Differential Dynamical Systems by James D. Meiss is a comprehensive textbook on the theory of differential equations and dynamical systems. It combines a thorough presentation of relevant notions and theorems with a wealth of well-thought-out examples and exercises. Often, key concepts can most easily be understood by looking at the proper example. This book takes this approach very seriously. I also particularly enjoyed the care with which the proofs are presented, even for less easily proved statements. Another distinguishing feature is the abundance of diagrams and illustrations which immensely help in understanding the theory.

The book is divided into nine chapters plus an appendix on mathematical software. The first chapter serves as a motivation for the theory developed throughout the rest of the book by presenting prominent example systems: population dynamics, mechanical systems, electrical circuits, fluids, as well as the famous Lorenz system. Chapter 2 presents the classical linear theory, including relevant notions and statements from linear algebra. It closes with a brief treatment of nonautonomous linear systems. In Chapter 3, the solution theory of general nonlinear ordinary differential equations is formulated. Here the author recalls important mathematical concepts such as contraction mappings on general metric spaces. In Chapter 4, the author turns to the qualitative theory of dynamical systems by introducing basic notions like flow, linearization, hyperbolicity, stability, Lyapunov function, conjugacy or attractor. Beginning with Chapter 5, more advanced dynamical systems concepts are discussed. This begins with stable, unstable and center manifolds in this chapter. In Chapter 6, topology plays a prominent role and, correspondingly, ideas and theorems related to Poincaré are presented: basics of index theory, the Poincaré-Bendixson theorem and the Poincaré sphere. Chapter 7 makes a brief excursion into the realm of chaotic dynamical systems, discussing concepts such as Lyapunov exponents and ("strange") attractors. The book closes with two comprehensive chapters on bifurcation theory and Hamiltonian dynamical systems.

Differential Dynamical Systems should serve as an excellent foundation for dynamical systems courses at the advanced undergraduate or graduate level. The first chapters lay a solid foundation of the basic notions and tools of differential equations and dynamical systems theory, while the latter ones give a thorough treatment of more advanced concepts such as chaos, bifurcations and Hamiltonian systems. While a textbook of this type certainly cannot (and should not) touch upon every possible subject in the area, Chapter 7 could benefit from a few more pages on topics such as basic notions from ergodic theory. Also, the appendix on mathematical software and numerical experiments is just whets one's appetite. (Both of these remarks are probably triggered by my personal view of the subject.)

In summary, Meiss has put together an excellent textbook which does not compromise on mathematical depth or rigor and at the same time carefully guides the reader through the material. Many examples and exercises clarify the matter, and lovingly designed illustrations make the theory come alive. This book definitely has the potential to become a classic in the field.