(look all the way to the bottom of the page
for older books that are listed here for the first time). For the
time being, only books that have appeared since 1990 are listed
here, though this could change later if demand warrants.The Books Corner
The importance of knowing about new publications in a timely manner needs no stressing. This page is devoted to reviews, summaries and/or outlines of recent texts and monographs. The on-line features include links to publishers' websites for detailed examinations of many titles and for ordering copies from the publishers, if desired. Publishers may update or change specific book links periodically, so if a particular book link does not work, try the publisher's link at the bottom of this page, and then look for the book information on the publisher's own website. Sometimes, you may find a good bargain offered by one of the booksellers listed below. For the budget conscious, it could pay to check them before ordering.
For convenience, new listings (though not necessarily
new books) since the last update of this page are indicated by
the image (look all the way to the bottom of the page
for older books that are listed here for the first time). For the
time being, only books that have appeared since 1990 are listed
here, though this could change later if demand warrants.
Information and links about new books on (or containing a significant treatment and/or applications of) difference equations and discrete dynamical systems should be forwarded to the editor.
Books in 2003:
Nonlinear
Difference Equations: Theory with Applications to Social Science
Models, by H. Sedaghat, Kluwer
This book provides rigorous treatments of models from various
social science disciplines and it also contains detailed
expositions of some the latest theory behind the mathematical
analysis. Topics covered in depth include stability and chaos in
dimensions one and greater, the coexistence and stability of
cycles in one dimension, the use of semiconjugacy as a means of
bridging the wide gap between dimension 1 and the higher
dimensions, weak contractions and expansions, global attractivity
and more. Many theoretical results are new and some are presented
for the first time in this book; endnotes and an extensive
bibliography relate the various topics discussed to the existing
literature. The book builds a comprehensive theoretical framework
for the study of nonlinear models in discrete time (many with no
continuous-time analogs). This aspect of the book makes it a
useful resource for graduate students as well as a reference for
researchers. To see the preface, sample chapters and other
details go to the Internet home of the
book.
Partial
Difference Equations, by S.S. Cheng, Taylor and Francis
Partial difference equations are a major class of functional relations with recursive structures so that the usual concepts of increments are important. This book describes mathematical methods that help in dealing with recurrence relations governing the behavior of variables such as population size and stock price and will be useful to anyone who has mastered the usual sophomore mathematical concepts. It offers a concise introduction to the tools and techniques that have proven successful in obtaining results in partial difference equations. For a table of contents and more information, click here.
Chaos: A
Mathematical Introduction, by J. Banks, V. Dragan, A. Jones,
Cambridge (May 03)
This is an introductory book on chaos in the discrete-time setting that may be accessible to people who have taken a first course in undergraduate calculus. The book evolved from a one-semester middle level undergraduate course over a period of several years and has therefore been well class-tested. For details and table of contents, click here.
A First Course
in Dynamics, by B. Hasselblat and A. Katok, Cambridge (June
03)
This introductory text covers topological and probabilistic
notions in dynamics. The authors use a progression of examples to
present the concepts and tools for describing asymptotic behavior
in dynamical systems, gradually increasing the level of
complexity. Subjects include contractions, logistic maps,
equidistribution, symbolic dynamics, mechanics, hyperbolic
dynamics, strange attractors, twist maps, and KAM-theory. Click here
for a table of contents and more details.
Books in 2002:
Stochastic
Finance: An Introduction in Discrete Time, by H. Follmer and
A. Schied, de Gruyter
This book presents an introduction to financial mathematics for
mathematicians. In contrast to many textbooks on mathematical
finance, only discrete-time stochastic models are considered.
This has the advantage that the text can concentrate from the
beginning on typical problems which are suggested by financial
applications. It is intended both for graduate students with a
certain background in probability theory as well as for
professional mathematicians in industry and academia. Moreover,
certain principles, such as the general incompleteness of
realistic market models, become thus more transparent and
visible. On the other hand, all models are based on general
probability spaces, and so the text captures the interplay
between probability theory and functional analysis which is
typical for modern mathematical finance. For more information,
click here.
The Handbook of
Brian Theory and Neural Networks, (2nd ed.) Ed. by M.A.
Arbib, MIT Press
This book contains a set of almost 300 articles covering topics
in brain theory and neural networks. Some of these articles seem
to be of potential interest to discrete dynamics folks. Part I
provides general background on brain modeling and on both
biological and artificial neural networks. Part II consists of
"Road Maps" to help readers steer through articles in
part III on specific topics of interest. The articles in part III
are written so as to be accessible to readers of diverse
backgrounds. They are cross-referenced and provide lists of
pointers to Road Maps, background material, and related reading.
The second edition increases the coverage of models of
fundamental neurobiology, cognitive neuroscience, and neural
network approaches to language and contains 287 articles,
compared to the 266 in the first edition. Articles on topics from
the first edition have been updated by the original authors or
written anew by new authors, and there are 106 articles on new
topics. Click here
for more details.
A New Kind of
Science, by Stephen Wolfram, Wolfram Media Inc.
