Book Description
Thirty years in the making, this revised text by three of the world's leading mathematicians covers the dynamical aspects of ordinary differential equations. it explores the relations between dynamical systems and certain fields outside pure mathematics, and has become the standard textbook for graduate courses in this area. The Second Edition now brings students to the brink of contemporary research, starting from a background that includes only calculus and elementary linear algebra.
The authors are tops in the field of advanced mathematics, including Steve Smale who is a recipient of the Field's Medal for his work in dynamical systems.
* Developed by award-winning researchers and authors
* Provides a rigorous yet accessible introduction to differential equations and dynamical systems
* Includes bifurcation theory throughout
* Contains numerous explorations for students to embark upon
NEW IN THIS EDITION
* New contemporary material and updated applications
* Revisions throughout the text, including simplification of many theorem hypotheses
* Many new figures and illustrations
* Simplified treatment of linear algebra
* Detailed discussion of the chaotic behavior in the Lorenz attractor, the Shil'nikov systems, and the double scroll attractor
* Increased coverage of discrete dynamical systems
Customer Reviews:
A new version of a classic book.......2007-02-21
I bought a copy of this new book and I have its old version with Hirsch and Smale as its only authors. Main differences between these books are some new chapters covering chaos and the exercises. Old version has better chapters dealing with linear algebra. I find this new version hard to read and it leaves many details to be filled by the reader. I would say that the new version is still a good choice for a second course in ODE or supplementary text for a graduate course. I gave it four stars.
Excellent Book.......2006-05-05
This is a great introduction to the next stage of differential equations after a first course. Devaney is a master of presenation, and makes everything seem easy. It is not as encyclopedic as some other books on this material, such as Arnold and Perko, but it is easier to read and still covers the most important advanced material.
good, not ideal.......2005-12-08
the two books by hirsch smale, one with devaney, seem like good books, but I am not crazy about either, at least from the few pages one can search online here.
the latter book with devaney just seems a dumbed down version of the earlier book by the two more famous authors. i expected that earlier book to be far better, but found to my regret that the two books actually share almost the same first page, and the main difference noticeable in the early going is that the 2 author work is poorly written, and the 3 author one is not written much better.
it is clearer but seems to be talking down to the reader in an annoying way. so neither is the absolute pleasure to read that the wonderfully written text of arnol'd is, or the classic of hurewicz. i would skip these books and get arnold and hurewicz instead.
New Edition.......2004-02-26
You should be aware that there are two similar books with similar titles by the same authors. The old edition is a hardcover all green book by Hirsch and Smale called:
"Differential Equations, Dynamical Systems and Linear Algebra"
The second with the lorenz attractors in yellow on the cover is by Hirsch, Smale and Devaney and is called:
"Differential Equations, Dynamical Systems and an Introduction to Chaos"
Now, that may be obvious to you, but it is important to note that because those are VERY different books (which I have both of right here). The 'old' one is a more theoretical text that mainly addresses linear systems and is organized more like a math monograph than a contemporary (i.e. with pictures and examples) textbook. It is difficult for most people. The newer version is COMPLETELY different and is written for a more diverse audience. It starts with linear systems but then goes into nonlinear systems and discrete systems. It is somewhat similar in character to Strogatz's Nonlinear Dynamics and Chaos. If you do not have a very strong abstract theoretical type of math background I would not recommend you start learning about differential equations from the "old" edition. You will find it very difficult. If you are used to a general abstract presentation of results you should be fine. For the NEW edition the level is very different. I would guess that courses in multi-variable calc, elementary diff eq, and linear algebra (if you understood them) would be sufficient preparation. Both books are excellent, just be clear on what you are looking for.
Book Description
What is life? Has molecular biology given us a satisfactory answer to this question? And if not, why, and how to carry on from there? This book examines life not from the reductionist point of view, but rather asks the question: what are the universal properties of living systems and how can one construct from there a phenomenological theory of life that leads naturally to complex processes such as reproductive cellular systems, evolution and differentiation? The presentation has been deliberately kept fairly non-technical so as to address a broad spectrum of students and researchers from the natural sciences and informatics.
Book Description
The study of nonlinear dynamical systems has exploded in the past 25 years, and Robert L. Devaney has made these advanced research developments accessible to undergraduate and graduate mathematics students as well as researchers in other disciplines with the introduction of this widely praised book. In this second edition of his best-selling text, Devaney includes new material on the orbit diagram fro maps of the interval and the Mandelbrot set, as well as striking color photos illustrating both Julia and Mandelbrot sets. This book assumes no prior acquaintance with advanced mathematical topics such as measure theory, topology, and differential geometry, Assuming only a knowledge of calculus, Devaney introduces many of the basic concepts of modern dynamical systems theory and leads the reader to the point of current research in several areas.
