Discrete dynamical models

398 Pages·2014·4.815 MB·English

by  Salinelli, Ernesto;Tomarelli, Franco

UNITEXT – La Matematica per il 3+2
Volume 76
Forfurthervolumes:
http://www.springer.com/series/5418
·
Ernesto Salinelli Franco Tomarelli
Discrete Dynamical Models
ErnestoSalinelli FrancoTomarelli
UniversitàdelPiemonteOrientale PolitecnicodiMilano
DipartimentodiStudiperl’Economiael’Impresa DipartimentodiMatematica
Novara,Italy Milano,Italy
UNITEXT–LaMatematicaperil3+2
ISSN2038-5722 ISSN2038-5757(electronic)
ISBN978-3-319-02290-1 ISBN978-3-319-02291-8(eBook)
DOI10.1007/978-3-319-02291-8
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013945789
TranslatedfromtheoriginalItalianedition:
ErnestoSalinelli,FrancoTomarelli:ModelliDinamiciDiscreti,3aedizione
©Springer-VerlagItalia2014
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Preface
Mathematical modeling plays a relevant role in many different fields: for in-
stance meteorology, population dynamics, demography, control of industrial
plants,financialmarkets, retirement funds. The few examplesjust mentioned
showhow, not onlythe expert inthe field butalso the commonman,in spite
of himself, has to take part in the debate concerning models and their impli-
cations, or even he is summoned to polls for the choice between alternative
solutions that may achieve some reasonable control of the complex systems
described by these models. So it is desirable that some basic knowledge of
mathematical modeling and related analysis is spread among citizens of the
“globalvillage”.
When models refer to time dependent phenomena, the modern terminol-
ogyspecifies the mathematical model with the notionof dynamical sys-
tem.
The word discrete, which appears in the title, points out the class of
models we deal with: starting from a finite number of snapshots, the model
allows to compute recursively the state of the system at a discrete set of
future times, e.g. the positive integer multiples of a fixed unit of time (the
time step), without the ambition of knowing the phenomenon under study
ateverysubsequent realvalueoftime,asinthethecaseof continuous-time
models.
Sometimes also the physical quantities under measure turn out to be
discrete (in this cases they are called “quantized”) due to experimental mea-
surement limits or to the aim of reducing the volume of data to be stored
or transmitted (signal compression). Since we focus our attention on models
where the quantization has no relevant effects, we willneglect it.
In recent years a special attention has been paid to nonlinear dynam-
ical systems. As a consequence, many new ideas and paradigms stemmed
from this field of investigation and allowed a deeper understanding of many
theoretical and applied problems. In order to understand the term nonlin-
ear, it is useful recalling the mathematical meaning of linear problem: a
problem is called linear ifthere is a direct proportionalitybetween the initial
vi Preface
conditionsandthe resultingeffects; informulae,summingtwoormore initial
dataleadstoaresponse ofthesystem whichisequaltothesumofthe related
effects.
In general the theory provides a complete description (or at least a sat-
isfactory numerical approximation) of solutions for linear problems, but the
relevant problems in engineering, physics, chemistry, biology and finance are
more adequately described by nonlinear models.
Unfortunately, much less is known about solutions ofnonlinear problems.
Often nonlinearity leads to qualitatively complex dynamics, instability and
sensitive dependence on initial conditions, to such an extent that some ex-
treme behavior is described by the term chaos.
Thisspellbindingandevocativebutsomehowmisleadingexpressionisused
to express the possible extreme complexity in the whole picture of the solu-
tions. This may happen even for the simplest nonlinear deterministic model,
as the iteration of a quadratic polynomial. So we are lead to an apparent
paradox: in such cases the sensitive dependence on initial conditions (whose
knowledge is always affected by experimental error) leads to very weak relia-
bilityof any long-term anticipation based on deterministic nonlinear models.
The phenomenon is well exemplified by weather forecast; nevertheless short-
term forecasting has become more and more affordable now, as everybody
knows.
These remarks do not call into question the importance andthe effective-
nessofnonlinearmodelsanalysis,buttheysimplyadvisenottoextrapolatein
the longterm the solutionscomputed by means of nonlinearmodels, because
of their intrinsic instability.
