As the authors state in their preface, ``This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure of linear operators on finite-dimensional spaces.'' The authors begin in a unique way—through a thorough investigation of x′=ax on R1 and x′=Ax in Rn where A is diagonal. They then illustrate the power of elementary theorems of differential equations by deriving Kepler's laws of planetary motions from Newton's law, after an introductory discussion of conservative and central force fields. The next five chapters are devoted to the authors' presentation of linear algebra. Keeping their approach as coordinate-free as possible, their main goals in these chapters are the computation of eigenvectors, the real Jordan canonical form of an operator, the semisimple+nilpotent decomposition of an operator, exponentials of operators, and the general solution of x′=Ax in Rn. Midway through this part, they provide the reader with a well-conceived introduction to the topology of Rn by showing that such concepts as continuity, convergence, and boundedness are independent of the particular norm chosen. They also present clear phase portraits for all of the types of planar linear systems. After this complete discussion of linear (autonomous) systems, the authors cover the fundamental existence and uniqueness theorem for nonlinear systems in Chapter 8, followed by the usual results on continuity of solutions with respect to initial conditions, maximal interval of existence, and the one-parameter flow property for autonomous systems. In Chapter 9, they give a fairly complete discussion of the criteria for the stability of an equilibrium point, using Ljapunov's first and second methods. As an application of Ljapunov functions, they discuss gradient systems. The next five chapters intersperse theory with applications. In Chapter 11, the authors discuss limit sets of orbits, the Poincaré-Bendixson theorem, and its corollaries. In Chapter 13, they treat criteria for the stability of periodic orbits in Rn, using the Poincaré first-return map on a transverse section. In Chapters 10, 12 and 14, the authors present interesting and illuminating applications of dynamical systems. In Chapter 10, they analyze the differential equations for electric circuits, ending with a complete discussion of the Brayton-Moser gradient-like equations for very general networks. They carefully derive the phase portrait for the van der Pol equation and introduce the Hopf bifurcation. In Chapter 12, they discuss the differential equations of ecology from the solution of the simple logistic equation of one species, to the phase portrait of the Volterra-Lotka equations for the predator-prey equations, to the proof of the theorem that ``the populations of two competing species always tend to one of a finite number of limiting populations'', in a model with a few, very general assumptions. In Chapter 14, they continue their earlier discussion of classical mechanics and Hamiltonian differential equations. In the last two chapters, the authors treat the fundamental existence, uniqueness, and continuity theorem for nonautonomous systems, mainly as a step on the way to proving the differentiability of the solution of autonomous systems with respect to initial conditions. They also treat the persistence of hyperbolic rest points and periodic orbits under perturbations of the differential equation and give an introduction to the study of the structural stability of vector fields—an area where both authors have made significant contributions. This book is definitely designed to be a course textbook and not an encyclopedic compendium of the most general results. The authors limit their treatment to autonomous systems and the geometry of their flows, briefly mentioning nonautonomous systems only in Sections 5.5 and 15.1. As a result, they do not even mention areas which some mathematicians consider to be fundamental topics in the study of differential equations, e.g., Wronskians and Green's functions, periodic systems and Hill's equation, power series solutions and Bessel equations, computational methods such as the Euler and Runge-Kutta methods, solvable one-dimensional equations, such as exact, separable, and ``homogeneous'' equations, averaging methods, Sturm-Liouville comparison theorems and boundary value problems, Laplace transforms, and equations in the complex domain and on nonlinear spaces. Of the earlier books on differential equations, this book is closest to Pontrjagin's text in emphasis and development. It differs from Pontrjagin's book in that most of the sections are followed by short problem sets. Unfortunately, this book is designed for a course which does not exist at most universities, since most math departments teach their advanced linear algebra and their differential equations in separate courses. The reviewer has used this book a number of times with some success in differential equations courses aimed at junior honors math majors and at first or second year graduate students. Both groups of students have usually had a solid course in linear algebra and need only a brief review of Jordan canonical forms. Their major frustrations with this book have been the length and complexity of the authors' presentation of linear algebra and also the mistakes which appear in a number of homework problems and statements of theorems. They have been most enthusiastic about the clarity of the sections which deal with the fundamental theory and especially the four sections of applications. They have also praised the method with which the authors work their way up from the simplest equation x′=ax to linear real-diagonalizable systems, and then to a complete theory of general linear autonomous systems on Rn before beginning a discussion of the existence and uniqueness of solutions to nonlinear autonomous equations.
Reviewer: Simon, Carl P. [form MathSciNet]