A Modern Introduction to Dynamical Systems

This text offers an introduction to the theory of dynamical systems for advanced undergraduates. It takes a fresh and somewhat unusual approach. While many books on the subject at this level focus on applications, this one treats it as a mathematical endeavor — one that is studied for the beauty of its structure. Applications are not ignored but are also not central.

The author’s view of the modern theory of dynamical systems is that it is “the study of the properties of one-parameter groups of transformations on a topological space, and what those transformations say about the properties of either the space or the acting group.” Translating this into material accessible to the intended audience of undergraduates in mathematics, engineering, and the natural sciences is something of a challenge.

The background expected for students includes vector calculus at least up to surface integration and the Gauss-Green-Stokes theorems, linear algebra, and differential equations including existence and uniqueness of first order-solutions and some analysis of nonlinear systems. The author wants this to be a proof-based course, but he realizes that some of the students might have little experience with proofs and mathematical writing, so he uses the content of the course to introduce it. No background in analysis or algebra is assumed either, but the author needs students to have a solid knowledge of general sequences in a space, the topological properties of a space, and the general idea of a group. So he introduces the ideas as needed — sometimes only as “here it is, and this is how it works”.

This results in a fairly demanding book that would probably have a lot of appeal to stronger students. It begins with the definition of a dynamical system and some quick sketches of examples (iterations of a function, symbolic dynamics and billiards). By the end of the book the author has developed enough background and sufficient tools to take up advanced topics related to dynamical complexity and topological entropy. The author develops both language and methods for investigating the dynamical content of a variety of dynamical systems, continuous and discrete. He chooses systems that exhibit many different phenomena, and he exploits these to show ways to classify and characterize behavior of the iterates of a map or solutions of an ordinary differential equation.

After the introductory sections there are seven chapters of roughly increasing complexity. The second chapter, on simple dynamics, already introduces one-parameter families of transformations of a phase space, contraction mappings and fixed points, interval maps and their bifurcations. Quite a start! The third chapter steps back to look more carefully at the idea of state space, its topology and metrics, some non-Euclidean state spaces, and then the Cantor set and some fractal structures.

The fourth chapter is the core of the book. It begins with a treatment of flows and maps in Euclidean space by reviewing linear first-order ordinary differential equations, and then quickly moves to linear planar maps and bifurcations of linear planar systems. The author then introduces local linearization for nonlinear systems and considers questions about the stability of equilibria. Stability of isolated periodic orbits and the Poincaré-Bendixson theorem come next, and then a nice application with the van der Pol oscillator. This chapter states without proof both the Hartman-Grobman theorem (the topological conjugacy near a hyperbolic critical point of a mapping and its linearization) as well as version of the Stable Manifold theorem. In the world of dynamical systems, we’re definitely getting into the real stuff.

Although the pace is not always fast, the author often tends to move very quickly through material that students will find difficult. Sometimes things appear without proof or even a reference (like the Stable Manifold theorem), but the author doesn’t say he won’t present the proof — he just moves on. This might well confuse students. Are these results so obvious they don’t need proof?

With all the topics and ideas that appear here, it might be easy for a student to get lost in the blitz. The author could probably do better helping the student understand the progression of topics, what is in scope and what is outside, and how all the pieces fit together.

The remaining chapters treat more complicated behavior beginning with recurrence (circle maps, Weyl’s theorem and equidistribution), then phase volume preservation (Newtonian mechanics and billiards), complicated orbit structure (leading to sensitive dependence on initial conditions and chaos), and finally topological conjugacy and topological entropy.

The book has over three hundred exercises integrated with the text. Very few are routine and several are pretty challenging. Overall this is a very attractive book, but an instructor using it as a text will likely have to provide some extra guidance.

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.