This is a textbook on ordinary differential equations aimed at upper level undergraduates and beginning graduate students. It is roughly midway in difficulty between Hirsch and Smale’s book and Coddington and Levinson’s. Students are expected to have completed multivariable calculus and an introduction to analysis as prerequisites. Although it begins in a fairly traditional manner, the text gradually moves into a broader treatment of differential dynamical systems with an emphasis on understanding the qualitative behavior of these systems.
The core of the book is the first six chapters. The first chapter introduces ordinary differential equations and the notion of a model, and offers several standard examples. Next, the author gives a fairly complete treatment of linear systems, including a discussion of eigenvalue methods and an extensive discussion of the matrix exponential. The third chapter is a very nice self-contained treatment of existence and uniqueness based on the contraction mapping theorem. Dynamical systems are formally introduced in Chapter 4. This is very meaty material: after some preliminaries, the student is introduced to topological conjugacy and hyperbolic fixed points and sees a proof of the Hartman-Grobman theorem. This naturally leads into the chapter on invariant manifolds, which includes proofs of the local stable and center stable manifold theorems. Chapter 6 focuses on the phase plane; it develops Poincaré-Bendixson theory and methods for obtaining global phase portraits in the plane.
The remaining three chapters are advanced material and any of them could follow the core material depending on the class and the instructor’s interests. One chapter is a brief treatment of chaotic dynamics. The other two chapters are more extensive; one of them takes up bifurcation theory and the other Hamiltonian dynamics. Bifurcation theory is treated thoroughly and competently, but it’s clear that the author is especially fond of Hamiltonian dynamics.
There is a short appendix with basic MATLAB, Maple and Mathematica code designed to give students some experience with computation applied to dynamical systems. Exercises are plentiful throughout the book at various levels of difficulty; they are well-designed and include a good range of theoretical and computational challenges.
This is an appealing book all around: it is visually attractive with good illustrations, and the development is clean and easy to follow.
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.