**Book Review
**

Source: Mathematical Reviews, Jan. 1998

** Chaos. **
**(English)**

An introduction to dynamical systems.

*Springer-Verlag, New York,* 1997. xviii+603 pp. $39.00. *ISBN* 0-387-94677-2

By Kathleen T. Alligood, Tim D. Sauer and James A. Yorke.

In recent years there has been a small but growing list of books
intended as texts for an undergraduate course in dynamical systems in
general, and chaos in particular. The present work, written by some
prominent contributors to the development of the field, is a nice
addition to that list. With regard to both style and content, the
authors succeed in introducing junior/senior undergraduate students to
the dynamics and analytical techniques associated with nonlinear systems,
especially those related to chaos.

There are several aspects of the book that distinguish it from some
other recent contributions in this area, such as one by R. Devaney
[A first course in chaotic dynamical systems,
Addison-Wesley, Reading, MA, 1992; MR 94a:58124]. First, the
treatment of discrete
systems here maintains a balanced emphasis between one- and two-
(or higher-) dimensional problems. This is an important feature since the
dynamics for the two cases and methods employed for their analyses may
differ significantly. Also, while most other introductory texts
concentrate almost exclusively upon discrete mappings, here at least
three of the thirteen chapters are devoted to differential equations,
including the Poincare-Bendixson theorem. Add to this a
discussion of omega-limit sets, including periodic and strange
attractors, as well as a chapter on fractals, and the result is one of
the most comprehensive texts on the topic that has yet appeared.

Such an inclusion of topics does come at a price. While definitions
and theorems are stated formally and in sufficient technical detail
to constitute a serious introduction to the field, analytic proofs of
most results are omitted. Among the few exceptions to this are proofs
of the stable manifold theorem, the Poincare-Bendixson theorem,
and the cascade theorem, which appear in later chapters of the book.
This absence of formal proofs is at present typical of all but a few
undergraduate texts in this area.

The book begins with two chapters that introduce respectively the two
types of systems that are to be the primary focus of the work.
Chapter 1 studies elementary concepts related to iteration of
one-dimensional maps: fixed and periodic points, stability, the logistic
equation, itineraries, sensitive dependence. Chapter 2 introduces
analogous issues for two-dimensional maps: linear and nonlinear maps,
the Jacobian matrix, eigenvalues, sinks, sources, saddles, stable
and unstable manifolds.

The treatment thus far stops just short of chaos, which is the study
of the next four chapters. Chapter 3 investigates chaos for
one-dimensional maps. Special emphasis is given to Lyapunov exponents,
the primary means used to determine chaotic orbits. Before
treating chaos in two dimensions, in the next chapter the authors
discuss fractals, including methods for computing the dimension.
Chapter 5 resumes the discussion of chaos, in this case for
two-dimensional maps. Lyapunov exponents are once again emphasized, along
with some purely qualitative issues, such as fixed point theorems,
shift maps, and horseshoe maps. Chaotic attractors of both one- and
two-dimensional systems are examined in Chapter 6. In this context
omega-limit sets and invariant measures are defined.

For the next three chapters the focus is shifted to the other type of
dynamical system considered in the text, differential equations.
Chapter 7 introduces one- and higher-dimensional, linear and nonlinear,
continuous systems. Limit sets and periodic orbits of planar flows
are the subject of the next chapter, which includes a proof of the
Poincare-Bendixson theorem. Chapter 9 studies chaos for continuous
systems, and considers some famous examples: the Lorenz attractor,
the Rossler attractor, Chua's circuit and forced oscillators.

The next chapter returns to discrete mappings with a more careful
analysis of stable and unstable manifolds, and implications for chaos
of homoclinic and heteroclinic points. A proof of the stable manifold
theorem also appears here.

Although previously introduced, a detailed examination of bifurcation
is saved for the next two chapters. Parameterized families of one- and
two-dimensional maps and of differential equations, and the types of
bifurcation associated with them, are considered in Chapter 11.
Cascades of period-doubling bifurcations and a proof of the cascade
theorem are the subject of Chapter 12.

The final chapter is a brief discussion of how the state of a system
can be reconstructed from experimental time series data. Delay plots
and embedology are described.

Overall, the book is well designed and typeset, and is written in a
casual, easy-to-read style with a generous assortment of clearly drawn
graphs and diagrams, as well as some full color plates. Suggested
exercises and computer experiments, intended to verify or further
elaborate certain points, appear regularly throughout the main body of
the text. These are supplemented at the end of each chapter with a
full set of exercises, as well as with two special sections. The
first, called a Challenge, allows the reader to develop and/or prove
some related mathematical observations with the use of supplied hints.
The other, called a Lab Visit, describes the results of laboratory
experiments or other scientific observations that demonstrate
real-world applications of concepts developed in the chapter. Answers to or
hints for selected exercises are included in the back of the book,
along with several appendices that review some matrix algebra and
the numerical solution of differential equations.

At nearly 600 pages the text may seldom be covered in its entirety in
a single semester, although the absence of detailed proofs may hasten
the pace. The wide range of topics allows some flexibility. Several
chapters could be partially of entirely bypassed at the discretion of
the instructor.

Besides the calculus sequence, prerequisites for such a course
would ordinarily include linear algebra and differential equations,
making it suitable primarily for undergraduate students of
mathematics, physical science, or engineering. The latter prerequisite
might be eliminated if the chapters on differential equations were
skipped either in the interests of time or to reach a wider audience.

In short, the book is a significant contribution to the increasing
collection of texts on this topic. The authors have succeeded in
taking the most important ideas from dynamical systems and chaos, and
presenting them to undergraduates in a serious but accessible manner.
The work should be a definite consideration for anyone contemplating
an introductory course in this area.