Customer Reviews:
Great Introduction to the topic.......2007-03-09
This is a very good book. Actually, Devaney's "First Course in Chaotic Dynamical Systems," is a good accompanying text. Fascinating subject...
Excellent book; unique in its accessibility and coverage of deep results.......2005-09-14
This book is an introduction to dynamical systems defined by iterative maps of continuous functions. It doesn't require much advanced knowledge, but it does require a familiarity and certain level of comfort with proofs. The basic idea of this book is to explore (in the context of iterative maps) the major themes of dynamical systems, which can later be explored in the messier setting of differential equations and continuous-time systems. While this book doesn't discuss differential equations directly, the techniques used here can be transferred (with considerable work and thought) to that setting. Someone wanting an elementary book covering differential equations as dynamical systems might want to check out the excellent multi-volume work by J. Hubbard; the combination of that work with this book would provide the background to tackle the tougher and less-accessible texts dealing with chaotic systems of differential equations.
Although this is a pure math book, the book does mention key applications and motivation behind the material; applied mathematicians will find this book quite useful, not necessarily because of the choice of topics but just because it greatly helps develop ones' intuition. The material is presented in a way that gives the student a sense of the big picture--what the theorems mean, how they fit together. Proofs are rigorous but as easy to follow as I have seen them in this subject.
The choice and order of subjects is also both practical and fun. The book begins with 1-dimensional systems and explores just about everything interesting that happens with them (including Sarkovski's Theorem, one of the most bizarre and surprising mathematical results), before moving into two-dimensions and then dynamics in the complex plane.
The bottom line? This book would be excellent both as a textbook and for self-study. If you're interested in this subject at all, this is a book you will want on your shelf. I know of no other book on the subject that covers such deep material while remaining as accessible.
Good introduction to the beginning student.......2001-08-11
This book gives a quick and elementary introduction to the field of chaotic dynamical systems that could be read by anyone with a background in calculus and linear algebra. The approach taken by the author is very intuitive, lots of diagrams are used to illustrate the major points, and there are many useful exercises throughout the book. It could serve well in an undergraduate mathematics course in dynamical systems, and in a physics undergraduate course in advanced mechanics. The author emphasizes the mathematical aspects of dynamical systems, and readers will be well prepared after finishing it to read more advanced books on dynamical systems.
Chapter 1 introduces one-dimensional dynamics, with the analysis of the quadratic map given particular attention. Called the logistic map in some circles, this very important dynamical system has been the subject of much study, and exhibits generically the properties of chaotic dynamical systems. The author also gives a brief review of some elementary notions in calculus needed for the chapter, making the book even more accessible to a wider readership. The important concept of hyperbolicity is discussed in the context of one-dimensional maps and a good discussion is given on symbolic dynamics. Structural stability, which is really useful only in dynamical systems in higher dimensions, is treated here. The intuition gained in one-dimension is invaluable though before moving on to higher-dimensional examples. Sarkovskii's theorem, which states that a one-dimensional dynamical system with a period three periodic orbit has periodic orbits for all other periods, is proved in detail. In addition, the Schwarzian derivative, so important in complex dynamics, is defined here. The author also gives an introduction to bifurcation theory, which again, is most interesting in high dimensions, and introduces the concept of homoclinicity in this discussion. Maps of the circle and the all-important Morse-Smale diffeomorphisms, are treated in this chapter also. The author introduces the reader briefly to the idea of genericity when discussing Morse-Smale diffeomorphisms. Kneading theory, so important in the mathematical theory of dynamical systems, is introduced here also.
In chapter 2, the author generalizes the results to higher dimensions, and begins with a review of linear algebra and some results from multivariable calculus, such as the implicit function theorem and the contraction mapping theorem. This is followed by a treatment of the dynamics of linear maps in two and three dimensions. Whereas the canonical example of one-dimensional dynamics is represented by the logistic map, in higher-dimensional dynamics this is represented by the Smale horseshoe map. The author carefully constructs this map and details its properties. Then he takes up the hyperbolic toral automorphisms (or Anosov systems as they are called in some books). Both the Smale horseshoe map and the toral automorphisms are excellent, easily understandable examples of higher dimensional dynamics and the associated symbolic dynamics.