This book uses the theory of cellular automata and various issues
related to their use in computing to advance a thesis (the
"principle of computational equivalence"). The book
aims broadly and its style is informal and philosophical with a
large number of helpful diagrams and pictures. However, the prose
spread over 800 pages of the main text is repetitive and
inefficient, so mathematicians interested in cellular automata,
their history and their relationship and applications to other
mathematical and scientific disciplines may find the end-notes
(which are substantial) of greater interest. This book comes up
oddly short on its underlying core, the theory of discrete
dynamical systems of which cellular automata are examples; see my
commentary (and references to web
resources on CA). For more details on the book, click here to go to its home
on the web.
Introduction to
Dynamical Systems, by M. Brin and G. Stuck, Cambridge
An introductory book, with topics in topological dynamics,
symbolic dynamics, ergodic theory, hyperbolic dynamics,
one-dimensional dynamics, complex dynamics, and measure-theoretic
entropy. Some applications to areas such as number theory, data
storage, and Internet search engines are included. For more
details, click here.
Advances in
Dynamic Equations on Time Scales, edited by M. Bohner and A.
Peterson, Springer-Verlag
A follow-up to the "Dynamic Equations on Time Scales: An
Introduction with Applications" (see the 2001 listing below)
in the new area of dynamic equations on time scales. The basic
concept is to express dynamic systems on time scales ranging from
the discrete (as in difference equations) to the continuous (as
in differential equations) as a way of unifying and extending
continuous and discrete analysis. For more details and a
table of contents, click here.
Chaos in
Ecology: Experimental Nonlinear Dynamics, by J. M. Cushing,
R. F. Costantino, B. Dennis, R. A. Desharnais, and S. M. Henson,
Academic Press
This book describes a decade long interdisciplinary project
investigating nonlinear (and, in particular, chaotic) dynamics in
a biological population. The project is an application of
discrete dynamical systems defined by a system of difference
equations. The book is the first in a new Series on Theoretical
Ecology to be published by Academic Press. For more details,
click here.
Theory of
Difference Equations: Numerical Methods
and Applications (Second Ed.) by V. Lakshmikantam and D.
Trigiante, Marcel-Dekker
This is a new, reworked and expanded edition of the familiar
1980's text. It explores classical problems such as orthogonal
polynomials, the Euclidean algorithm, roots of polynomials, and
well conditioning. For more information and a table of contents,
click here.
Discrete
Dynamical Systems and Difference Equations with Mathematica,
By M.R.S. Kulenovic and O. Merino, CRC Press;
While presenting the essential theoretical concepts and results,
the emphasis in this work is on using the software
"Dynamica". The authors present two sets of Dynamica
sessions: one that serves as a tutorial of the different
techniques, the other features case studies of well-known
difference equations. Dynamica and notebooks corresponding to
particular chapters are available for download from the Internet.
For more details, click here.
Managing
emergent phenomena: Nonlinear dynamics in work organizations,
by S.J. Guastello, Lawrence-Erlbaum Associates;
This book discusses applications of nonlinear dynamics to
organizational behavior and group dynamics. Basic concepts such
as chaos, fractals, bifurcations, catastrophies, etc. are
discussed together with their applications to diverse areas such
as organizational change and development, motivation theory,
economics, social networks and creativity.
Gaussian
Self-Affinity and Fractals, by B. Mandelbrot, Springer;
This is the third volume of Mandelbrot's Selected Works, focusing
on a detailed study of fractional Brownian motions. The fractal
themes of "self-affinity" and "globality" are
presented. Introductory material, written especially for this
book, precedes the papers and presents a number of new
observations and conjectures. For more, click this link.
Books in 2001:
Dynamic Asset
Pricing Theory, by. D. Duffie, Princeton Univ. Press;
The asset pricing results in this book are based on the three
increasingly restrictive assumptions: absence of arbitrage,
single-agent optimality, and equilibrium. These results are
unified with two key concepts, state prices and martingales.
Technicalities are given relatively little emphasis, so as to
draw connections between these concepts and to make plain the
similarities between discrete and continuous-time models. While
much of the continuous-time portion of the theory is based on
Brownian motion, this third edition introduces jumps--for
example, those associated with Poisson arrivals--in order to
accommodate surprise events such as bond defaults. Applications
include term-structure models, derivative valuation, and hedging
methods. Numerical methods covered include Monte Carlo simulation
and finite-difference solutions for partial differential
equations. For more information and a table of contents, click
this link.
Inequalities
for Finite Difference Equations, by B.G. Pachpatte,
Marcel-Dekker;
A reference-style book, containing linear and nonlinear
difference inequalities, with applications to various types of
finite difference and sum—difference equations. Focuses on
stability of finite difference systems, and considers
inequalities involving iterated sums. Click here
for more inforamtion and the table of contents.
Qualitative
Methods in Nonlinear Dynamics: Novel Approaches to Liapunov's
Matrix Functions, edited by A.A. Martynyuk, Marcel-Dekker;
This book focuses on exponential polystability of separable
motions as well as integral and Lipschitz stabilities, and
considers problems of dynamics of nonlinear systems in the
presence of impulsive perturbations. Click this link
for more information and a table of contents.
