This is the second edition of a comprehensive text on dynamical systems and nonlinear ordinary differential equations. It consists of two parts that are largely independent. The first part treats systems of nonlinear ordinary equations using a variety of qualitative and geometric methods. A second part focuses on those aspects of dynamical systems associated with the iteration of a function. The book is addressed to advanced undergraduates or beginning graduate students in mathematics or related fields. It should be accessible to any reader with a background that includes single and multivariable calculus, linear algebra, and introductory differential equations.
The author’s goal is to introduce the primary concepts of dynamical systems and then to amplify those ideas using examples, methods of calculation, and applications. There is ample material to support quite a variety of courses with different flavors — focusing on the concepts, the applications, or the theoretical foundation and proofs. Each chapter follows a regular pattern. The author begins by introducing individual concepts, proceeds to applications, and concludes with a final section on theory that includes proofs of the more difficult results. The practice of postponing more difficult proofs has the advantage of not interrupting the development in the main text, but it may tempt students to avoid the proofs altogether.
The first part of the book establishes basic material for the geometric approach to ordinary differential equations. It includes discussions of phase portraits and flow, stability, periodic orbits and chaotic attractors. The chapter on periodic orbits is particularly good and representative of the author’s strengths in presentation — it smoothly integrates concepts and examples to help the student build good intuition. A chapter on chaotic attractors introduces the idea of sensitive dependence on initial conditions and discusses several examples including the Lorenz and Rössler attractors.
The book’s second part explores iterations of functions as dynamical systems. The author looks first at one-dimensional maps (specifically at their periodic points, itineraries and invariant sets), before turning to higher dimensional maps and examples such as the famous horseshoe. A final chapter is a look at fractals from a dynamical systems perspective.
This is an appealing and readable introduction to dynamical systems that would serve the needs of a variety of courses or support self-study. One might have wished for more attention to the connections between continuous and discrete systems, but the book is already very long as it is.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.