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Discover a wide range of findings in quantitative complex system science that help us make sense of our complex world. Written at an introductory level, the book provides an accessible entry into this fascinating and vitally important subject.

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In the new edition of this classic textbook Ed Ott has added much new material and has significantly increased the number of homework problems. The most important change is the addition of a completely new chapter on control and synchronization of chaos. Other changes include new material on riddled basins of attraction, phase locking of globally coupled oscillators, fractal aspects of fluid advection by Lagrangian chaotic flows, magnetic dynamos, and strange nonchaotic attractors. ... Read more

Customer Reviews (6)

Echo...echo...echo...
I can only echo the comments made by other reviewers: excellent, well-paced introduction that focuses on the meat of the subject and leaves out all the pretty-picture stuff.Suitably pitched at non-mathematicians who are drawn to this fascinating subject, it eschews the formal theorem-proof format and carefully explains concepts, then applies them.If you are scientifically literate, I highly recommend this textbook.Genuinely useful.

good introduction to chaos

The book is a good introduction to chaos and the new edition has a chapter on synchronization that is a good review of the literature

no pretty fractal diagrams, but good explanations
Ott gives a very clear description of the concept of chaos or chaotic behaviour in a dynamical system of equations. Where often these equations are nonlinear. While containing rigour, the text proceeds at a pace suitable for a non-mathematician in the physical sciences. In other words, it is not at a very formal level, like the epsilon-delta approach to teaching calculus. The concepts are also backed by well drawn diagrams, that illustrate key points.

The book does not have the lovely diagrams of Julia sets and fractals, that you often see in other books on this subject. Those are certainly pretty and useful. But Ott's book concentrates on the ideas.

Very good for Physicists
The best book on chaos in Dynamical Systems for physicists: clear, well written, contains the right things and does not waste time treating less necessary sections on the subject. Particularly valuable is the part on Entropy, Information and strange attractors. A good choice is to use it together with V.I. Arnold's CM. Contains also a final part on connections between QM and chaos.

Good for physicists
A good introduction to chaos in dynamical systems for physicists. The emphasis is not on time-series analysis or nonlinear systems, but chaos in "physical" systems (in the sense of applications in physics). A good reading for undergrads in physics and maths. One of the best starters for getting deeper into chaos theory... ... Read more

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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
... Read more

Customer Reviews (4)

A new version of a classic book
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
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
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
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. ... Read more

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Customer Reviews (3)

A must-have book
I think, this is a must-have for graduate students who want to research in the area of nonlinear dynamical systems and control. It covers plenty of topics in the area, but, I need to say that this book is mainly about analysis part of nonlinear system, not control of nonlinear system. So, if you want to learn much more about nonlinear control, then you may consider Sastry's book or a more applied one, Slotine's book. Prof. Haddad uses this book as the text book of his Aerospace Nonlinear Control class in Georgia Tech and he teaches too much things that is not covered in the book. One more thing, there are a few typos and errors in the book, so before reading to it, it is better to download the errata from Prof. Haddad's homepage.

Only one word: Excellent
It is just an excellent book. It is the best in its area. I strongly suggest this book to any control scientist and engineer.

a masterpiece

This is a monumental work. If you're dealing with dynamical systems this is a must. Of course, it's full of maths but this is not a book intended for the beginner. It assumes a solid knowledge of algebra and analysis. In some aspects it's a true encyclopaedic work. It contains an excellent statement of Zubov's theorem that I haven't found nowhere else. ... Read more

Product Description
Here is an introduction to dynamical systems and ergodic theory with an emphasis on smooth actions of noncompact Lie groups. The main goal is to serve as an entry into the current literature on the ergodic theory of measure preserving actions of semisimple Lie groups for students who have taken the standard first year graduate courses in mathematics.The author develops in a detailed and self-contained way the main results on Lie groups, Lie algebras, and semisimple groups, including basic facts normally covered in first courses on manifolds and Lie groups plus topics such as integration of infinitesimal actions of Lie groups.He then derives the basic structure theorems for the real semisimple Lie groups, such as the Cartan and Iwasawa decompositions and gives an extensive exposition of the general facts and concepts from topological dynamics and ergodic theory, including detailed proofs of the multiplicative ergodic theorem and Moore's ergodicity theorem. This book should appeal to anyone interested in Lie theory, differential geometry and dynamical systems. ... Read more

Product Description
In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics. This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience. It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology.

