Discovering Discrete Dynamical Systems

This text is designed to support a student-led and discovery-based introduction to dynamical systems for advanced mathematics majors. It primarily addresses discrete dynamical systems.

The book comprises fourteen modules. Each of them begins with an exploration section that asks students to tackle an open-ended question with little or no guidance. Following the exploration is a short expository section that typically includes definitions and some theorems. Its main function is to provide some background information without explicitly answering the exploration question. The module concludes with a collection of exercises — usually ten or fewer — and a project that calls for a deeper investigation of a topic from the module.

The first module, for example, studies fixed points of dynamical systems. The exploration asks students to examine behavior of some logistic equations under iteration. In the expository section students learn about attracting and repelling fixed points, basins of attraction and some related concepts. They are also introduced to (or reminded of) the definition of the convergence of a sequence and the monotone convergence theorem for sequences. Then the exercises ask for examples, calculation of fixed points, application of the definitions in specific instances, and the like. The module’s project is then to explain a well-known card trick using the idea of a fixed point.

The first three modules are prerequisites for the remaining eleven. This gives the instructor a fair amount of flexibility in constructing the course. The authors expect each module to require three or four class periods to complete. If most modules could be completed in a week, there would probably be barely enough time in a semester to finish all fourteen. So selection and sequencing may be important.

After the basics (fixed points and cycles and their classifications) are covered in the first three modules, the other topics range broadly from fractal dimension, chaos and iterated function systems to conjugacy of dynamical systems, Sharkovskii’s theorem and symbolic dynamics.

The authors have clearly given a lot of thought to the structure of the book, the contents of the modules and their sequencing. They remind instructors of the importance of stepping back and letting students struggle on their own with the ideas and concepts. I would, however, have some concern about continuity of the course across modules; these are tightly focused and there is not much explicit connecting material. A natural role of the instructor would be to provide a unified overview without interfering with students’ investigations. It might also worthwhile for the instructor to spend a little time discussing the interplay between the discrete and the continuous. Discrete systems get the attention here, but there are occasions when their relation to continuous systems shouldn’t be neglected.

The authors are not specific about prerequisites, though they say that part of their goal is to “apply principles of real analysis to the study of dynamical systems”, and they aim the book at “advanced mathematics majors”. The level of maturity required to use the book is probably at the junior-senior level.

The book provides an attractive, carefully structured tool for exploring dynamical systems. It is well worth a look for anyone considering such as course.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.