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Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications) [Paperback]

Anatole Katok , Boris Hasselblatt
4.8 out of 5 stars  See all reviews (5 customer reviews)

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Book Description

December 28, 1996 0521575575 978-0521575577 1
This book provides a self-contained comprehensive exposition of the theory of dynamical systems. The book begins with a discussion of several elementary but crucial 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. The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate and up.

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Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications) + Geometrical Methods in the Theory of Ordinary Differential Equations (Grundlehren Der Mathematischen Wissenschaften (Springer Hardcover))
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Editorial Reviews

Review

"...well written and clear...a valuable reference for engineers and mechanicians." H.W. Haslach Jr., Applied Mechanics Reviews

"The book is a pleasure to read." Edoh Amiran, Mathematical Reviews

"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

Book Description

Beginning with a discussion of several elementary but crucial examples, this study provides a self-contained comprehensive exposition of the theory of dynamical systems. It is aimed at students and researchers in mathematics at all levels from advanced undergraduate and up.

Product Details

  • Series: Encyclopedia of Mathematics and its Applications (Book 54)
  • Paperback: 824 pages
  • Publisher: Cambridge University Press; 1 edition (December 28, 1996)
  • Language: English
  • ISBN-10: 0521575575
  • ISBN-13: 978-0521575577
  • Product Dimensions: 9.3 x 6.3 x 2 inches
  • Shipping Weight: 2.6 pounds (View shipping rates and policies)
  • Average Customer Review: 4.8 out of 5 stars  See all reviews (5 customer reviews)
  • Amazon Best Sellers Rank: #763,355 in Books (See Top 100 in Books)

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4.8 out of 5 stars
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Most Helpful Customer Reviews
15 of 15 people found the following review helpful
5.0 out of 5 stars Great, advanced intro to dynamical systems December 19, 2003
Format:Paperback
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.
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14 of 14 people found the following review helpful
By A Customer
Format:Paperback
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.
Read more ›
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Was this review helpful to you?
12 of 13 people found the following review helpful
4.0 out of 5 stars Great book with lots of detail May 16, 2001
Format:Paperback
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.
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Was this review helpful to you?
6 of 7 people found the following review helpful
By A Customer
Format:Hardcover
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.
Read more ›
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