The concept of an attractor is also treated in chapter 2 in the context of the solenoid and the Plykin attractor. Both of these are of purely mathematical interest, but by studying them the physicist reader can get a better understanding of what to look for in actual physical examples of attractors (or the more exotic concept of a strange attractor). The author also gives a proof of the stable manifold theorem in dimension two. This is the best part of the book, for this theorem is rarely proved in textbooks on chaotic dynamics, the proof being delegated to the original papers. However, the proof in these papers is extremely difficult to get through, and so the author has given the reader a nice introduction to this important result, even though it is done only in two dimensions. This is followed by a very understandable discussion of Morse-Smale diffeomorphisms. In addition, the author introduces the Hopf bifurcation, of upmost importance in applications, and introduces the Henon map as an application of the results obtained so far.
The last chapter of the book is a brief overview of complex analytic dynamics. Complex dynamical systems are very important from a mathematical point of view, and they have fascinating connections with number theory, cryptography, algebraic geometry, and coding theory. The author reviews some elementary complex analysis and then reintroduces the quadratic maps but this time over the complex plane instead of the real line. The Julia set is introduced, and the reader who has not seen the computer graphical images of this set should peruse the Web for these images, due to their beauty. The geometry of the Julia set and the associated complex polynomial maps are given a fairly detailed treatment by the author in the space provided.
The best starting point........2000-06-25
This book covers almost every aspect of theory of discrete dynamical systems and by far the easiest explains and proofs with useful exercises, anyone with solid calculus and linear algebra background shouldn't have any problem absorbing this material and is highly recommended to whom wants to know about the theory of chaos from the scratch.
The best starting point........2000-06-25
This book covers almost every aspect of theory of discrete dynamical systems and by far the easiest explains and proofs with useful exercises, anyone with solid calculus and linear algebra background shouldn't have any problem absorbing this material and is highly recommended to whom wants to know about the theory of chaos from the scratch.
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
Largely self-contained, this is an introduction to the mathematical structures underlying models of systems whose state changes with time, and which therefore may exhibit "chaotic behavior." The first portion of the book is based on lectures given at the University of London and covers the background to dynamical systems, the fundamental properties of such systems, the local bifurcation theory of flows and diffeomorphisms and the logistic map and area-preserving planar maps. The authors then go on to consider current research in this field such as the perturbation of area-preserving maps of the plane and the cylinder. The text contains many worked examples and exercises, many with hints. It will be a valuable first textbook for senior undergraduate and postgraduate students of mathematics, physics, and engineering.
Customer Reviews:
Excellent Introduction to the Subject.......2001-05-24
covers most important areas of the subject with a clear yet rigorous approach. Advanced text better suited for graduate students in applied math. It promises as a must for anyone serious about the subject
Excellent focus on what is important.......1997-11-29
Dynamical systems is a vast subject to which no single book can provide an adequate introduction, but the authors do an excellent job of focusing on what is important and avoiding the temptation to go off on enticing tangents. Their treatment is clear, and this book is highly recommended for any student seeking a solid foundation for further work.
Average customer rating:
- First Aid to those outside the EE community
- Systems theory done right!
|
Introduction to Mathematical Systems Theory : A Behavioral Approach (Texts in Applied Mathematics, Vol. 26)
Jan Willem Polderman , and
Jan C. Willems
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover
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ASIN: 0387982663 |
Book Description
This is a book about modelling, analysis and control of linear time- invariant systems. The book uses what is called the behavioral approach towards mathematical modelling. Thus a system is viewed as a dynamical relation between manifest and latent variables. The emphasis is on dynamical systems that are represented by systems of linear constant coefficients. In the first part of the book the structure of the set of trajectories that such dynamical systems generate is analyzed. Conditions are obtained for two systems of differential equations to be equivalent in the sense that they define the same behavior. It is further shown that the trajectories of such linear differential systems can be partitioned in free inputs and bound outputs. In addition the memory structure of the system is analyzed through state space models. The second part of the book is devoted to a number of important system properties, notably controllability, observability, and stability. An essential feature of using the behavioral approach is that it allows these and similar concepts to be introduced in a representation-free manner. In the third part control problems are considered, more specifically stabilization and pole placement questions. This text is suitable for advanced undergraduate or beginning graduate students in mathematics and engineering. It contains numerous exercises, including simulation problems, and examples, notably of mechanical systems and electrical circuits.
Customer Reviews:
First Aid to those outside the EE community.......2006-12-02
The book starts off with an extremely useful modeling term: "Exclusion Law." This term describes a notion that any closed system of physical laws/equations (Newton's, etc.) state what's included (possible) and what's excluded (impossible) from its (a certain set of physical laws/equations) interpretation of reality.
