When I was asked to teach differential equations a while ago, I asked my colleagues for recommendations of textbooks and references. One of them pointed me to Steven Strogatz’s Nonlinear Dynamics and Chaos. It was, she said, her “if only” book: it would have been her first choice for a textbook if our students had had an introduction to differential equations as part of their calculus sequence. Since that isn’t the case at Colby, she recommended a different textbook, but suggested consulting Strogatz for ideas and examples.
She was right: this is a readable and exciting book. I learned a lot from it.
Strogatz takes a thoroughly “dynamical” point of view, organizing his treatment by the dimension of the flows determined by differential equations. The first part, on one-dimensional flows, covers autonomous first-order differential equations, focusing from the beginning (as the title indicates) on nonlinear equations. Bifurcations show up already in chapter three, followed by a treatment of flows on the circle.
The two-dimensional theory occupies the next four chapters, which include a discussion of limit cycles and the Poincaré-Bendixon theorem. The final three chapters move on to chaotic dynamical systems, with chapters on the Lorenz equations, one-dimensional chaotic maps, and fractals.
This is not a theoretical book, as there are few proofs. It is also not a plug-and-chug book of recipes in the old style. Instead, it uses well-chosen examples to illustrate the main ideas. The exercises are excellent, and often quite difficult. Anyone who works through the book will emerge with a solid appreciation for the power of mathematics in science, and will also know quite a bit about the general theory of dynamical systems. Strogatz is very good at building the right intuitions and sharing the way the experts think about these things.
Strogatz assumes his readers know how to solve simple differential equations and know some elementary physics at the “force is mass times acceleration” level. If your students know what a differential equation is and know how to solve separable equations, they are probably ready for a course based on this book. If not, you probably won’t be able to use the book as a text, but you’ll still want to read it and have it on your desk. It is also a fantastic resource for independent study.
Nonlinear Dynamics and Chaos is an excellent book that effectively demonstrates the power and beauty of the theory of dynamical systems. Its readers will want to learn more.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He taught differential equations to engineering students in São Paulo in the 1980s and to Colby students in the 1990s and 2010s.