An Introduction to Difference Equations

Reviews

• Reviewer: Michael Berg, The MMA Online book review column, August 2005

Saber Elaydi suggests in his Preface to the third edition of An Introduction to Difference Equations that this textbook for “advanced undergraduate and beginning graduate [students]” be used for a two-semester course, following one of four options as a function of taste and interest: stability theory, asymptotic theory, oscillation theory, or control theory. The book seems to fit the bill perfectly. It is very well-written and thorough in its coverage of topics (which are fascinating and numerous, notwithstanding the author’s insistence that he is not interested in encyclopedic coverage). Additionally the book is full of good exercises at all levels replete with hints and answers (although Elaydi even goes so far at to sprinkle in an occasional unsolved problem!) and is rich in good examples. It is impossible not to admire Elaydi’s achievement in putting together a textbook of such quality.

Difference equations are of course to some extent analogous to differential equations and this parallel is already evident in the familiar appearance of the general solution of a homogeneous linear difference equation with constant coefficients: just as with homogeneous linear ordinary differential equations (also with constant coefficients, of course) one immediately goes over to an associated characteristic polynomial whose roots are intimately involved in the manufacture of the according general solution. In this connection Elaydi presents e.g. the example of the difference equation realizing the Fibonacci numbers, the upshot being that, qua general solution, the general Fibonacci number is expressible in terms of powers of the golden section and its conjugate, the roots of one of the most famous characteristic polynomials of all.

Proceeding to the non-homogeneous case we encounter further parallels with ordinary differential equations: the methods of undetermined coefficients (which is actually misspelled as “undetermined coefficients” — an isolated singularity) and variation of parameters appear, soon to be followed by a study of the limiting behavior of solutions. Elaydi goes on to play with some nice examples, featuring an elegant treatment of the gambler’s ruin problem.

There is of course a thorough treatment of difference equations per se, culminating in a discussion of Markov chains. Then we encounter stability theory, including coverage of Lyapunovâ’s second method, and a discussion of Z-transforms and Volterra difference equations featuring a section titled, “The Z-transform Versus the Laplace Transform.”

After a chapter on oscillation theory, Elaydi hones in on the asymptotic behavior of difference equations in greater detail, with the focus falling on Poincaré’s Theorem and its extension by O. Perron. This is beautiful mathematics: even in the case of homogeneous linear difference equations with possibly non-constant coefficients, the general solution, x(n), generally has the property that the quotient x(n+1)/x(n) tends (as n → ∞) to a root of an associated (still “characteristic”) polynomial, with the case of constant coefficients a leitmotiv (as exemplified dramatically by, again, the Fibonacci numbers with F(n+1)/F(n) → λ, the golden section).

Subsequently the theory of continued fractions is given a systematic going-over and Elaydi’s treatment of the connection between continued fractions and infinite series is particularly evocative: it even includes some very nice material on the Riemann ζ-function (if k is an integer ≥ 2, ζ(k) itself is rendered as a continued fraction with wonderfully suggestive numerators and denominators).

Thus, An Introduction to Difference Equations is a terrific book almost every page of which contains marvelous things. It will serve all the pedagogical purposes Elaydi delineates, even though the wealth of material in the book will often tempt the reader to go off in tangent or orthogonal directions at the risk of destroying the pace of the coverage. But wherever one ends up, it will be a trip well worth taking.

• Reviewer: Shandelle M. Henson, Journal of Difference Equations and Applications Vol. 7, 2001

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 Cushing for an example of a noninvertible map for which the conjugacy fails). Readers may be interested to know that Cushing proved a weaker version of the Hartman-Grobman Theorem 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.

This is an electronic version of Shandelle Henson's own version and not the published version of this article published in the [Journal of Difference Equations and Applications] @ [2001] Copyright Taylor & Francis; [Journal of Difference Equations and Applications] is available online.

• Reviewer: Ronald E. Mickens, The American Mathematical Monthly 104(8), October 1997

The Newtonian revolution in physics led to the use of differential equations as the fundamental basis for the mathematical modeling of dynamical systems. Paradoxically, the great success of the paradigm also led to the widespread use of difference equations as discrete models of differential equations, primarily for the purposes of numerical integration of these equations. More recently, the easy access to digital computers provided a strong incentive for their users to have a knowledge of at least the elementary properties of difference equations. Modern books on the subject began to appear in the 1950's. One of the first and most popular of them was written by Goldgerg [3]. It gave a general introduction to the linear difference equations and applied them to the formulation and solution of problems in economics, psychology, and sociology. During the decade of the 1960's several books appeared for the mathematically more sophisticated reader, for example, those by Levy and Lessman [10], Brand [2], Hildebrand [4], and Miller [14].