Dynamics of
Second Order Rational Difference equations with Open Problems and
Conjectures, by M.R.S. Kulenovic and G. Ladas, CRC;
This self-contained monograph provides a systematic analysis of a
class of second-order rational difference equations whose maps
consist of a ratio of two linear functions. After classifying the
various special cases of these equations and introducing some
preliminary results, the authors investigate each equation for
semicycles, invariant intervals, boundedness, periodicity, and
global stability. For more information and the table of contents,
click here.
Dynamic
Equations on Time Scales: An Introduction with Applications, by
M. Bohner and A.C. Peterson, Birkhauser;
The study of dynamic equations on a measure chain (time scale)
goes back to its founder S. Hilger (1988), and is a new area of
still fairly theoretical exploration in mathematics. Requiring
only a first semester of calculus and linear algebra, Dynamic
Equations on Time Scales may be considered as an interesting
approach to differential equations via exposure to continuous and
discrete analysis. For more information and the table of
contents, click this link.
Laminations and
Foliations in Dynamics, Geometry and Topology, by M.
Lyubich, J. Milnor, Y. Minsky (editors), AMS;
This volume is based on a conference held at SUNY, Stony Brook
(NY). The concepts of laminations and foliations appear in a
diverse number of fields, such as topology, geometry, analytic
differential equations, holomorphic dynamics, and renormalization
theory. For list of contributions and more information, click here.
Stability and
Complexity in Model Ecosystems, By R.M. May, Princeton
University Press;
A paperback reprisal of the influential 1973 classic, with a new
introduction by May. For more information and a table of
contents, click this link.
The Theory of
Difference Schemes, by A.A. Samarskii, Marcel-Dekker;
Illustrates how to solve boundary problems with a unique
methodical approach emphasizing the application and creation of
difference schemes. For more information, click here.
Qualitative
Theory of Dynamical Systems, 2nd ed, A.N. Michel, K. Wang
and B. Hu, Marcel Dekker;
Includes a discussion of the Lyapunov and Lagrange stability
theory for a general class of dynamical systems in a metric space
independently of equations, inequalities, or inclusions; applies
the general theory to specific classes of equations; and presents
new and expanded material on the stability analysis of hybrid
dynamical systems and dynamical systems with discontinuous
dynamics. Second Edition adds several case studies and specific
examples; for more information and a table of contents, click
this link.
Topics in
Functional Differential and Difference Equations, T. Faria
and P. Freitas, Editors, AMS;
This book contains papers written by participants at the
Conference on Functional Differential and Difference Equations
held at the Instituto Superior Técnico in Lisbon, Portugal. The
authors work in a wide range of topics, including qualitative
properties of solutions, bifurcation and stability theory,
oscillatory behavior, control theory and feedback systems,
biological models, state-dependent delay equations, Lyapunov
methods, etc. so the book may be of interest to both theoretical
and applied mathematical scientists. Click this AMS
link for a list of authors and articles.
Difference
Equations: An Introduction with Applications (2nd ed.) by A.
Peterson and W. Kelley, Academic Press;
Some of the techniques discussed in this introductory book are
summation methods, generating functions, z-transforms, theory of
linear equations, matrix methods, stability, chaos, asymptotic
methods, Green's functions, finite Fourier analysis, variational
methods, fixed point theorems, and connections with differential
equations. Applications of difference equations to combinatorics,
geometry, epidemiology, special functions, economics, population
biology, numerical analysis, circuit analysis, differential
equations, and other fields have been included. For more
information, click this link.
Dynamical
Systems with Applications using Maple, by S. Lynch,
Birkhauser;
Introductory differential equations as well as real and complex
maps studied with the aid of the software Maple. For detailed
information, go to the book's website.
Books in 2000:
Difference
Equations and Inequalities, 2nd ed., by Ravi P. Agarwal,
Marcel-Dekker;
In this new, revised and expanded edition of the sizable 1992
monograph, one finds 25% all new material, including new
problems, additional references, and a new chapter on the
qualitative properties of solutions of neutral difference
equations. A good resource containing discussions of over 4000
difference equations and inequalities. For more information and a
table of contents, click here.
Single-Orbit
Dynamics, by B. Weiss, AMS;
An approach to dynamical systems based on "the dynamical
study of single orbits as opposed to the global study of the
system as a whole". Many common ideas of measure-theoretical
dynamics are discussed from this vantage point, including
ergodicity and entropy. For more, click this AMS
link.
Measure-Preserving
Homeomorphisms, by S. Alpern and V.S. Prasad, Cambridge
University Press;
An introduction to typical properties of volume preserving
homeomorphisms, examples of which include transitivity, chaos and
ergodicity. The first part of the book is more concrete, focusing
on volume preserving homeomorphisms of the unit n-dimensional
cube. Also included are the fixed point theorems of Conley,
Zehnder, Franks. For additional information, click this link.