The present volume aims to provide a self-contained introduction to discrete
mathematicalmodelingtogetherwithadescriptionofbasictoolsfortheanal-
ysis of discrete dynamical systems.
The techniques for studying discrete dynamical models are scattered in
the literature of many disciplines: mathematics, engineering, biology,demog-
raphyandfinance.Here, startingbyexamplesandmotivationandthenfacing
the study of the models, we providea unitary approachby mergingthe mod-
elingviewpointwiththe perspective ofmathematicalanalysis,system theory,
linear algebra, probabilityand numerical analysis.
Severalqualitativetechniquestodealwithrecursivephenomenonareshown
andthe notionof explicit solutionis given(Chap. 1);the general solutions of
multi-step linear difference equations are deduced (Chap. 2); global and lo-
cal methods for the analysis of nonlinear systems are presented, by focussing
upontheissueofstability(Chap.3).Thelogisticdynamicsisstudiedindetail
(Chap. 4).
The more technical vector-valued case is studied in the last chapters, by
restricting the analysis to linear problems.
The theory is introduced step-by-step together with recommended ex-
ercises of increasing difficulty. Additional summary exercises are grouped in
Preface vii
specific sections. The worked solutions are available for most of them in a
closingchapter. Hintsandalgorithmsfornumerical simulationsare proposed.
We emphasize that discrete dynamical systems essentially consist in the
iterationofmaps,whilecomputers are very efficient inthe implementationof
iterative computations. The reader is invited to do the exercises of the text,
either by looking for a closed solution (if possible), or by finding qualitative
properties of the solution via graphical method and exploitation of the theo-
retic results provided by the text, or by numerical simulations.
Presently, manysoftware packages oriented to symbolic computationand
computer graphics are easily available; this allows to produce on the com-
puter screen those entangling and fascinating images which are called frac-
tals. It can be achieved by simple iterations of polynomials, even without
knowingthe underlying technicalities of Geometric Measure Theory, the dis-
cipline which describes and categorizes these non-elementary geometrical ob-
jects.
At any rate some basic notions for the study of these objects are given in
Chap. 4 and Appendix F: fractal dimension and fractal measure.
Forevery proposed exercise, the reader isasked toponder notonlyonthe
formal algebraic properties of the wanted solution, but also on the modeled
phenomenon and its meaning, on the parameter domain where the model
is meaningful, on the reliability of model predictions and on the numerical
computations starting from initial data which are unavoidably affected by
measurement errors.
In this perspective Mathematics can play as a feedback loop between the
study and the formulationof models. If an appropriate abstraction level and
arigorous formulationofthe model are achieved for one singleproblem, then
innovativeideas maydevelop and be applied to different problems, thanks to
the comprehension of the underlying general paradigms in the dynamics of
solutions. In short: understanding the whole by using the essential.
We thank Maurizio Grasselli, Stefano Mortola and Antonio Cianci for
their useful comments and remarks on the text and Irene Sabadini for the
careful editing of many figures. We wish to express our sincere acknowledge-
ments to Francesca Bonadei for her technical support, to Alberto Perversi
for his helpful graphic advises and to the students of the program Ingegneria
Matematica for their enthusiasm and valued comments. We wish to thank
Debora Sesana for her revising work on the figures. Last but not least, we
wouldlike to thank Simon Chiossi for his linguisticversion of the translation
of the Third Italian edition of this book.
Milano Ernesto Salinelli
October 2013 Franco Tomarelli
Reader’s Guide
Thisbookconsistsofeightchapters,thefirstsevenofwhichunfoldtheoretical
aspects of the subject, show many examples of applications and offer a lot of
exercises. Thelastchaptercollects theworkedsolutionsofmostexercises pro-
posed earlier. The text ends with a collection of appendices containing main
prerequisites and several summarizing tables.
Chapters 1and2are self-contained:the onlyprerequisite isthe familiarity
with elementary polynomial algebra.
Chapter 3 requires the knowledge of the notions of continuity and deriva-
tive for functions of one variable.
Chapter 4 relies on some basic topologicalnotions.
