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Introduction to the Modern Theory of Dynamical Systems

By Anatole Katok and Boris Hasselblatt
with a supplement by Anatole Katok and Leonardo Mendoza

Encyclopedia of Mathematics and Its Applications 54, Cambridge University Press, 1995.
ISBN 0-521-34187-6, Paperback, 1997: ISBN 0-521-57557-5.

Call 1-800-872 7423 or order online.
Russian translation: Publishing House Factorial, 1999Portuguese translation: Fundação Calhouste Gulbenkian, 2005Samples


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 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 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for anybody working on dynamical systems in almost any context, and even experts will find interesting new proofs and historical references throughout the book.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 exceptional reference. 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 mechanicians. H. Haslach, Applied Mechanics Review
The book is a pleasure to read. E. Amiran, Mathematical Reviews

The table of contents and preface can be read here. Some corrigenda are available, including a major correction on page 254ff. Please report 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 forms, etc.).

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.