Topics in
Symbolic Dynamics and Applications, Ed.s F. Blanchard, et
al. Cambridge University Press;
This book is devoted to recent developments in symbolic dynamics,
and it comprises eight chapters contributed by various authors.
These include the study of symbolic sequences of "low
complexity" and of "high complexity" systems,
results on asymptotic laws for the random times of occurrence of
rare events, diophantine problems and combinatorial Ramsey
theory, dynamics of symbolic systems arising from numeration
systems, and the symbolic dynamics of Lorenz maps. For contents
and more information, click this CUP
link.
Holomorphic
Dynamics, by S. Morosawa, et al. Cambridge University Press;
The mathematical treatment emphasizes the substantial role of
classical complex analysis in understanding holomorphic dynamics
and offers up-to-date coverage of the modern theory. The authors
cover entire functions, Kleinian groups and polynomial
automorphisms of several complex variables such as complex Hénon
maps, as well as the case of rational functions. For contents and
more information, click this CUP
link.
Chaos and
Nonlinear Dynamcis, 2nd ed., by R. Hilborn, Oxford
University Press;
Introductory book discusses both difference and differential
equations. For table of contents, click this OUP link.
Difference
Equations with Applications to Queues, by D.L. Jagerman,
Marcel Dekker;
This monograph presents a theory of difference and functional
equations with continuous argument based on a generalization of
the Riemann integral introduced by N. E. Nörlund, allowing
differentiation with respect to the independent variable and
permitting greater flexibility in constructing solutions and
approximations. "Solves the nonlinear
first order equation by a variety of methods, including an
adaptation of the Lie-Gröbner theory!" For contents and
more information, click this Dekker link.
Bifurcations
and Catastrophes: Geometry of Solutions to Nonlinear Problems,
by M. Demazure, Springer-Verlag;
This text gives a rigorous introduction to many ideas in
nonlinear analysis, dynamical systems and bifurcation theory
including catastrophe theory. Wherever appropriate it emphasizes
a geometrical or coordinate-free approach which allows a clear
focus on the essential mathematical structures. Taking a unified
view, it brings out features common to different branches of the
subject while giving ample references for more advanced or
technical developments.
Optimization
and Chaos, Edited by M. Majumdar, K. Mitra and T. Nishimura,
Springer-Verlag;
This volume brings together a number of advanced research papers
on complex behavior of dynamic economic models. These make it
clear that complexity cannot be dismissed as
"exceptional" or "pathological" and, for
explanation and prediction of economic variables, it is
imperative to develop models with special structures suggested by
empirical studies. Graduate students will find the book valuable
for an introduction to optimization and chaos. Specialists will
find new directions to explore themes like robustness of chaotic
behavior and the role of discounting in generating cycles and
complexity. For more information, click this Springer
link.
Symmetries and
Integrability of Difference Equations, Edited by D. Levi and
O. Ragnisco, AMS;
In this collection the reader finds new developments in a number
of areas, including: Lie-type symmetries of
differential-difference and difference-difference equations,
integrability of fully discrete systems such as cellular
automata, the connection between integrability and discrete
geometry, difference and q-difference equations and orthogonal
polynomials, difference equations and quantum groups, and
integrability and chaos in discrete-time dynamical systems.
Special emphasis on the systems that can be integrated by
analytic methods or at least admit special explicit solutions.
For more information, click this AMS
link.
The Mandelbrot
Set, Theme and Variations, Edited by T. Lei, Cambridge
University Press;
Contains latest research, some as yet unpublished, on the old bug
and related topics. Suitable for researchers and graduate
students. Click this CUP
link for more information.
Oscillation
Theory for Difference and Functional Differential Equations,
by Ravi P. Agarwal, Said R. Grace and Donal O'Regan, Kluwer
Academic Publishers;
This book is for professional mathematicians and graduate
students - for summary and contents, click this Kluwer link.
Discrete Chaos,
by Saber N. Elaydi, CRC Press;
This book is similar in its focus to the well-known 1989 text
"An Introduction to Chaotic Dynamical Systems" by
Robert Devaney (Addison-Wesley). The book is suitable for upper
level undergraduate students and non-mathematicians looking for
an introduction to discrete dynamical systems. For specific
details (table of contents, etc) click this CRC link.
Books in 1999:
Controlling
Chaos and Bifurcations in Engineering Systems, by G. Chen,
CRC Press;
This book features contributions from experts, highlights open
problems in both fundamental theory and potential applications,
shows designers how to use chaos and control it to provide a
wider variety of properties and greater flexibility in the design
process and details some applications from electrical,
mechanical, and biomedical engineering. For more information
click this link.
Dynamics in One
Complex Variable, by J. Milnor, AMS;
The text studies the dynamics of iterated holomorphic mappings
from a Riemann surface to itself, concentrating on the classical
case of rational maps of the Riemann sphere. It is based on
introductory lectures given by Milnor at SUNY, Stony Brook (NY),
over the past 10 years. There are many computer generated
illustrations. Click here
for more details.
