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1999 | Buch | 2. Auflage

Dynamics of First-Order Difference Equations Kapitel lesen Erstes Kapitel lesen

An Introduction to Difference Equations

verfasst von: Saber N. Elaydi

Verlag: Springer New York

Buchreihe : Undergraduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

The second edition has greatly benefited from a sizable number of comments and suggestions I received from users of the book. I hope that I have corrected all the er­ rors and misprints in the book. Important revisions were made in Chapters I and 4. In Chapter I, we added two appendices (global stability and periodic solutions). In Chapter 4, we added a section on applications to mathematical biology. Influenced by a friendly and some not so friendly comments about Chapter 8 (previously Chapter 7: Asymptotic Behavior of Difference Equations), I rewrote the chapter with additional material on Birkhoff's theory. Also, due to popular demand, a new chapter (Chapter 9) under the title "Applications to Continued Fractions and Orthogonal Polynomials" has been added. This chapter gives a rather thorough presentation of continued fractions and orthogonal polynomials and their intimate connection to second-order difference equations. Chapter 8 (Oscillation Theory) has now become Chapter 7. Accordingly, the new revised suggestions for using the text are as follows. The diagram on p. viii shows the interdependence of the chapters The book may be used with considerable flexibility. For a one-semester course, one may choose one of the following options: (i) If you want a course that emphasizes stability and control, then you may select Chapters I, 2, 3, and parts of 4, 5, and 6. This is perhaps appropriate for a class populated by mathematics, physics, and engineering majors.

Inhaltsverzeichnis

Frontmatter
1. Dynamics of First-Order Difference Equations
Abstract
Difference equations usually describe the evolution of certain phenomena over the course of time. For example, if a certain population has discrete generations, the size of the (n + 1)st generation x(n + 1) is a function of the nth generation x(n). This relation expresses itself in the difference equation
$$x(n + 1) = f(x(n))$$
(1.1.1)
Saber N. Elaydi
2. Linear Difference Equations of Higher Order
Abstract
In this chapter we examine linear difference equations of high order, namely, those involving a single dependent variable.1 Such equations arise in almost every field of scientific inquiry, from population dynamics (the study of a single species) to economics (the study of a single commodity) to physics (the study of the motion of a single body). We will become acquainted with some of these applications in this chapter. We start this chapter by introducing some rudiments of difference calculus that are essential in the study of linear equations.
Saber N. Elaydi
3. Systems of Difference Equations
Abstract
In the last chapter we concerned ourselves with linear difference equations, namely, those equations with only one independent and one dependent variable. Since not every situation that we will encounter will be this simple, we must be prepared to deal with systems of more than one dependent variable.
Saber N. Elaydi
4. Stability Theory
Abstract
In Chapter 1 we studied the stability properties of first-order difference equations. In this chapter we will develop the theory for k-dimensional systems of first-order difference equations. As shown in Chapter 3, this study includes difference equations of any order. Here we are interested in the qualitative behavior of solutions without actually computing them. Realizing that most of the problems that arise in practice are nonlinear and mostly unsolvable, this investigation is of vital importance to scientists, engineers, and applied mathematicians.
Saber N. Elaydi
5. The Z-Transform Method
Abstract
In the last four chapters we used the so-called time domain analysis. In this approach we investigate difference equations as they are, that is, without transforming them into another domain. We either find solutions of the difference equations or provide information about their qualitative behavior.
Saber N. Elaydi
6. Control Theory
Abstract
In the last three decades, control theory has gained importance as a discipline for engineers, mathematicians, scientists, and other researchers. Examples of control problems include landing a vehicle on the moon, controlling the economy of a nation, manufacturing robots, and controlling the spread of an epidemic. Though a plethora of other books discuss continuous control theory [5, 56, 72], we will present here an introduction to discrete control theory.
Saber N. Elaydi
7. Oscillation Theory
Abstract
In previous chapters we were mainly interested in the asymptotic behavior of solutions of difference equations both scalar and nonscalar. In this chapter we will go beyond the question of stability and asymptoticity. Of particular interest is to know whether a solution x(n) oscillates around an equilibrium point x*, regardless of its asymptotic behavior. Since we may assume without loss of generality that x* = 0, the question that we will address here is whether solutions oscillate around zero or whether solutions are eventually positive or eventually negative.
Saber N. Elaydi
8. Asymptotic Behavior of Difference Equations
Abstract
In Chapters 4 and 5 we were mainly interested in stability questions. In other words, we wanted to know whether solutions of a difference equation converge to zero or to an equilibrium point. In asymptotic theory, we are concerned rather with obtaining asymptotic formulae for the manner in which solutions tend to zero or a constant. We begin this chapter by introducing the reader to the tools of the trade.
Saber N. Elaydi
9. Applications to Continued Fractions and Orthogonal Polynomials
Abstract
Continued fractions are intimately connected with second-order difference equations. Every continued fraction may be associated with a second-order difference equation; and conversely, every second-order difference equation may be derived from some continued fraction. The first point of view is useful for computing continued fractions, the second for computing the minimal solutions.
Saber N. Elaydi
Backmatter
Metadaten
Titel
An Introduction to Difference Equations
verfasst von
Saber N. Elaydi
Copyright-Jahr
1999
Verlag
Springer New York
Electronic ISBN
978-1-4757-3110-1
Print ISBN
978-1-4757-3112-5
DOI
https://doi.org/10.1007/978-1-4757-3110-1