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Publisher:

Chapman & Hall/CRC

Publication Date:

2008

Number of Pages:

381

Format:

Hardcover

Edition:

3

Series:

Pure and Applied Mathematics 292

Price:

89.95

ISBN:

9780824723378

Category:

Textbook

[Reviewed by , on ]

Henry Ricardo

08/27/2010

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 4^{th}, although we are now up to the 9^{th} 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 (henry@mec.cuny.edu) 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.

**Prefaces**

**Introduction**

**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

Examples

**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**

Introduction

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

**Stability**

Introduction

Definition of Stability

Examples

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**

Introduction

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**

Introduction

**Using the Averaging Method: An Introduction**

Introduction

Periodic Solutions

Almost Periodic Solutions

**Appendix**

Ascoli’s Theorem

Principle of Contraction Mappings

The Weierstrass Preparation Theorem

Topological Degree

**References**

**Index**

*Exercises appear at the end of each chapter.*

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