Exploring
Chaos: Theory and Experiment, by B. Davies, Perseus
Publishing Company;
A textbook suitable for undergraduate students and others
interested in learning about complex dynamical systems through
numerical methods and computer simulations. There is a detailed
discussion of dynamics and the various familiar bifurcationsof
one dimensional maps, as well as some exposition of two
dimensional dynamics and complex plane dynamics and fractals.
Proofs are largely omitted and the emphasis is on computer
simulations and experiments. Comes with the software Chaos for
Java, which may be downloaded from the internet. Read more about
the book and the software (and view an interesting Java applet of
the Lorenz attractor, if your browser permits it) on the author's
website; click Davies
to go there.
An Introduction
to Difference Equations, (2nd ed.) by Saber N. Elaydi,
Springer-Verlag;
This textbook updates the original 1996 edition and is accessible
to undergraduates at the level of US college seniors. It is a
well written contemporary text and its focus is different from
Discrete Chaos mentioned above. It discusses a range of topics,
including stability theory (linear and nonlinear), elements of
control theory and some classical topics such as the Z-transform
and oscillations. Click this Springer
link for contents and summary.
Books in 1998:
An Introduction
to Structured Population Dynamics, by J.M. Cushing, CBMS-NSF
series, 71;
This monograph introduces the theory of structured population
dynamics and its applications, focusing on the asymptotic
dynamics of deterministic models. Applications that illustrate
both the theory and a variety of biological issues are given,
along with an interdisciplinary case study that illustrates the
connection of models with the data and the experimental
documentation of model predictions. The author also discusses the
use of discrete and continuous models and presents a general
modeling theory for structured population dynamics. Click here for
more details and table of contents.
Dynamical
Systems: Stability, Symbolic Dynamics, and Chaos (2nd Ed.),
by C. Robinson, CRC press;
This book treats the dynamics of both iteration of functions and
solutions of ordinary differential equations. Many concepts are
first introduced for iteration of functions where the geometry is
simpler, but results are interpreted for differential equations.
The dynamical systems approach of the book concentrates on
properties of the whole system or subsets of the system rather
than individual solutions. The more local theory discussed deals
with characterizing types of solutions under various hypothesis,
and later chapters address more global aspects. Click here
for more information.
Analysis of
Observed Chaotic Data, by H.D.I. Abarbanel, M.E.Gilpin and
M. Rotenberg, Springer-Verlag
This book develops a systematic treatment of time series of data,
regular and chaotic, that one finds in observations of nonlinear
systems. The emphasis throughout is on the use of the modern
mathematical tools for investigating chaotic behavior to uncover
properties of physical systems. The methods require knowledge of
dynamical systems at the advanced undergraduate level and some
knowledge of Fourier transforms and other signal processing
methods. For more information, click this link.
Applied
Symbolic Dynamics and Chaos, by B-L Hao and W-M Zheng, World
Scientific;
This book discusses symbolic dynamics for one dimensional maps
(unimodal, multiple critical points, circle maps) and for some
two dimensional maps and ODE's. Applications to grammatical
complexity, knot theory and counting the number of critical
points are presented. Also see the 1996 book by Xie below. For
more information, go to the publisher's website, then click
"nonlinear science and chaos" and find the title in the
list provided.
The Complex
Matters of the Mind, by Franco Orsucci, World Scientific;
A book for a wide audience (Neuroscientists, psychologists,
applied mathematicians, physicists, biologists and computer
scientists). Includes works in the areas of psychology,
linguistics, anthropology, etc. For more information, go to the
publisher's website, then click "nonlinear science and
chaos" and find the title in the list provided.
Analysis and
Modelling of Discrete Dynamical Systems, edited by Daniel
Benest and Claude Froeschle, Gordon and Breach Publishers;
This book is interesting for its applications in physics and
mechanics and it is directed at mathematicians and graduate
students - for summary and contents, click this GBP
link.
Nonlinear
Dynamics and Endogenous Cycles, edited by Gilbert
Abraham-Frois, Springer-Verlag;
A series of articles on economic applications of nonlinear
difference and differential equations; primarily for researchers
and graduate students. Click this Springer
link for more information.
Nonlinear
Physics for Beginners, edited by Lui Lam, World Scientific;
Topics in chaos, fractals, cellular automata, etc. There are
student projects, but the "beginner" should be serious
and motivated. For more information, go to the publisher's
website, then click "nonlinear science and chaos" and
find the title in the list provided.
Books in 1997:
Nonlinear
Economic Dynamics (4th ed.), by T. Puu, Springer-Verlag;
Topics from both micro- and macroeconomics that exhibit various
types of dynamical behavior, such as limit cycles,
quasiperiodicity, bifurcations, and chaos are studied in this
book. Models appear in both discrete time and in continous time
in this completely rewritten fourth edition which has an
extensive mathematical introduction, dealing with both
differential equations, and with discrete maps that make the book
self-contained. For more information click here.
