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Differential Equations: Theory and Applications

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This textbook is intended as a comprehensive introduction to ordinary differential equations for graduate students. It is designed to be flexible enough to suit the needs of three different kinds of courses: those with a theoretical focus, those that emphasize applications, and those that propose to mix theoretical and applied topics.

The feel of this book is distinctly different from traditional books at this level such as Coddington and Levinson’s Theory of Ordinary Differential Equations. The current book sees differential equations from the perspective of dynamical systems and thus a fair portion of the book is devoted to applications in Newtonian mechanics and Hamiltonian systems. In this respect it is similar in approach to Arnold’s Ordinary Differential Equations, although the pace is slower in the book under review and fewer topics are included.

The first two chapters introduce the basic notions of dynamical systems and provide examples of particular systems of differential equations (e.g., heat flow in a cube, ball in a hoop, pendulum, and two-body problem). Existence and uniqueness results are proved in Chapter 3. A long Chapter 4 takes up linear systems. Characteristic of this chapter, and indeed of the whole book, are a series of worked-out examples in two and three dimensions with a level of detail not common at this level.

Chapter 5 describes the critically important linearization technique for analyzing the behavior of a nonlinear system in the neighborhood of a hyperbolic fixed point. The proof of the Hartman-Grobman theorem that underlies that technique is provided in an appendix. This chapter also discusses the transformation of vector fields, beginning with polar and spherical coordinate transformations and then moving on to more general transformations. Chapter 6 addresses stability theory; it begins with linear systems and then uses the linearization theorems to extend the results to nonlinear systems. Lyapunov stability is introduced, primarily for later use in the chapters on mechanics and Hamiltonian systems. This chapter also includes a treatment of the stability of periodic orbits and the Poincaré map. Once again, there are several worked-in-detail examples.

The final three chapters apply the basic theory to integrable systems, Newtonian mechanics and Hamiltonian systems. The chapter on Newtonian mechanics proceeds from Newton’s second law and takes up central force problems as well as the two-, three- and n-body problems. The motion of a rigid body gets an extended treatment. After an introduction to Hamilton’s equations, conservation laws and the Poisson bracket, the chapter on Hamiltonian systems focuses on the statement and proof of three theorems. These are Arnold’s theorem (on the conditions under which the reduced phase space of a Hamiltonian system is diffeomorphic to an n-torus), Liouville’s theorem (that the volume form is preserved under a Hamiltonian flow), and Poincaré’s recurrence theorem.

This is a very readable text that is enhanced with good supporting figures. Especially striking are hand-drawn figures reproduced to look like blackboard sketches. The proofs throughout the book are particularly (and unusually) detailed. There are also well-designed exercises for every section in the text. This is one graduate-level graduate differential equations text that really would support self-study.

Bill Satzer ( 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.

Date Received: 
Tuesday, December 8, 2009
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David Betounes
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William J. Satzer

Introduction.- Techniques, Concepts and Examples.- Existence and Uniqueness:The Flow Map.-Linear Systems.- Linearization & Transformation.- Stability Theory.- Integrable Systems.- Newtonian Mechanics.-Hamiltonian Systems.- Appendix A: Elementary Analysis.- Appendix B: Lipschitz Maps and Linearization.- Appendix C: Linear Algebra.-Appendix D: Electronic Contents.

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Thursday, February 18, 2010