Most of the concepts in this text will be nothing new to those who already have a bachelor of science in electrical engineering. Still, learning from the author's choice of words and his arrangement of materials will provide one with a very effective way of communicating modeling ideas to those outside the EE community or to those who are somehow academically juvenile (for example, speaking with someone who had to suffer a poor signals and systems instructor or with anyone who had to deal with a similar academic situation).
Here's a quote from page 12:
"Now it becomes necessary to consider two cases:"
From page 1:
"We view a mathematical model as an exlcusion law. A mathematical model expresses the OPINION that some things can happen, are possible, while others cannot, are declared impossible. Thus Kepler claims that planetary orbits that do not satisfy his three famous laws are impossible. In particular, he judges nonelliptical orbits as unphysical. ... Economic production functions tell us that certain amounts of raw materials, capital, and labor are needed in order to manufacture a finished product: it prohibits the creation of finished products unless the required resources are available."
And from page 8:
"Dynamical Systems.... The adjective dynamical refers to phenomena with a delayed reaction, phenomena with an aftereffect, with transients, oscillations, and, perhaps, an approach to equilibrium. ...a mathematical model in which the objects of interest are functions of time..."
Systems theory done right!.......2004-04-06
This is an excellent book on mathematical control theory. As opposed to the classical state-space approach, the behavioural theory gives a more fundamental and natural way of looking at physical systems. The book deals with the notions of controllability, observability, stability and feedback in a beautiful mathematical framework. A more appropriate title would be "Systems theory done right"!
Average customer rating:
|
Introduction to Hamiltonian Dynamical Systems and the N-Body Problem (Applied Mathematical Sciences)
Kenneth R. Meyer , and
Glen R. Hall
Manufacturer: Springer
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Binding: Hardcover
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ASIN: 038797637X |
Book Description
This book provides a broad introduction to the subject of dynamical systems, suitable for a one or two-semester graduate course. In the first chapter, the authors introduce over a dozen examples, and then use these examples throughout the book to motivate and clarify the development of the theory. Topics include topological dynamics, symbolic dynamics, ergodic theory, hyperbolic dynamics, one-dimensional dynamics, complex dynamics, and measure-theoretic entropy. The authors top off the presentation with some beautiful and remarkable applications of dynamical systems to areas such as number theory, data storage, and internet search engines.
Download Description
This book provides a broad introduction to the subject of dynamical systems, suitable for a one- or two-semester graduate course. In the first chapter, the authors introduce over a dozen examples, and then use these examples throughout the book to motivate and clarify the development of the theory. Topics include topological dynamics, symbolic dynamics, ergodic theory, hyperbolic dynamics, one-dimensional dynamics, complex dynamics, and measure-theoretic entropy. The authors top off the presentation with some beautiful and remarkable applications of dynamical systems to such areas as number theory, data storage, and Internet search engines. This book grew out of lecture notes from the graduate dynamical systems course at the University of Maryland, College Park, and reflects not only the tastes of the authors, but also to some extent the collective opinion of the Dynamics Group at the University of Maryland, which includes experts in virtually every major area of dynamical systems.
Customer Reviews:
Great Survey Book.......2006-02-19
So far I have found this book to include many of the classic and meaningful examples of Dynamical Systems, and has appropriate exercise for a graduate level course. So, I'll give it a good rating based soley on its use for researchers and advanced math students. However, it is very terse. Do not pick this up expecting a lot of discussion about the topics. Most of the time it is basically a list of definitions and theorem/proof's with exercises. Because of that it is great for use in a class, but might be a little difficult for independent learning. It could use a few more pictures, too.
Book Description
This volume is intended for advanced undergraduate or first-year graduate students as an introduction to applied nonlinear dynamics and chaos. The author has placed emphasis on teaching the techniques and ideas that will enable students to take specific dynamical systems and obtain some quantitative information about the behavior of these systems. He has included the basic core material that is necessary for higher levels of study and research. Thus, people who do not necessarily have an extensive mathematical background, such as students in engineering, physics, chemistry, and biology, will find this text as useful as students of mathematics.
This new edition contains extensive new material on invariant manifold theory and normal forms (in particular, Hamiltonian normal forms and the role of symmetry). Lagrangian, Hamiltonian, gradient, and reversible dynamical systems are also discussed. Elementary Hamiltonian bifurcations are covered, as well as the basic properties of circle maps. The book contains an extensive bibliography as well as a detailed glossary of terms, making it a comprehensive book on applied nonlinear dynamical systems from a geometrical and analytical point of view.
