An Introduction to Difference Equations

This book is intended to be used as a textbook for a course on difference
equations at both the advanced undergraduate and beginning graduate
levels. It may also be used as a supplement for engineering courses on
discrete systems and control theory.
The main prerequisites for most of the material in this book are calculus
and linear algebra. However, some topics in later chapters may require some
rudiments of advanced calculus and complex analysis. Since many of the
chapters in the book are independent, the instructor has great flexibility in
choosing topics for a one-semester course.
This book presents the current state of affairs in many areas such as stability,
Z-transform, asymptoticity, oscillations, and control theory. However,
this book is by no means encyclopedic and does not contain many important
topics, such as numerical analysis, combinatorics, special functions
and orthogonal polynomials, boundary value problems, partial difference
equations, chaos theory, and fractals. The nonselection of these topics is
dictated not only by the limitations imposed by the elementary nature of
this book, but also by the research interest (or lack thereof) of the author.
Great efforts were made to present even the most difficult material in
an elementary format and to write in a style that makes the book accessible
to students with varying backgrounds and interests. One of the main
features of the book is the inclusion of a great number of applications in
economics, social sciences, biology, physics, engineering, neural networks,
etc. Moreover, this book contains a very extensive and carefully selected
set of exercises at the end of each section. The exercises form an integral
part of the text. They range from routine problems designed to build basic
skills to more challenging problems that produce deeper understanding
and build technique. The asterisked problems are the most challenging, and
the instructor may assign them as long-term projects. Another important
feature of the book is that it encourages students to make mathematical
discoveries through calculator/computer experimentation.



1
Dynamics of First-Order
Difference Equations
1.1 Introduction
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)

After this discussion one may conclude correctly that difference equations
and discrete dynamical systems represent two sides of the same coin.
For instance, when mathematicians talk about difference equations, they
usually refer to the analytic theory of the subject, and when they talk
about discrete dynamical systems, they generally refer to its geometrical
and topological aspects.

If the function f in (1.1.1) is replaced by a function g of two variables,
that is, g : Z+ ×R→ R, where Z+ is the set of nonnegative integers and R
is the set of real numbers, then we have
x(n + 1) = g(n, x(n)). (1.1.2)
Equation (1.1.2) is called nonautonomous or time-variant, whereas (1.1.1)
is called autonomous or time-invariant. The study of (1.1.2) is much more
complicated and does not lend itself to the discrete dynamical system
theory of first-order equations.

1.2 Linear First-Order Difference Equations
In this section we study the simplest special cases of (1.1.1) and (1.1.2),
namely, linear equations. A typical linear homogeneous first-order equation
is given by
x(n + 1) = a(n)x(n), x(n0) = x0, n ≥ n0 ≥ 0, (1.2.1)
and the associated nonhomogeneous equation is given by
y(n + 1) = a(n)y(n) + g(n), y(n0) = y0, n ≥ n0 ≥ 0, (1.2.2)
where in both equations it is assumed that a(n) = 0, and a(n) and g(n)
are real-valued functions defined for n ≥ n0 ≥ 0.

One may obtain the solution of (1.2.1) by a simple iteration:
x(n0 + 1) = a(n0)x(n0) = a(n0)x0,
x(n0 + 2) = a(n0 + 1)x(n0 + 1) = a(n0 + 1)a(n0)x0,
x(n0 + 3) = a(n0 + 2)x(n0 + 2) = a(n0 + 2)a(n0 + 1)a(n0)x0.
And, inductively, it is easy to see that
x(n) = x(n0 + n − n0))
= a(n − 1)a(n − 2) ··· a(n0)x0,
The unique solution of the nonhomogeneous (1.2.2) may be found as
follows:
y(n0 + 1) = a(n0)y0 + g(n0),
y(n0 + 2) = a(n0 + 1)y(n0 + 1) + g(n0 + 1)
= a(n0 + 1)a(n0)y0 + a(n0 + 1)g(n0) + g(n0 + 1).