Book Review

An Introduction to Difference Equations. Second Edition by Saber N. Elaydi, New York: Springer—Verlag, 1999. ISBN 0-387-98830-0. $54.95. Gone are the days when difference equations arose mainly in the context of sections of flows or as finite difference approximations to PDE's. Today difference equations have come into their own, both as objects of intrinsic mathematical interest and as dynamical models in their own right. Discrete models form an important part of dynamical systems theory independently from their continuous cousins. In Saber Elaydi's book dynamicists have the long awaited discrete counterpart to standard textbooks such as Hirsch and Smale (“Differential Equations, Dynamical Systems, and Linear Algebra”). The first edition of this book appeared in 1996. The second edition includes substantial new material including appendices on global stability and periodic solutions, a section on applications to mathematical biology, and a new chapter entitled “Applications to Continued Fractions and Orthogonal Polynomials”. Additional material on Birkhoff's theory now appears in the chapter on asymptotic behavior. The initial chapter covers first order equations, including equilibria, cobwebbing, stability, cycles, and the bifurcations of the discrete logistic equation. Chapter 2 moves on to higher order linear equations and briefly treats the difference calculus (for an in—depth treatment, see “Difference Equations: Theory and Applications. Second Edition” by Ronald E. Mickens, New York: Van Nostrand Reinhold, 1990). The subsequent chapters include systems of difference equations, stability theory, Z—transforms, control theory, oscillation theory, asymptotic behavior, and applications to continued fractions and orthogonal polynomials.

The chapters are composed of short sections, each of which ends with a nice selection of exercises. Answers to the odd—numbered problems appear in the back of the book. The core chapters include sections of applications to various fields such as population biology, economics, and physics. Several famous examples and topics are treated in the applications, including Gambler's Ruin, the Nicholson—Bailey host/parasitoid model, the heat equation, and Markov chains. Many discrete models are noninvertible, yet as many frustrated modelers know, most of the old standard treatments of linearization and the Stable Manifold Theorem., coming as they do from the context of sections of flows, require invertibility. Commendably, Elaydi avoids the needless assumption of invertibility in his stability theorems, and also in the Stable Manifold Theorem. However, invertibility is assumed in the Hartman—Grobman Theorem, where indeed it is necessary to establish conjugacy between the map and its linearization (see “An Introduction to Structured Population Dynamics”, CBMS—NSF Regional Conference Series in Applied Mathematics, Vol. 71, SIAM, Philadelphia, 1998 by J. M. Gushing, for an example of a noninvertible map for which the conjugacy fails. Readers may be interested to know that in this reference a weaker version of the Hartman—Grobman Theorem is proved that does not require invertibility but does establish the desired correspondence between types of hyperbolic equilibria in maps and their linearizations.)

This book is in Springer's Undergraduate Texts in Mathematics series and is indeed a very readable and appropriate text for advanced undergraduates or beginning graduate students. According to the author, the main prerequisites for such a course are calculus and linear algebra, with basic advanced calculus and complex analysis needed only for some topics in the later chapters. This is true; however in most situations the book would be best appreciated by students with a bit more mathematical maturity than is engendered by today's calculus and beginning linear algebra courses.Elaydi's book is a valuable reference for anyone who models discrete systems. It is so well written and well designed, and the content is so interesting to me, that I had a difficult time putting it down. I am pleased to own a copy for reference purposes, and am looking forward to using it to teach a senior topics course in difference equations.