Exotic
Attractors, J. Buesco; Birkhauser;
This book on attractors in dynamical systems will appeal
primarily to researchers and advanced postgraduate students
working in the area of dynamical systems. The study is divided
roughly into two parts: The first part discusses several
different notions of attractor and illustrates by examples and
counterexamples. The second part of the book deals with two
different problems in discrete dynamics to which the author has
contributed. For more, click this Birkhauser
link.
Nonlinear
Dynamics and Time Series, Edited by C.D. Cutler, AMS,
This book is a collection of research and expository papers
reflecting the interfacing of two fields: nonlinear dynamics (in
the physiological and biological sciences) and statistics. The
papers highlight current areas of research in statistics that
might have particular applicability to nonlinear dynamics and new
methodology and open data analysis problems in nonlinear dynamics
that might find their way into the toolkits and research
interests of statisticians. For more information, click this AMS
link.
Economic
Dynamics, by G. Gandolfo, Springer-Verlag;
Economic modelling using difference and differential equations -
good for theoretical economists as well as mathematicians and
graduate students looking for applications to theoretical
economic models. Click this Springer
link for more information.
Nonlinear
Economic Dynamics, by T. Puu, Springer-Verlag;
Nonlinear discrete and continuous time business cycles; for
researchers and graduate students. Click this Springer
link for more information.
Laws of Chaos,
by Abraham Boyarsky and Pawel Gora, Birkhauser;
This graduate level textbook is a detailed and updated
introduction to the stochastic aspects of discrete
one-dimensional dynamics, and as such is suitable for specialists
who wish to gain a thorough understanding of the probabilisitic
and ergodic aspects of one-dimensional maps. Topics include
ergodic theory, absolutely continuous invariant measures, the
Ferobnius-Perron operator, Markov transformations and
applications to certain chaotic systems. For more information and
table of contents, click this Birkhauser
link.
Chaos in
Discrete Dynamical Systems, by Ralph H.Abraham, Laura
Gardini and Christian Mira, Springer-Verlag;
This book comes with a CD-ROM and gives an introductory account
of the works of Mira and others using critical curves for the
analysis of attractors and bifurcations in the plane (see the
1996 entry below). Ralph Abraham's visual approach is likely to
appeal to broad audiences without a significant mathematical
background, including college freshmen. Click this Springer
link for more information.
Advanced Topics
in Difference Equations, by Ravi P. Agarwal and Patricia
J.Y. Wong, Kluwer Academic Publishers;
This book contains some of the authors' own results and it is for
professional mathematicians and graduate students - for summary
and contents, click this Kluwer link.
Books in 1996:
Discretization
of Homoclinic Orbits, Rapid Forcing and Invisible Chaos, by
B. Fiedler and J. Schurle, AMS
The authors study the behavior of a homoclinic orbit under
discretization. Under generic assumptions they show that this
orbit becomes transverse for positive step size e.
Likewise, the region where complicated "chaotic"
dynamics prevail is under certain conditions estimated to be
exponentially small. These results are illustrated by high
precision numerical experiments. Click this AMS
link for more information.
Renormalization
and Geometry in One Dimensional and Complex Dynamics, by Y-P
Jiang, World Scientific;
This book discusses various aspects of renormalization for one
dimensional maps. Suitable for physics and mathematics
researchers and graduate students. For more information, go to
the publisher's website, then click "nonlinear science and
chaos" and find the title in the list provided.
Chaotic
Dynamics in Two-Dimensional Noninvertible Maps, by Christian
Mira, Laura Gardini, Alexandre Barugola and Jean-Claude Cathala,
World Scientific;
The concept of critical curves and their use in the study of
complex dynamics exhibited by endomorphisms of the plane, is the
main subject of this well-illustrated book. Although working with
critical curves is not easy, the method can yield significant
infomation about attractor structure and bifurcations, especially
if adequate computing resources are available. For more
information, go to the publisher's website, then click
"nonlinear science and chaos" and find the title in the
list provided.
Chaos: An
Introduction to Dynamic Systems, by K. Alligood, T. Sauer
and J.A. Yorke, Springer-Verlag;
This textbook is accessible to post calculus undergraduate
students (level of US sophomores or above), by three authors who
are known experts in the field. Click this Springer
link for contents and summary.
Grammatical
Complexity and One Dimensional Dynamical Systems, by H.M.
Xie, World Scientific;
An unusual application of symbolic dynamics, automata, etc. For
more information, go to the publisher's website, then click
"nonlinear science and chaos" and find the title in the
list provided.
A First Course
in Discrete Dynamical Systems, by Richard Holmgren,
Springer-Verlag;
Many are familiar with this concise, undergraduate level
introduction to discrete dynamical systems. The second edition
contains Mathematica programs. Click this Springer link
for more information.
Books in 1995:
Nonlinear
Dynamics in Economics, by B. Finkenstädt, Springer-Verlag;
This book deals with nonlinear economic dynamics and chaotic
motion where a specific approach is taken to the evolution of
prices in agricultural markets. Topics covered include
correlation integral diagnostics, testing for nonlinear
dependencies in a time series, nearest neighbor prediction and a
robust nonparametric methodology. For more information, click here.
