Book Description
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.
Customer Reviews:
Great, advanced intro to dynamical systems.......2003-12-20
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.
Great book with lots of detail.......2001-05-17
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.
Excellent rigorous introduction to chaotic dynamical system.......1997-04-17
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.
Excellent rigorous introduction to chaotic dynamical systems.......1997-04-17
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.
Book Description
Differential Equations: An Introduction to Modern Methods and Applications is a textbook designed for a first course in differential equations commonly taken by undergraduates majoring in engineering or science. It emphasizes a systems approach to the subject and integrates the use of modern computing technology in the context of contemporary applications from engineering and science. Section exercises throughout the text are designed to give students hands-on experience in modeling, analysis, and computer experimentation. Optional projects at the end of each chapter provide additional opportunitites for students to explore the role played by differential equations in scientific and engineering problems of a more serious nature.
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Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (Ergebnisse der Mathematik und ihrer Grenzgebiete. ... / A Series of Modern Surveys in Mathematics)
Michael Struwe
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Calculus of Variations (Cambridge Studies in Advanced Mathematics)
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Functional Analysis (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts)
ASIN: 3540664793 |
Book Description
Hilbert's talk at the second International Congress of 1900 in Paris marked the beginning of a new era in the calculus of variations. A development began which, within a few decades, brought tremendous success, highlighted by the 1929 theorem of Ljusternik and Schnirelman on the existence of three distinct prime closed geodesics on any compact surface of genus zero, and the 1930/31 solution of Plateau's problem by Douglas and Radó. The book gives a concise introduction to variational methods and presents an overview of areas of current research in the field. The third edition gives a survey on new developments in the field. References have been updated and a small number of mistakes have been rectified.
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Modern Methods in the Calculus of Variations: L^p Spaces (Springer Monographs in Mathematics)
Irene Fonseca , and
Giovanni Leoni
Manufacturer: Springer
ProductGroup: Book
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ASIN: 038735784X |
Book Description
This book addresses fundamental questions related to lower semicontinuity and relaxation of functionals within the unconstrained setting, mainly in L^p spaces. It prepares the ground for the second volume where the variational treatment of functionals involving fields and their derivatives will be undertaken within the framework of Sobolev spaces.
This book is self-contained. All the statements are fully justified and proved, with the exception of basic results in measure theory, which may be found in any good textbook on the subject. It also contains several exercises. Therefore, it may be used both as a graduate textbook as well as a reference text for researchers in the field.
Book Description
The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed, leading up to Trépreau’s theorem on sufficiency of condition (capital Greek letter Psi) for microlocal solvability in the analytic category.
At the beginning of the book, no prerequisites are assumed beyond calculus and linear algebra. Later on, basic facts from distribution theory and functional analysis are needed. In a few places, a more extensive background in differential geometry or pseudodifferential calculus is required, but these sections can be bypassed with no loss of continuity. The major part of the book should therefore be accessible to graduate students so that it can serve as an introduction to complex analysis in one and several variables. The last sections, however, are written mainly for readers familiar with microlocal analysis.
Book Description
The second in a series of three volumes surveying the theory of theta functions, this volume gives emphasis to the special properties of the theta functions associated with compact Riemann surfaces and how they lead to solutions of the Korteweg-de-Vries equations as well as other non-linear differential equations of mathematical physics.
This book presents an explicit elementary construction of hyperelliptic Jacobian varieties and is a self-contained introduction to the theory of the Jacobians. It also ties together nineteenth-century discoveries due to Jacobi, Neumann, and Frobenius with recent discoveries of Gelfand, McKean, Moser, John Fay, and others.
A definitive body of information and research on the subject of theta functions, this volume will be a useful addition to individual and mathematics research libraries.
Book Description
The purpose of this book is to demonstrate that complex numbers and geometry can be blended together beautifully. This results in easy proofs and natural generalizations of many theorems in plane geometry, such as the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem. The book is self-contained - no background in complex numbers is assumed - and can be covered at a leisurely pace in a one-semester course. Many of the chapters can be read independently. Over 100 exercises are included. The book would be suitable as a text for a geometry course, or for a problem solving seminar, or as enrichment for the student who wants to know more.
