelius Lanczos bemoans the authors of typical mathematical physics texts who write “as though to impress the reader with the uncomfortable feeling that he is in the permanent presence of a superman.” In contrast, Professor Feldman comes down to our level and lends a human voice to his prose, sharing the excitement and fun of seeing unexpected complexity and beauty arise from seemingly simple beginnings.
John L. Bohn is a Research Professor in the Department of Physics at the University of Colorado and a Fellow of JILA. His research is in the area of ultracold atoms and molecules. He is the author of A Student's Guide to Analytical Mechanics and is happy to plug that book here.
