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Introduction to the Modern Theory of Dynamical Systems [Anglais] [Broché]

Anatole Katok , Boris Hasselblatt

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28 décembre 1996 Encyclopedia of Mathematics and its Applications (Livre 54)
This book provided the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with most of the main areas of mathematics. The authors introduce and rigorously develop the theory while providing researchers interested in applications with fundamental tools and paradigms. The book begins with a discussion of several elementary but fundamental examples. These are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods. The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. The third and fourth parts develop the theories of low-dimensional dynamical systems and hyperbolic dynamical systems in depth. Over 400 systematic exercises are included in the text. The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate up.

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Revue de presse

' … 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working on dynamical systems in almost any context, and even experts will find interesting new proofs, insights and historical references throughout the book.' Monatshefte für Mathematik

'… contains detailed discussion … presents many recent results … The text is carefully written and is accompanied by many excercises.' European Mathematical Society Newsletter

'This book provides the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline.' L'Enseignement Mathématique

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Première phrase
The most general and somewhat vague notion of a dynamical system includes the following ingredients: (i). A "phrase space" X, whose elements or "points" represent possible states of the system. Lire la première page
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Concordance
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Couverture | Copyright | Table des matières | Extrait | Index | Quatrième de couverture
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Amazon.com: 4.8 étoiles sur 5  5 commentaires
15 internautes sur 15 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 Great, advanced intro to dynamical systems 20 décembre 2003
Par David Elder - Publié sur Amazon.com
Format:Broché
This is really one of the very best books on dynamical systems available today. Nearly every topic in modern dynamical systems is treated in detail. The authors have provided many important comments and historical notes on the material presented in the main text. The writing is clear and the many topics discussed are given appropriate motivation and background.

There are only two potential drawbacks. First, the prerequisites for this book are quite high. The reader should be familiar with real and functional analysis, differential geometry, topology, and measure theory, for starters. Fortunately a well-organized appendix collects the key results of each of the branches of math for the reader's reference. Second, many dynamical systems of interest to applied mathematicians, scientists, and engineers arise from differential equations. This book does not discuss in much detail the connection between ODEs and continuous dynamical systems. Other books (e.g. Perko) treat this connection more thoroughly.

For completeness, clarity, and rigor, Katok and Hasselblatt is without equal. If you work in dynamical systems, you should definitely have this excellent text on your bookshelf. Highly recommended.
14 internautes sur 14 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 Excellent rigorous introduction to chaotic dynamical system 17 avril 1997
Par Un client - Publié sur Amazon.com
Format:Broché
This remarkable book is by far the best rigorous introduction to many facets of the modern theory of (chaotic) dynamical systems. It introduces and rigorously develops the central concepts and methods in dynamical systems in a hands-on and highly insightful fashion. The authors are world experts in smooth dynamical systems and have played a major role in the development of the modern theory and this shows througout the book.

The book starts with a comprehensive discussion of a series of elementary but fundamental examples. These examples 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.). This chapter alone is worth the price of the
book.

