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Ordinary Differential Equations: Introduction and Qualitative Theory

Jane Cronin
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Pure and Applied Mathematics 292
[Reviewed by
Henry Ricardo
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Cronin’s book, with a prerequisite of the first semester of advanced calculus and a semester of linear algebra, is aimed at advanced undergraduates and beginning graduate students — undergraduates would have to be rather mature mathematically to appreciate this book. It is a classic treatment of many of the topics an instructor would want in such a course, with particular emphasis on those aspects of the qualitative theory that are important for applications to mathematical biology. In general, the number of exercises at the end of the chapters is adequate, although the last two chapters have no exercise sets. There is a lengthy appendix on topics from real analysis and metric spaces, especially the concept of topological degree.

The book starts with a thorough treatment of the classical existence and uniqueness theorems, illustrating a variety of proof methods. The first chapter concludes with examples and exercises showing the application of these results to predator-prey equations (Volterra), the Hodgkin-Huxley equations, the Field-Noyes model for the Belousov-Zhabotinsky reaction, and the Goodwin equations for a chemical reaction system.

As the TOC indicates, most of the rest of the book deals with systems of differential equations (especially two-dimensional systems, with emphasis on the Bendixson theory) and stability results. The treatment is mathematically rigorous throughout. Some potential users of this book may be discouraged by the “brief and uneven” discussion of boundary value problems and the Sturm-Liouville theory. Another criticism is that, although Lyapunov functions are defined and various stability/instability criteria are proved, there is no explanation of how to construct such functions.

A nice feature of this edition is an extended and unified treatment of the perturbation problem for periodic solutions. The author demonstrates that use of the averaging method to study periodic solutions is a special case of a method of Poincaré.

The book concludes with seven and a half pages of references, mostly to the classic literature. I think it’s significant that there are only two books on this list with publication dates later than 2000 — and both deal with applications. The latest edition of the popular text by Boyce and DiPrima referenced is the 4th, although we are now up to the 9th edition.

Overall, this book is a solid graduate-level introduction to ordinary differential equations, especially for applications. It is reminiscent of the classic texts of Birkhoff and Rota and of Coddington and Levinson, rather than, say, the recently updated book by Hirsch, Smale, and Devaney.

Henry Ricardo ( has retired from Medgar Evers College (CUNY), but continues to serve as Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations (Second Edition). His linear algebra text was published in October 2009 by CRC Press.

Existence Theorems
What This Chapter Is About
Existence Theorem by Successive Approximations
Differentiability Theorem
Existence Theorem for Equation with a Parameter
Existence Theorem Proved by Using a Contraction Mapping
Existence Theorem without Uniqueness
Extension Theorems
Linear Systems
Existence Theorems for Linear Systems
Homogeneous Linear Equations: General Theory
Homogeneous Linear Equations with Constant Coefficients
Homogeneous Linear Equations with Periodic Coefficients: Floquet Theory
Inhomogeneous Linear Equations
Periodic Solutions of Linear Systems with Periodic Coefficients
Sturm–Liouville Theory
Autonomous Systems
General Properties of Solutions of Autonomous Systems
Orbits near an Equilibrium Point: The Two-Dimensional Case
Stability of an Equilibrium Point
Orbits near an Equilibrium Point of a Nonlinear System
The Poincaré–Bendixson Theorem
Application of the Poincaré–Bendixson Theorem
Definition of Stability
Stability of Solutions of Linear Systems
Stability of Solutions of Nonlinear Systems
Some Stability Theory for Autonomous Nonlinear Systems
Some Further Remarks Concerning Stability
The Lyapunov Second Method
Definition of Lyapunov Function
Theorems of the Lyapunov Second Method
Applications of the Second Method
Periodic Solutions
Periodic Solutions for Autonomous Systems
Stability of the Periodic Solutions
Sell’s Theorem
Periodic Solutions for Nonautonomous Systems
Perturbation Theory: The Poincaré Method
The Case in which the Unperturbed Equation Is Nonautonomous and Has an Isolated Periodic Solution
The Case in which the Unperturbed Equation Has a Family of Periodic Solutions: The Malkin–Roseau Theory
The Case in which the Unperturbed Equation Is Autonomous
Perturbation Theory: Autonomous Systems and Bifurcation Problems
Using the Averaging Method: An Introduction
Periodic Solutions
Almost Periodic Solutions
Ascoli’s Theorem
Principle of Contraction Mappings
The Weierstrass Preparation Theorem
Topological Degree
Exercises appear at the end of each chapter.