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Salinelli Ernesto - Discrete Dynamical Models

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Discrete Dynamical Models
Author:
Salinelli Ernesto / Tomarelli Franco
Publisher:
Springer International Publishing
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2014
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This book provides an introduction to the analysis of discrete dynamical systems. The content is presented by an unitary approach that blends the perspective of mathematical modeling together with the ones of several discipline as Mathematical Analysis, Linear Algebra, Numerical Analysis, Systems Theory and Probability. After a preliminary discussion of several models, the main tools for the study of linear and non-linear scalar dynamical systems are presented, paying particular attention to the stability analysis. Linear difference equations are studied in detail and an elementary introduction of Z and Discrete Fourier Transform is presented. A whole chapter is devoted to the study of bifurcations and chaotic dynamics. One-step vector-valued dynamical systems are the subject of three chapters, where the reader can find the applications to positive systems, Markov chains, networks and search engines. The book is addressed mainly to students in Mathematics, Engineering, Physics, Chemistry, Biology and Economics. The exposition is self-contained: some appendices present prerequisites, algorithms and suggestions for computer simulations. The analysis of several examples is enriched by the proposition of many related exercises of increasing difficulty; in the last chapter the detailed solution is given for most of them.

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Ernesto Salinelli and Franco Tomarelli UNITEXT Discrete Dynamical Models 10.1007/978-3-319-02291-8_1

Springer International Publishing Switzerland 2014

1. Recursive phenomena and difference equations

Ernesto Salinelli 1 and Franco Tomarelli 2

(1)

Dipartimento di Studi per lEconomia e lImpresa, Universit del Piemonte Orientale, Novara, Italy

(2)

Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

Abstract

In this first chapter we set the notations used in the text and present the main background motivations to study discrete mathematical models using examples. The systematic analysis of the latter is postponed to subsequent chapters.

The abstract definitions of the first paragraph provide a unified framework for the study of a wide class of problems, some examples of which are presented in the second paragraph.

1.1 1.1 Definitions and notation

In the sequel we will consider sequences of complex numbers. A sequence is a function F : , where = {0,1,2,3,}.

The reader not familiar with complex numbers can, at a first reading, replace with . For convenience the basic properties of complex numbers are recalled in Appendix B.

A generic sequence will be indicated with a capital letter, for example F, while its k -th term will be denoted with a capital letter with a subscript, F k . In other words F k = F ( k ).

Following a common practice, we will identify a sequence F with the ordered set of values it takes: F = {F k } k or, briefly F = {F k }.

Although the student interested in the description of variables in a succession of steps or temporal instants may find it restrictive to deal only with functions defined on the set of natural numbers, this is not the case. In fact, if t 0 is any initial instant and h > 0 is an increment, the sequence {t 0 , t 0 + h, t 0 + 2 h , t 0 + 3 h,} is mapped one-to-one to the sequence {0,1,2,3 ,} by the transformation

Since the object of our study are quantities that depend on natural numbers, it is not surprising that the induction principle, recalled below, turns out to be very useful for the analysis:

Let P (k) be a proposition about the natural number k. If:

there exists

such that

is true;

ii)

for all

, if P (k) is true, then P (k + 1) is true;

then the proposition P (k) is true for all For any subset I of in most cases I will be an interval in we set ie - photo 5

For any subset I of (in most cases I will be an interval in ), we set:

ie I n denotes the set of the n -tuples x 1 x 2 x n such that x j I - photo 6

i.e. I n denotes the set of the n -tuples ( x 1 , x 2 ,, x n ) such that x j I for all j = 1,2 , ,n .

Definition 1.1. We call difference equation of order n, the set of equations

(1.1)

where n is a positive integer and g is a given scalar function in n+2 variables, defined on I n +1

A difference equation of order n is said to be in normal form if it is expressed in the form

(1.2)

where is a given function

(1.3)

Formula is a recursive relationship that starting from the knowledge of the first k consecutive values of the sequence Y allows, with a lot of patience, to evaluate step by step all values of the sequence. In many cases, this calculation can be prohibitive even with automatic procedures. Hence, it is useful to know an explicit formula (say, a non-recursive one) for Y.

Definition 1.2. A solution of is a sequence X e xplicitly defined via

(1.4)

where f : I is a given function that renders an identity when f (k) replaces Y k , for any k .

The set of all solutions of is called general solution of the equation.

Theorem 1.3 (existence and uniqueness) . If is a function as in has solutions .

For each choice of the n-tuple ( 0, 1 ,, n 1) I n , the problem with initial data associated with the difference equation in normal form :

has a unique solution.

Proof. By substituting the initial conditions 0 , 1 , , n 1 in , ( 1 , 2 , , n 1 , n ) belongs to I n : substituting the conditions

in equation .

When it is not possible to find an explicit expression for the solution X, it is still useful to show whether the sequence X exhibits a particular property or not: periodicity, asymptotic behavior to a certain value, or even a more complex trend.

1.2 1.2 Examples

In this section we give motivations for the study of discrete mathematical models through the discussion of some examples. Their systematic analysis is deferred to later chapters.

The first examples are basic. They are presented in an increasing order of difficulty and they are grouped by subject. The last examples, not elementary and quite technical at first sight, are intended to illustrate the variety of situations in which difference equations play an important role in applied mathematics.

Example 1.4. A mobile phone call costs as connection charge and for each minute of conversation. The total cost of call C k +1 after k + 1 minutes of conversation is the solution of

which is an example of a problem with initial data for a difference equation of - photo 14

which is an example of a problem with initial data for a difference equation of the first order.

Example 1.5 (Simple interest). Suppose that the amount Y k +1 of a deposit of money calculated at time k+ 1 (the date for the computation of the interests) is given by the amount Y k of the preceding time plus the interest calculated on the basis of a constant rate r on the initial deposit D 0. The difference model describing the behavior of the deposit corresponds to the problem:

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