The authors ... have a definite idea what dynamical systems theory is all
about. A first rate text with more than enough dynamics to suit just about
anyone's taste...carefully and masterfully written...a classic
compendium. It is a must-have for any researcher in the field.
R. Devaney, Mathematical Intelligencer
A comprehensive exposition. Seemingly every topic is covered in depth.
M. Richey, American Mathematical Monthly
The book...is unique in giving a detailed presentation of a large part of
smooth dynamics in a consistent style...unrivalled as a comprehensive
introduction at an advanced level.
D. Ruelle, Ergodic Theory and Dynamical Systems
...even specialists will find original aspects and new points of view...the
mathematical examples play a prominent role, which I found very
attractive...The treatment of hyperbolic systems, including their ergodic
properties...is in my opinion really excellent. It is the most accessible
treatment of this theory.
F. Takens, Bulletin of the American Mathematical Society
Of the current flood of books on the subject, this one distinguishes itself
in many ways...I recommend it also as an important source to all those
involved in the interface between the mathematical theory and its
increasingly pervasive role in the scientific world.
R. MacKay, Bulletin of the London Mathematical Society
...there is no other treatment coming close in terms of comprehensiveness
and readability. It is indispensable for anybody working on dynamical
systems in almost any context, and even experts will find interesting new
proofs and historical references throughout the book.
K. Schmidt, Monatshefte für Mathematik The notes section at the end of the book is complete and quite helpful. There are hints and answers provided for a good
percentage of the problems in the book. The problems range from fairly straightforward ones to results that I remember reading
in research papers over the last 10-20 years....I recommend the text as an
Richard Swanson, SIAM Review Meines Erachtens stellt Katok und Hasselblatts "Introduction to the modern
theory of dynamical systems" eine äußerst wertvolle Bereicherung
der Literatur über die Theorie dynamischer Systeme dar, und ich kann
das Buch jedem uneingeschränkt empfehlen, der diese Theorie in Lehre
oder Forschung behandelt oder anwendet. G. Sorger, International Mathematical News
...well written and clear...a valuable reference for engineers and
H. Haslach, Applied Mechanics Review
The book is a pleasure to read.
E. Amiran, Mathematical Reviews
The table of
can be read here. Some
are available, including a major correction on
page 254ff. Please
any errors you notice in the book to .
Note one serious omission: The first three print runs
(up to the first paperback printing) fail to acknowledge that Section
20.6. reproduces work of Charles Hansen Toll (A multiplicative asymptotic of
the prime geodesic theorem, Thesis, University of Maryland 1984). Our
sincere apologies for this failure to give due credit.
The theory of dynamical systems is a major mathematical
discipline closely intertwined with all main areas of
mathematics. Its concepts, methods and paradigms greatly
stimulate research in many sciences and gave rise to the
vast new area variously called applied dynamics, nonlinear
science, or chaos theory. This book provides the first
self-contained coherent comprehensive exposition of the
theory of dynamical systems as a core mathematical
discipline while providing researchers interested in
applications with fundamental tools and paradigms. It
introduces and rigorously develops the central concepts and
methods in dynamical systems in a hands-on fashion.
It starts with a comprehensive discussion of a series of
elementary but fundamental examples. These are used to
formulate the general program of the study of asymptotic
properties as well as to introduce the principal notions
(differentiable and topological equivalence, moduli,
asymptotic orbit growth, entropies, ergodicity, etc.) and,
in a simplified way, a number of important methods (fixed
point methods, coding, KAM-type Newton method, local normal
The main theme of the second part is the interplay between
local analysis near individual (e.g., periodic) orbits and the
global complexity of the orbit structure. This is achieved
by exploring hyperbolicity, transversality, global
topological invariants, and variational methods. The methods
include study of stable and unstable manifolds,
bifurcations, index and degree, and construction of orbits
as minima and minimaxes of action functionals.
In the third and fourth part the general program is carried
out for low-dimensional and hyperbolic dynamical systems
which are particularly amenable to such analysis. In
addition these systems have interesting particular
properties. For hyperbolic systems there are structural
stability, theory of equilibrium (Gibbs) measures, and
asymptotic distribution of periodic orbits, in
low-dimensional dynamical systems classical
Poincaré-Denjoy theory, and Poincaré-Bendixson theories
are presented as well as more recent developments, including
the theory of twist maps, interval exchange transformations
and noninvertible interval maps.
This book provides a large number of systematic exercises in
order to be the principal source for the professional
training of future researchers. On the other hand the book
may be used by advanced undergraduates in mathematics,
graduate students in any area of the mathematical sciences
and graduate students in science and engineering with a
strong mathematical background as well as researchers in any
area of mathematics, science or engineering. Since a
considerable part of the material of the book is either
previously unpublished or presented in an essentially new
way it is also of interest to experts in dynamical systems.
Each of the four parts of the book can be the base of
a course roughly at the second year graduate
level. They are accessible to students having taken standard
US first year courses in analysis, geometry and topology. In
fact, the background material beyond multivariable calculus
and linear algebra and ordinary differential equations is
covered in appendices. This allows to use certain parts of
the book, especially parts 1 and 3, as the basis for more
elementary courses starting from advanced undergraduate
(junior or senior) level. Many courses dedicated to more
specialized topics can be tailored from this book, such as
variational methods in classical mechanics, hyperbolic
dynamical systems, twist maps and applications, introduction
to ergodic theory and smooth ergodic theory, the
mathematical theory of entropy.
In the US any university with a graduate program as well as
good undergraduate institutions would be
able to thus use the book. In continental Europe the book is
appropriate for courses to students at any level above
undergraduate, as well as to undergraduate students
specializing in mathematics.
This book has been used for courses at institutions world-wide. It is among
the 50 most cited mathematics books, and virtually every 21st-century PhD
in dynamical systems has been trained using it.
You can read
more about the book.
The picture shows the authors in Oberwolfach
(1997). Photograph by Krystyna Kuperberg.