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 Poincare-Denjoy theory, and Poincare-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 should be on the desk (not bookshelf!) of any serious student of dynamical systems or any mathematically sophisticated scientist or engineer interested in using tools and paradigms of dynamical systems to model or study nonlinear systems.
12 internautes sur 13 ont trouvé ce commentaire utile 
4.0 étoiles sur 5 Great book with lots of detail 17 mai 2001
Par Dr. Lee D. Carlson - Publié sur Amazon.com
Format:Broché
This book is a comprehensive overview of modern dynamical systems that covers the major areas. The authors begin with an overview of the main areas of dynamics: ergodic theory, where the emphasis is on measure and information theory; topological dynamics, where the phase space is a topological space and the "flows" are continuous transformations on these spaces; differentiable dynamics where the phase space is a smooth manifold and the flows are one-parameter groups of diffeomorphisms; and Hamiltonian dynamics, which is the most physical and generalizes classical mechanics. Noticeably missing in the list of references for individuals contributing to these areas are Churchill, Pecelli, and Rod, who have done interesting work in the area of both topological and Hamiltonian mechanics. No doubt size and time constraints forced the authors to make major omissions in an already sizable book.
Some elementary examples of dynamical systems are given in the first chapter, including definitions of the more important concepts such as topological transitivity and gradient flows. The authors are careful to distinguish between topologically mixing and topological transitivity. This (subtle) difference is sometimes not clear in other books. Symbolic dynamics, so important in the study of dynamical systems, is also treated in detail.
The classification of dynamical systems is begun in Chapter 2, with equivalence under conjugacy and semi-conjugacy defined and characterized. The very important Smale horseshoe map and the construction of Markov partitions are discussed. The authors are careful to distinguish the orbit structure of flows from the case in discrete-time systems.
Chapter 3 moves on to the characterization of the asymptotic behavior of smooth dynamical systems. This is done with a detailed introduction to the zeta-function and topological entropy. In symbolic dynamics, the topological entropy is known to be uncomputable for some dynamical systems (such as cellular automata), but this is not discussed here. The discussion of the algebraic entropy of the fundamental group is particularly illuminating.
Measure and ergodic theory are introduced in the following chapter. Detailed proofs are given of most of the results, and it is good to see that the authors have chosen to include a discussion of Hamiltonian systems, so important to physical applications.
The existence of invariant measures for smooth dynamical systems follows in the next chapter with a good introduction to Lagrangian mechanics.
Part 2 of the book is a rigorous overview of hyperbolicity with a very insightful discussion of stable and unstable manifolds. Homoclinicity and the horseshoe map are also discussed, and even though these constructions are not useful in practical applications, an in-depth understanding of them is important for gaining insight as to the behavior of chaotic dynamical systems. Also, a very good discussion of Morse theory is given in this part in the context of the variational theory of dynamics.
The third part of the book covers the important area of low dimensional dynamics. The authors motivate the subject well, explaining the need for using low dimensional dynamics to gain an intuition in higher dimensions. The examples given are helpful to those who might be interested in the quantization of dynamical systems, as the number-theoretic constructions employed by the author are similar to those used in "quantum chaos" studies. Knot theorists will appreciate the discussion on kneading theory.
The authors return to the subject of hyperbolic dynamical systems in the last part of the book. The discussion is very rigorous and very well-written, especially the sections on shadowing and equilibrium states. The shadowing results have been misused in the literature, with many false statements about their applicability. The shadowing theorem is proved along with the structural stability theorem.
The authors give a supplement to the book on Pesin theory. The details of Pesin theory are usually time-consuming to get through, but the authors do a good job of explaining the main ideas. The multiplicative ergodic theorem is proved, and this is nice since the proof in the literature is difficult.
6 internautes sur 7 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 Excellent rigorous introduction to chaotic dynamical systems 17 avril 1997
Par Un client - Publié sur Amazon.com
Format:Relié
This remarkable book is by far the best rigorous introduction to many facets of the modern theory of (chaotic) dynamical systems. It introduces and rigorously develops the central concepts and methods in dynamical systems in a hands-on and highly insightful fashion. The authors are world experts in smooth dynamical systems and have played a major role in the development of the modern theory and this shows througout the book.

The book starts with a comprehensive discussion of a series of elementary but fundamental examples. These examples 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.). This chapter alone is worth the price of the
book.

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 Poincare-Denjoy theory, and Poincare-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 should be on the desk (not bookshelf!) of any serious student of dynamical systems or any mathematically sophisticated scientist or engineer interested in using tools and paradigms of dynamical systems to model or study nonlinear systems.
0 internautes sur 1 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 a comprehensive introduction 22 novembre 2010
Par Narada - Publié sur Amazon.com
Format:Broché|Achat vérifié
Almost everything you want to know about dynamical systems is introduced in this book. The writing style is clear (if, of necessity, terse), plenty of examples are given, what's not to like?

The one problem with this book is that it is at a level where it is really aimed at people starting to do research in the field, and those people are often better off looking up original references, especially as dynamical systems is a very young field, and so there is not the generation gap you would get by learning about determinants from Sylvester or equations from Legendre, or whatever (not that Sylvester and Legendre should not be read...), so perhaps a more useful book would be an "annotated bibliography", such as (in a different field) A History of the Theory of Investments: My Annotated Bibliography (Wiley Finance)
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