Customer Reviews:
New ways to do old things.......2000-03-28
Proofs in "pure" geometry are easy to understand but difficult to conceive. When presented with the opportunity to do such proofs, many people suffer from a brain cramp similar to that experienced by writers. One of the most common phrases heard when I was teaching is similar to the following, "I can follow the proof once it is done, but how do you think of trying those steps?" In this book, the author performs a marriage of complex numbers and geometry that can sometimes serve to point the aspiring geometer in the proper direction.
Contrary to their name, complex numbers are easy to understand and manipulate. Only basic knowledge of algebra is essential. In this case, the author uses all of Chapter One to introduce the fundamental ideas of their use. After that, things get exciting. Applying this knowledge to geometry, we see new ways to do old, sometimes very old things. In many cases, the approach is general, in that it is easy to see how such ideas can be used to attack other problems. Large numbers of exercises are included at the end of each chapter.
Worthy of inclusion in any library, this author shows that it is always possible to develop new ways to solve old problems.
Published in Journal of Recreational Mathematics, reprinted with permission.
Book Description
If you want top grades and thorough understanding of differential equations, this powerful study tool is the best tutor you can have! It takes you step-by-step through the subject and gives you 563 accompanying problems with fully worked solutions. You also get plenty of practice problems to do on your own, working at your own speed. (Answers at the back show you how you're doing.) Famous for their clarity, wealth of illustrations and examples, and lack of dreary minutiae, Schaum’s Outlines have sold more than 30 million copies worldwide—and this guide will show you why!
Customer Reviews:
Really Nice.......2006-03-08
This book is quite useful at times. I am taking a Differential Equations course and so far it as helped me out in this sometimes tricky subject. Although it has left out a few things that are in my textbook, I would recommend it for anyone that needs a little help with the process of solving problems.
Practice, Practice , Practice Problems.......2006-03-02
If you need a helpful outline with lots of solved practice problems to learn Differential Equations, this is your book!
Good Review Outline.......2006-01-28
This is a well-written and concise review of ODEs. As a review, it's awesome, and I would highly recommend this for a refresher or a reference for improving your skill level. However, I am not entirely sure it's a sure thing for introductory-level students. In other words, if you want to just pick it up and learn, be aware; it does not derive results in detail as some textbooks do. Nonetheless, it is one of the better Schaum's Outlines.
Misguiding.......2005-08-29
I feel that the original reviews as to the content and audience are misguiding. Firstly, this book does not cover enough the basics before dealing with more complex stuff, unlike for example Schaum's Outline on Matrix Operations. Secondly, there aren't enough practical applications (only two chapters out of 30), all the other worked examples are based on abstract problems. Thirdly, when buying a book on differential equations, I would have expected PDEs to be covered, at least at the basic level- this is not the case and to get this you have to buy the 'Partial Differential Equations' Outline, which I found quite annoying.
Saved Me Big Time.......2005-06-05
If you're a really dedicated student, you want to understand all the theory behind differential equations. However, if you're a slacker like me, you just want to know how to do the problem without really understanding what you're doing. This book does just that. I did not understand the method of Variation of Parameters with my college textbook, but totally got it using Schaum's. The three chapters that were spent on Laplace transforms were also very helpful, essential brushup using partial fractions and learning how to manipulate the inverse Laplace, which was not covered in my college textbook. Everyday applications such as the brine water tank and the spring problems and temperature were also covered in detail here. However, I found that Fourier Series were not covered extensively, and actually found my textbook to be better. Nonetheless, this book is essential and a big time saver when you don't have time. It definitely does not replace a professor, or a college textbook, but will quickly become you're best friend the night before the midterm.
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Modern Dynamical Systems and Applications
Manufacturer: Cambridge University Press
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ASIN: 0521840732 |
Book Description
Presenting a wide cross-section of current research in the theory of dynamical systems, this collection consists of articles by leading researchers (and several Fields medallists) in a variety of specialties. Surveys featuring new results, as well as research papers, are included because of their potentially high impact. Major areas covered include hyperbolic dynamics, elliptic dynamics, mechanics, geometry, ergodic theory, group actions, rigidity, and applications. The target audience is dynamicists, as well as mathematicians from other disciplines.
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- MATLAB: An Introduction with Applications 2nd Edition
- MATLAB: An Introduction with Applications 2nd Edition
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