Customer Reviews:
Great reference or grad school level course text on general nonlinear dynamics.......2007-04-15
This book served as the "hidden basis" for a course in nonlinear dynamics by the late John David Crawford back at the University of Pittsburgh (the overt basis was Glendinning's book, which has proved less appealing as a reference). It's subsequently been useful to me in its treatment of Melnikov's method, and to review ideas in bifurcation theory.
As the other reviewer pointed out, it is weak in the section on symbolic dynamics. In its defense, I only know of one book which treats symbolic dynamics in a way that isn't utterly confusing, so perhaps leaving a lot of it out helps keep the student on track towards what the author is trying to present. Certainly, if he stuck to his theorem heavy style, one could get very lost in symbolic dynamics land. I'll also complain he never mention's Painleve's property. There are probably deep "theorist" reasons I'll never understand for his not mentioning this weird little thing. I hear the full treatment of Painleve's property is pretty complex, but I have always found it very helpful in understanding what integrability really is, in my "seat of the pants" way. I also would have liked more detail on Peixoto's theorem. Sure it's only useful in R2; if you're on the 'applied' side of things (or a student, learning by examining practical examples) -how often will you leave R2-land?
These complaints are minor, and they're probably effectively complaints that the book's author has a different purpose in mind than I would for writing such an introductory text, were I actually qualified to do such a thing. Wiggins writes very clearly, and he writes for physicists rather than mathematicians, and brings an amateur in the subject to a fairly high level of sophistication by the end of the text. The problem sets are also excellent.
Effective overview of a useful subject.......2001-06-10
The subject of dynamical systems has been around for over a century now, having been defined by Henri Poincare in the early 1900s, but having its roots in Hamiltonian and Lagrangian mechanics in the 19th century. In this book ths author has done a fine job of overviewing the subject of dynamical systems, particularly with regards to systems that exhibit chaotic behavior. There are 292 illustrations given in the book, and they effectively assist in the understanding of a sometimes abstract subject.
After a brief introduction to the terminology of dynamical systems in Section 1.1, the author moves on to as study of the Poincare map in the next section. Recognizing that the construction of the Poincare map is really an art rather than a science, the author gives several examples of the Poincare map and discusses in detail the properties of each. Structural stability, genericity, transversality are defined, and, as preparation for the material later on, the Poincare map of the damped, forced Duffing oscillator is constructed. The later system serves as the standard example for dynamical systems exhibiting chaotic behavior.
The simplification of dynamical systems by means of normal forms is the subject of the next part, which gives a thorough discussion of center manifolds. Unfortunately, the center manifold theorem is not proved, but references to the proof are given.
Local bifurcation theory is studied in the next part, with bifurcations of fixed points of vector fields and maps given equal emphasis. The author defines rigorously what it means to bifurcate from a fixed point, and gives a classification scheme in terms of eigenvalues of the linearized map about the fixed point. Most importantly, the author cautions the reader in that dynamical systems having time-dependent parameters and passing through bifurcation values can exhibit behavior that is dramatically different from systems with constant parameters. He does give an interesting example that illustrates this, but does not go into the singular perturbation theory needed for an effective analysis of such systems.
An introduction to global bifurcations and chaos is given in the next part, which starts off with a detailed construction of the Smale horseshoe map. Symbolic dynamics, so important in the construction of the actual proof of chaotic behavior is only outlined though, with proofs of the important results delegated to the references. The Conley-Moser conditions are discussed also, with the treatment of sector bundles being the best one I have seen in the literature. The theory is illustrated nicely for the case of two-dimensional maps with homoclinic points. The all-important Melnikov method for proving the existence of transverse homoclinic orbits to hyperbolic periodic orbits is discussed and is by far one of the most detailed I have seen in the literature. The author employs many useful diagrams to give the reader a better intuition behind what is going on. He employs also the pips and lobes terminology of Easton to study the geometry of the homoclinic tangles. Homoclinic bifurcation theory is also treated in great detail. This is followed by an overview of the properties of orbits homoclinic to hyperbolic fixed points. A brief introduction to Lyapunov exponents and strange attractors is also given.
This book has served well as a reference book and should be useful to students and other individuals who are interested in going into this area. It is a subject that has found innumerable applications, and it will continue to grow as more tools and better computational facilities are developed to study the properties of dynamical systems.
Book Description
This book gives an introduction into the ideas of dynamical systems. Its main emphasis is on the types of behavior which nonlinear systems of differential equations can exhibit.
It is divided into two parts which can be read in either order: the first part treats the aspects coming from systems of nonlinear ordinary differential equations, and the second part is comprised of those aspects dealing with iteration of a function.
For professionals with a strong mathematics background.
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