Second Order
Sturm-Liouville Difference Equations and Orthogonal Polynomials,
by A. Jirari, AMS
This book develops a theory for regular and singular
Sturm-Liouville boundary value problems for difference equations,
generalizing many of the known results for differential
equations. The book is suitable as a text for an advanced
graduate course on Sturm-Liouville operators or on applied
analysis. Click this AMS
link for more information.
Computational
Analysis of One-Dimensional Cellular Automata, by B.H.
Voorhees, World Scientific;
Not exactly difference equations, but certainly discrete
dynamics! Seems like a good introduction to the curious topic of
cellular automata that is mathematically within the reach of US
undergraduate levels. For more information, go to the publisher's
website, then click "nonlinear science and chaos" and
find the title in the list provided.
Chaos,
Catastrophe and Human Affairs, by Stephen J. Guastello,
Lawrence Erlbaum Associates;
Applications of catastrophe theory and nonlinear systems to
psychology: work, organizations and social evolution.
Books in 1994:
Coping with
Chaos, edited by E. Ott, T. Sauer, J.A. Yorke, Wiley;
A collection of mathematical ideas about the analysis of chaotic
phenomena encountered in phyiscs. However, many of these ideas
have been found useful in other fields including biology and life
sciences. The topics covered include Liapunov exponents,
embeddings, noise reduction and chaos control. The introductory
"backgound" section is very helpful. For more, click
this link.
Fractal
Analysis Software Package: A Fractal Generator for Windows 3.x, by
P. Ferland, C. Tricot and A. van de Walle, AMS
The software features an accessible geometrical approach and
user-friendly environment. The user can create and render a
famous family of fractal images: iterated function systems of
affine application attractors. Several methods of fractal
dimension estimation, such as the box counting method and the
Minkowski sausage method, are included. The software makes
complete use of the user-friendly environment and interfacing
capabilities of Microsoft WindowsTM 3.x. Click this AMS
link for more information.
Business
Cycles: Theory and Empirical Methods, edited by Willi
Semmler, Kluwer Academic Publishers;
Applications of difference and differential equations to economic
business cycle models; for researchers and graduate students.
Click this Kluwer
link for contents and more information.
Books in 1993:
Nonlinear
Dynamics in Economics and Social Sciences, edited by F.
Gori, L. Geronazzo and M. Galeotti, Springer-Verlag;
This volume constitutes the Proceedings of the "Nonlinear
Dynamics in Economics and Social Sciences" Meeting held at
the Certosa di Pontignano, Siena, 1991 and includes contributions
by both economists and mathematicians. It includes 13 contributed
papers covering endogenous cycles, imperfectly competitive
economics, real wage dynamics, keynesian business cycle theory,
complexity of optimal paths, incomplete markets, walrasian and
non-walrasian equilibria and financial dynamics. For more
information, click here.
The General
Topology of Dynamical Systems, by E. Akin, AMS;
A general theory of topological dynamics in metric spaces is
presented based on closed relations rather than maps or flows.
The author likens this work to John Kelley's classic
"Topology" and the style of presentation is indeed
similar (one might also recall Paul Halmos's approach in
"Measure Theory"). As such, it is not a quick read but
important areas are covered thoroughly; they include Liapunov
functions, chain recurrence, invariant measures, attractors and
hyperbolic sets/axiom A homeomorphism. Certain aspects are
presented within the context of maps or flows, as pertinent. For
more information, click this AMS
link.
Continuum
Theory and Dynamic Systems, Edited by T. West,
Marcel-Dekker;
Based on a Conference/Workshop on Continuum Theory and Dynamical
Systems held in Lafayette, Louisiana. Illustrates the current
expansion of knowledge on the relationship between these
subjects. For contents and more information, click this Dekker link.
Combinatorial
Dynamics and Entropy in Dimension One, by L. Alseda, J.
Llibre, M. Misiurewicz, World Scientific;
This book is suitable for mathematicians and graduate students.
It covers Sharkovski's theorem for the line, and its analogs for
the circle, and discusses topological entropy. The exposition is
detailed, and some of the topics (e.g., the results on the
circle) are rarely seen in book form. For more information, go to
the publisher's website, then click "nonlinear science and
chaos" and find the title in the list provided.
Global Behavior
of Nonlinear Difference Equations of Higher Order with
Applications, by V.L. Kocic and G. Ladas, Kluwer Academic
Publishers;
An influential book in the area of nonlinear difference
equations, with applications to problems in biology. The authors
present and discuss in detail, several general and specific types
of nonlinear, higher order scalar difference equations. The
methodology is often equation-specific, a fact that accounts for
rather sharp results in many cases, but which also limits
applicability in some cases. Introduction of certain concepts
(e.g., semicycles) is timely and relevant as the young field
expands and defines itself. Results include some general theorems
on permanence, oscillations, global attractivity and/or stability
of equilibria and cycles, as well as specific results for certain
equations motivated by biological models. A list of open problems
and conjectures motivates further research. A good working book
for researchers and graduate students. For a table of contents
and more information, click this Kluwer link.
