The “Texts in Applied Mathematics” series addresses the needs of advanced undergraduates and beginning graduate students. This book is written to be used after one of the usual undergraduate differential equation courses, as an introduction to the advanced theory. The book succeeds in fulfilling this intention. The emphasis is on acquiring the skills needed to compute examples that give insight into the phenomena encountered in nonlinear dynamics. The tools required should be known to any undergraduate mathematics student and are used efficiently.

The first chapter is a short review of linear systems of differential equations, with emphasis on the skills needed to effectively treat the cases of multiple roots, including actually computing Jordan normal forms (no problems of numerical linear algebra are addressed here). The second chapter treats the basic theory of nonlinear systems, with an introduction to stability theory and a presentation of the basic topological notions. Most of the assertions in the first two chapters are proven.

The core of the book is the last two chapters, on the global theory of nonlinear systems (periodic orbits, limit cycles, Poincaré maps, Poincaré-Bendixson, Liénard equations, global phase portraits, and rudimentary index theory) and bifurcation (structural stability, bifurcations of several kinds, global behavior of one-parameter families, and homoclinic bifurcations). All these topics are illustrated by the detailed computation of many two- and three-dimensional examples; most of the theorems are only quoted and not proven. The treatment of the examples is good.

The one major criticism I have to make concerns the references. The bibliography contains a few historical papers and a number of recent advanced texts, but many advanced theorems quoted are left dangling without a proof or any references. In a book of this kind it should not be forbidden to quote recent research papers (even if they are quite unreadable compared to the classics of Poincaré and Bendixson). The book is produced according to the usual Springer standards, and I failed to detect any typos. The book contains many exercises, and almost all of them are demanding and require computation (with pencil and paper).