The past ten years has seen the arrival of books that go beyond just giving an introduction to the various elementary properties of difference equations. Collectively [1, 5, 6, 7, 8, 9, 12, 13, 15], they consider the generalization to difference equations of many of the techniques usually associated with linear and nonlinear differential equations. They raise questions, and provide (partial) answers, to such questions as

• How can the asymptotic behavior of the solutions be determined?
• For difference equations with a nonlinear term multiplied by a “small” parameter, can perturbation methods be devised to provide uniform approximations to the solutions?
• For difference equations, what corresponds to the Hopf bifurcation theorem?
• Can “exact” difference equation models of differential equations be constructed?
• How can accurate numerical solutions be determined computationally for all the solutions of a given linear difference equation?

The recent blossoming of research in difference equations and their application to problems in nonlinear discrete dynamics led Gerry Ladas and Saber Elaydi to found the Journal of Difference Equations and Applications (published by Gordon and Breach), whose first issue was published in January 1995. A major reason for this new journal was the need for a place where researchers in difference equations could publish excellent papers on their results and have them read by colleagues in the field. Another factor was the lack of any mathematics journal that emphasized work in difference equations. In the past, researchers in this area usually published papers in the journals devoted mainly to differential equations or in journals dealing with specific discipline-based applications.

Much of the current interest in difference equations within the general scientific community had its genesis in a review paper written by Robert May in 1976 [11]. He did an analytical and computational investigation of the properties of the solutions to

(1)

xk+1 = λxk (1 - xk),   λ≥1.

This equation can be considered a discrete model for the growth of a single population using a finite-difference approximation to the so-called logistic differential equation

(2)

dy/dt = y (1 - y).

That is, if y_k is an approximation to y (t_k), where t_k = hk with Δt = h, then using a forward Euler approximation for the derivative give the scheme

(yk+1 - yk) / h = yk (1 - yk) .

The substitution

xx = (h / (1 + h)) yk,   λ = 1 + h,

leads to (1). Now for positive initial data, y(0) = y₀ < 0, for all the solutions of (2) go monotonically to the value y = 1. However, it is rather easy to demonstrate, as May [11] does, that (1) exhibits a variety of solution behaviors (periodic, chaotic, etc.) depending on the value selected for λ.

While (1) has a simple mathematical form, it turns out that many phenomena in the sciences can be modeled by simple difference equations whose solutions describe complex dynamical behavior [8, 9, 15]. A major advantage of discrete models for such systems is that, while the equations of motion cannot, in general be solved analytically in terms of elementary functions, the solutions can be determined easily with the aid of digital computers. The results can then be displayed in a variety of visual formats for study and analysis. These facts have led many colleges and universities to create courses on discrete dynamical systems and/or difference equations. One consequence of these new courses was the publication of several books that could be used as introductions to discrete dynamical systems and difference equations or to more advanced topics in the theory of difference equations. The book under review combines both of these features.

The author, Saber Elaydi, is Professor of Mathematics at Trinity Unversity in San Antonio, Texas. He has made important contributions in difference equations, especially in the areas of the discrete Levinsonżs theorem and the theory of Volterra difference equations. His book consists of eight chapters and is based on a course that he teaches at Trinity. The students in the course are upper-level undergraduates and come largely from mathematics and the physical and engineering sciences. The background required of the students is rather minimal, calculus and linear algebra, but certain topics from advanced calculus are needed for material near the end of the book.

The first three chapters introduce the reader to the fundamental concepts needed to understand both linear and nonlinear difference equations. In particular, the author does an excellent job in his presentation of the criteria for the asymptotic stability of fixed points (Chapter 1) and in its generalization in Chapter 4, where he discusses determination of stability by both linear approximation and Liapunov's second method.

