Preface to the Book
This book is a visual introduction to chaos and bifurcations in
noninvertible discrete dynamical systems in two dimensions, by the
method of critical curves.
HISTORICAL BACKGROUND
Dynamical systems theory is a classical branch of mathematics
which began with Newton around 1665. It provides mathematical
models for systems which evolve in time according to a rule,
originally expressed in analytical form as a system of
ordinary differential equations. These models are called continuous dynamical
systems. They are also called flows, as the points of the system
evolve by flowing along continuous curves.
In the 1880s, Poincare studied continuous dynamical systems in
connection with a prize competition on the stability of the
solar system. He found it convenient to replace the continuous flow of time
with a discrete analogue, in which time increases in regular,
saltatory jumps. These systems are now called discrete dynamical
systems. So, for over a century, dynamical systems have come in two
flavors: continuous and discrete. Discrete dynamical systems are
usually expressed as the iteration of a map (also called an
endomorphism) of a space into itself. In these systems, points of the system
jump along dotted lines with a regular rhythm.
In the context of a discrete dynamical system, in which a given
map is iterated, that map might be invertible (because of being
one-to-one and onto) or noninvertible (failing one or the other or both of
these conditions). So, discrete dynamical systems come in two
types, invertible and noninvertible. The invertible maps were
introduced by Poincare, and have been extensively studied ever since.
The studies of noninvertible maps have been more sparse until
recently, when they became one of the most active areas on the
research frontier because of their extraordinary usefulness in
applications.
Chaos theory is a popular pseudonym for dynamical systems
theory. This new name became popular about 20 years ago, when
its applicability to chaotic systems in nature became widely known
through the advent of computer graphics. As there are two flavors
of dynamical systems, continuous and discrete, there are also two
chaos theories. The first to develop, in the work of Poincare about a
century ago, was the theory of chaotic behavior in continuous
systems. He also studied chaotic behavior in discrete dynamical
systems generated by an invertible map.
Discrete chaos theory for noninvertible maps began some years
after Poincare. Its development has been accelerated particularly
since the computer revolution, and today it is a young and active
field of study. The earliest development of the theory came in the
context of one-dimensional maps, that is, the iteration of a real
function of a single real variable. One
of the key tools in the one-dimensional theory was the
calculus of critical points, such as local
maxima and minima. The two-dimensional context is the current
research frontier, and, it is the subject of this book.
For two-dimensional noninvertible maps, the critical curve is a
natural extension of the classical notion of critical point for
one-dimensional noninvertible maps. The first introduction of the
critical curve, as a mathematical tool for two-dimensional noninvertible
maps, appeared in papers by Gumowski & Mira in the 1960s (see
the bibliographies at the end of the book for references.)
THE IMPORTANCE OF OUR SUBJECT
Chaos theory generally is crucially important in all the sciences
(physical, biological, and social) because of its unique capability
for modeling those natural systems which behave chaotically. It is
for this reason that there is a chaos revolution now ongoing in the
sciences. For those systems which present continuous, evolving,
data (such as the solar system) P continuous chaos theory provides
models. And for those which present discrete data (such as
economics) P discrete chaos theory provides models. One advantage of
discrete dynamical models is the ease and speed of simulating the
models with digital computers, as compared with continuous
dynamical models. Discrete models are sometimes advantageous,
even in the context of natural systems presenting continuous data.
UNIQUENESS OF THIS PUBLICATION
The book component of this book/CD-ROM/Website package
is not a conventional text book, and yet its purpose is pedagogic. It
intends to provide any interested person having a minimal
background in mathematics, but with a basic understanding of the
language of set-theory, to become an initiate in this new field. It is
unique in providing both an elementary and a visual approach to the
subject. While chaos theory is mathematically sophisticated, by
focusing on examples and visual representations P there are about
one hundred computer graphics in the book P and minimizing the
symbols and jargon of formal mathematics P they are relegated to
a set of appendices P the text provides the reader with an easy
entry into this important and powerful theory. The primary focus of
the package is the concept of bifurcation for a chaotic attractor.
These are introduced in four exemplary bifurcation sequences, each
defined by a family of very simple noninvertible maps of the plane
into itself. Each family, the subject of an entire chapter in the book,
exhibits many bifurcations.
And as dynamics involves motion, computer graphic
animations provide a particularly appropriate medium for communicating
dynamical concepts. The CD-ROM contains 12 animations which
bring life to the basic ideas of the theory, literally animating the still
images of the book. For each of the four map families there is one
long, fast movie which is a fast forward through the entire chapter,
as well as two *zooms: which expand a brief piece of the action
into a slow motion movie. The movies can be understood only by
reading along in the book while viewing the movie. The motion
controls of the movie players (in both the Windows and the
Macintosh environments) allow easy stop, play, fast-forward, reverse, and
slow-motion, by dragging a slider. This makes the CD-ROM ideal
for studying in conjunction with the book.
INTENDED AUDIENCE
While many devotees of pure mathematics may enjoy this
package for the novelty of its fresh ideas and the mathematical challenge
of a new subject, with most of its main problems unsolved, the
intended audience for this book is the large and heterogeneous
group of science students and working scientists who must, due to
the nature of their work, deal with the modeling and simulation of
data from complex dynamical systems of nature which are
intrinsically discrete. This means, for example, applied scientists,
engineers, economists, ecologists, and students of these fields.
HOW TO USE THE BOOK
The book is divided in three parts, which are almost
independent, and which can be utilized in parallel. The first part provides
the simplest introduction to the basic concepts of discrete chaos
theory, with many drawings and examples. The second part is a
detailed analysis of computer experiments with four families of
discrete chaotic systems, with emphasis on the method of critical
curves, and the phenomena of bifurcation. The third part is a set of
appendices which provide more official definitions for readers
having a stronger background in abstract mathematics. Here, is also
found extensive historical material by Professor Mira, some made
available in English for the first time. It is proposed that the second
part be regarded as a *guided tour: through a very difficult terrain,
and each example studied repeatedly, with recourse as necessary
(using the index) to the first and third parts, and to the CD-ROM.
ACKNOWLEDGMENTS
Thanks to Raymond Adomaitis, Gian Italo Bischi, Robert
Devaney, and Daniel Lathrop, for their very helpful comments on the
text, and Karen Acker for her FrameMaker artistry. Our publisher,
TELOS, has been extraordinarily supportive and patient with our
process: a thousand thanks to Allan Wylde, Paul Wellin, and Jan
Benes. And we are deeply indebted to Scott Hotton for his
extraordinary illustrations in the book and corresponding MAPLE
programs on the CD-ROM.
A video by John Dorband of NASA Goddard Space Flight
Center originally sparked our collaboration presented in Chapter 7 here,
and we are grateful to him for sharing his work with us.
The exceptional figure on the cover, and Figures 7-24, 7-25,
7-28, and 7-29, very difficult to compute, are the work of our
colleague Daniele Fournier-Prunaret, and we are grateful to her for
contributing them to this book. Her beautiful drawings have
inspired us.
Finally, we are very grateful to Peter Broadwell for the loan of a
Silicon Graphics Indigo computer, which ran continuously for
several months cranking out the frames for the movies on our CD-
ROM. And without our mathematical copy editor, Paul Green, this
book would be a mine field for the novice reader. We are deeply in
his debt.
Revised 24 September 1996 by Ralph Abraham,
<abraham@vismath.org>