Difference
Equations and their Applications, by A.N. Sharkovski, Yu.L.
Maistrenko and E.Yu. Romanenko, Kluwer Academic Publishers;
The most significant part of this book for most readers is likely
to be the first part concerning first order difference equations.
It contains several important contributions, some well known
(like the first author's theorem on coexistence of cycles) and
others lesser known, though quite interesting. The book contains
many good examples and diagrams, and the authors should be
commended for taking care to put many of the non-transparent
concepts in perspective with suitable remarks and elaborations
judiciously inserted in between lemmas, theorems, etc. The main
complaint about the book would be the tiny index and the fact
that definitions are not properly highlighted. Although
challenging technical details abound and coverage is in-depth and
at an advanced level, it is recommended for both researchers and
graduate students; click this Kluwer link
for contents and more information.
Discrete
Dynamical Modeling, by J.T. Sandefur, Oxford University
Press;
A well-written introduction full of examples and applications.
Requires only a semester of calculus, and otherwise
self-contained. A recommended introductory text for
undergraduates. Click this OUP link
for contents and more details.
Books in 1992:
Economic
Evolution and Demographic Change, edited by G. Haag, U.
Mueller, K.G. Troitzsch, Springer-Verlag;
The articles collected in this volume relate to economics,
demography and geography. The book is subdivided into three
parts, where Part I focuses on economic evolution, Part II on
geographical development and Part III is related to demographic
change. The book is addressed to social scientists in general,
and those in particular with a background in economics,
geographics and demographics. It should also be of interest to
mathematicians, physicists, and systems analysts interested in
model building and applications of nonlinear dynamics. For more
information, click here.
A First Course
in Chaotic Dynamical Systems, by R.L. Devaney, Perseus
Publishing Company;
The undergraduate level version of the author's influential 1989
text. To read more, click this Perseus
link. The author's website contains information about this
and his other books; click Devaney to go
there.
Dynamics in One
Dimension, by L. Block and W.A. Coppel, Springer-Verlag;
This book, suitable for graduate students and mathematicians,
covers certain aspects of the dynamics of one dimensional maps in
detail. Topics covered include basics of maps of the line and the
circle, Sharkovski's theorem, chaotic maps, chain recurrence and
topological entropy. For more information, click here.
Cycles and
Chaos in Economic Equilibrium, edited by Jesse Benhabib,
Princeton University Press;
A collection of papers, many original contributions, make this a
valuable resource for the difference equations specialist
interested in the use of difference and differential equations in
economic modeling. Highly recommended for researchers and
graduate students. For a table of contents and more information,
click this PUP link.
Nonlinear
Systems, by P.G. Drazin, Cambridge University Press;
A well-written textbook at the level of US upperlevel
undergraduates, this book presents introductory topics in
difference and differential equations. Click this CUP
link for contents and more information.
Chaotic
Dynamics: Theory and Practice, edited by T. Bountis,
Pluenum/Kluwer Academic;
This NATO ASI Series volume contains some interesting papers on
difference equations/discrete dynamical systems and their
applications; click this Kluwer link
for more information.
Books in 1991:
Difference
Equations, Theory and Applications, (2nd ed.) by Ronald
Mickens, CRC Press;
Classical difference equations theory with applications from
natural and social sciences; suitable for graduate students and
upper level undergraduate students as well as those looking for
an introduction to difference equations. Book information and a
table of contents may be found here.
Dynamics of
Fractal Surfaces, edited by Freydoon Family and Tamas
Vicsek, World Scientific;
Growth of rough, fractal surfaces, for researchers and graduate
students. For more information, go to the publisher's website,
then click "nonlinear science and chaos" and find the
title in the list provided.
Books in 1990:
Chaos and
Socio-Spatial Dynamics, by D. Dendrinos and M. Sonis,
Springer-Verlag;
Applications of difference equations to spatially dependent
sociological models; here is a Springer
link for this book.
An Introduction
to Dynamical Systems, by D.K. Arrowsmith and C.M. Place,
Cambridge University Press;
This book covers both maps and flows. It is has many exercies,
some with hints, and covers most of the standard results (up to
its publication date) but perhaps the best feature is the way it
is so well-illustrated. Many parts of it are accessible to upper
level undergraduate students (US college standard), although like
many other books on dynamical systems, it also contains
sophisticated results, often presented without proof. Here is the
CUP
link for this book.
Links to publishers and booksellers:
Publishers:
Academic Press
American Mathematical
Society (AMS)
Birkhauser
Cambridge University Press
CRC Press
De Gruyter
John Wiley and Sons, Inc.
Kluwer Academic Publishers
Lawrence Erlbaum
Associates
Marcel Dekker, Inc.
MIT
Press
Oxford University Press
Perseus
Publishing Company
Princeton University
Press
Springer-Verlag
Taylor and Francis
Group
World
Scientific
Booksellers:
Reiters Scientific and Professional Books
Last update: 2003(87)