Chapter 5 gives a thorough discussion of the Z-transform method, which serves exactly the same function for difference equations as the Laplace transform does for differential equations. In addition to showing how this technique can be used to solve linear difference equations the author applies it to the scalar case of Volterra difference equations of convolution type.

Chapter 6 is on control theory. The chapter considers only time-invariant (autonomous) discrete systems and covers the basic concepts needed for an introduction to this topic: controllability, observability , and stabilizability by feedback. Chapter 7 gives an excellent introductions to various techniques that can be applied to determine the asymptotic behavior of solutions to both linear and nonlinear difference equations. In addition to discussing the well-known theorems of Poincaré and Perron, the author includes some of his own recently discovered results in this area. The final Chapter 8 is on oscillation theory. Let a nontrivial solution to a difference equations be denoted by x_k. The solution is is said to be oscillatory (around zero) if for every positive integer N there exists k ≥ N such that x_kx_k+1 ≤ 0. If this is not the case, the solution is said to be non-oscillatory. The author presents an introduction to this topic and includes references where a more advanced treatment can be found.

There are several features that I especially like about the book. First, it contains a very extensive set of exercises at the end of each section. They are used not only as applications of the previously given theory but also in many cases, help the reader extend the theoretical discussion given earlier in the section. Second, many applications are given for a variety of disciplines. For example, the materials in Sections 3.5.1-3.5.3 on Markov chains are excellent. Third, the book also includes several programs written especially for the TI-85 calculator. This feature will certainly help the reader discover various interesting features of difference equations through actual experimentation with the calculator.

The book does contain some typos and misprints. However, the alert reader can locate them rather easily. (My copy of the book came with an errata sheet.) I found the book to be well-written and very suitable for a good solid introduction to the fundamentals of difference equations, some of their applications, and several related advanced topics.

REFERENCES

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1. Agaerwall, R. P. Difference Equations and Inequalities, Marcel Dekker, 1992.
2. Brand, L., Differential and Difference Equations, Wiley, 1966.
3. Goldberg, S., Introduciton to Difference Equations, Wiley, 1958.
4. Hildebrand, F. B., Finite-Difference Equations and Simulations, Prentice-Hall, 1968.
5. Jerri, A. J., Linear Difference Equations with Discrete Transform Methods, Kluwer, 1996.
6. Immink, G. K., Asymptotics of Analytic Difference Equations, Springer-Verlag, 1983.
7. Kelley, W. G. and A. C. Peterson, Difference Equations, Academic Press, 1991.
8. Kocic, V. L. and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order and Applictions, Kluwer, 1993.
9. Lakshmikantham, V. and D. Trigante, Theory of Difference Equations, Academic Press, 1988.
10. Levy, H. and F. Lessman, Finite Difference Equations, Macmillan, 1961.
11. May, R. M., Simple mathematical models with very complicated behavior, Nature 261 (1976) 459-467.
12. Mickens, R. E., Difference Equations, Van Nostrand Reinhold, 1987.
13. Mickens, R. E., Mickens, R. E., : Theory and Applications, Chapman and Hall, 1990.
14. Sandefur, J. E., Discrete Dynamical Systems, Oxford, 1990.
15. Wimp, J., Computation and Recurrence Relations, Pitman, 1984.

Department of Physics
Clark Atlanta
Atlanta, GA 30314
rohrs@math.gatech.edu

• Reviewer: Martin Bohner, Amazon.com, August 2000

One of the Very Best Introductions on Difference Equations.

Among the few introductory texts to difference equations this book is one of the very best ones. It has many features that the other texts don't have, e.g., stability theory, the Z-transform method (including a study of Volterra systems), and asymptotic behavior of solutions of difference equations (including Levinson's lemma) are studied extensively. It also contains very nice examples that primarily arise in applications in a variety of disciplines, including neural networks, feedback control, biology, Markov chains, economics, and heat transfer. Unfortunately, among differential and difference equations, students usually only learn about differential equations, although many problems they encounter later can be solved using difference equations. Hence it is very useful for students to study difference equations (or even, dynamic equations, which can accomodate differential and difference equations at the same time), and for such an undergraduate or beginning graduate course, Saber's book can be used as a textbook. It is particularly useful for such a purpose since it contains many exercises and also many programs for the calculator TI-85. Although the first edition did contain several misprints, the second edition has eliminated most of them. Saber's superb writing style makes reading this book very enjoyable.