Chapsters 5, 6 and7study vector-valued dynamicalsystems bydeeply ex-
ploitingLinear Algebra techniques.
Allprerequisites, evenwhenelementary,arebrieflyrecalledinthetextand
can be found in the appendices.
Proofs, always printed in a smallertypographical font, are useful for afull
comprehension, but can be omitted at first reading.
Chapters 1, 2 and 3 may be taken as syllabus for an elementary teaching
modulus (half-semester course): in this perspective, the Sects. 2.5, 2.6, 2.7,
2.8,3.3,3.4 and 3.9 can be omitted.
Chapters4,5,6and7mayconstituteanotheradvancedteachingmodulus.
Allchapters, except the first one, are rather independent fromeach other,
so to meet the reader’s diversified interests. This allows for a big choice in
which path to follow when reading, reflecting the many applications of the
theory presented. A few possible “road maps” through the various chapters
are shown below:
Contents
Preface ........................................................ v
Reader’s Guide ................................................ ix
Contents....................................................... xi
Symbols and notations......................................... xv
1 Recursive phenomena and difference equations ............ 1
1.1 Definitions and notation.................................. 1
1.2 Examples............................................... 3
1.3 Graphical method ....................................... 18
1.4 Summary exercises ...................................... 20
2 Linear difference equations ................................ 25
2.1 One-step linear equations with constant coefficients .......... 25
2.2 Multi-step linear equations with constant coefficients......... 31
2.3 Stability of the equilibria of multi-step linear equations with
constant coefficients ..................................... 40
2.4 Finding particular solutions when the right-hand side is an
elementary function...................................... 47
2.5 The Z-transform ........................................ 50
2.6 Linear equations with non-constant coefficients.............. 61
2.7 Examples of one-step nonlinear equations which can be
turned into linear ones ................................... 65
2.8 The Discrete Fourier Transform ........................... 71
2.9 Fast Fourier Transform (FFT) Algorithm................... 76
2.10 Summary exercises ...................................... 81
3 Discrete dynamical systems:
one-step scalar equations .................................. 85
3.1 Preliminary definitions ................................... 85
xii Contents
3.2 Back to graphical analysis ................................ 93
3.3 Asymptotic analysis under monotonicityassumptions ........ 95
3.4 Contraction mapping Theorem ............................ 99
3.5 The concept of stability ..................................100
3.6 Stabilityconditions based on derivatives....................105
3.7 Fishing strategies........................................110
3.8 Qualitative analysis and stability of periodic orbits ..........114
3.9 Closed-form solutions of some nonlinear DDS ...............117
3.10 Summary exercises ......................................122
4 Complex behavior of nonlinear dynamical systems:
bifurcations and chaos .....................................125
4.1 Logistic growth dynamics.................................125
4.2 Sharkovskii’sTheorem ...................................129
4.3 Bifurcations of a one-parameter familyof DDS ..............134
4.4 Chaos and fractal sets....................................146
4.5 Topologicalconjugacy of discrete dynamicalsystems .........156
4.6 Newton’s method........................................160
4.7 Discrete dynamical systems in the complex plane ............163
4.8 Summary exercises ......................................177
5 Discrete dynamical systems:
one-step vector equations..................................179
5.1 Definitions and notation..................................179
5.2 Applications to genetics ..................................181
5.3 Stabilityof linear vector discrete dynamical systems .........191
5.4 Strictly positive matrices and the Perron–Frobenius Theorem .199
5.5 Applications to demography ..............................205
5.6 Affine vector-valued equations.............................208
5.7 Nonlinear vector discrete dynamical systems ................212
5.8 Numerical schemes for solving linear problems ..............215
5.9 Summary exercises ......................................225
6 Markov chains.............................................227
6.1 Examples, definitions and notations........................227
6.2 Asymptotic analysis of models described by absorbing
Markov chains ..........................................240
6.3 Random walks,duels and tennis matches ...................244
6.4 More on asymptotic analysis ..............................250
6.5 Summary exercises ......................................254
7 Positive matrices and graphs ..............................257
7.1 Irreducible matrices......................................257
7.2 Graphs and matrices.....................................266
7.3 More on Markov Chains..................................273