Dynamical Systems in Neuroscience presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons. It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties.

The book introduces dynamical systems, starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems. Each chapter proceeds from the simple to the complex, and provides sample problems at the end. The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians. Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines.

Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum—or taught by math or physics department in a way that is suitable for students of biology. This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience.

An additional chapter on synchronization, with more advanced material, can be found at the author's website, www.izhikevich.com.

Computational Neuroscience series ... Read more

Customer Reviews (6)

Great book for the mathematics of neuroscience
Being a biomathematician and neuroscientist, I found that Izhikevich's book "Dynamical Systems in Neuroscience" is a great reference to broaden my understanding of mathematical neuroscience and neurophysiology, and in particular, neural modeling, nonlinear dynamics and the mathematics involved between the brief bursts of neural activity. I recommend it to every neuroscientist in the field.

Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Computational Neuroscience)
I could know the new model of human brain which I have not ever seen. It is very interesting.

Beautiful book of dynamical system of neurons
This is an excellent book on application of 2-D dynamical system theory to (minimal) spiking neuron models. I highly recommend it for electrophysiologist who wants to learn more about what they observe, and to computational neuroscientists in general.
Prior exposure to dynamical systems and neuroscience is helpful.

Book on the dynamcis of neurons
This book gives an understanding of how the dynamics of neurons work. It gives an insight to many different types of models. However, you would require a neuroscience background before understanding it more in depth.

So you think you are afraid of some math?
This book encapsulates in a single text a large body of knowledge by the author and others over the past two decades on the use of geometrical techniques to both classify and study a large range of single neuron models. While much of this material is known to "experts" in the field, the value of this text is i1) teaching this dynamical systems perspective on single neuron dynamics to generations of new students and 2) educating non-mathematicians into both the utility and use of these theories.Many other texts and papers on this topic leave non-mathematicians "in the dust" shortly after the introduction, but Eugene's excellent use of figures to explain concepts geometrically as well as mathematically enables a PhD student in engineering or quantitative biology to fully appreciate what is going on, not to mention seasoned experimentalists. ... Read more

Product Description
This book presents the first comprehensive treatment of a rapidly developing area with many potential applications: the theory of monotone dynamical systems and the theory of competitive and cooperative differential equations. The primary aim is to provide potential users of the theory with techniques, results, and ideas useful in applications, while at the same time providing rigorous proofs. The main result of the first two chapters, which treat continuous-time monotone dynamical systems, is that the generic orbit converges to an equilibrium. The next two chapters deal with autonomous, competitive and cooperative, ordinary differential equations: every solution in the plane has eventually monotone components, and the Poincare-Bendixson theory in three dimensions is discussed. Two chapters examine quasimonotone and nonquasimonotone delay differential equations, and the book closes with a discussion of applications to quasimonotone systems of reaction-diffusion type. Throughout, Smith discusses applications of the theory to many mathematical models arising in biology. An extensive guide to the literature is provided at the end of each chapter. Requiring a background in dynamical systems at the level of a first graduate course, this book would be suitable as a graduate text for a topics course. ... Read more

Product Description
This text grew out of notes from a graduate course taught to students in mathematics and mechanical engineering. The goal was to take students who had some basic knowledge of differential equations and lead them through a systematic grounding in the theory of Hamiltonian systems, an introduction to the theory of integrals and reduction. Poincaré’s continuation of periodic solution, normal forms, and applications of KAM theory. There is a special chapter devoted to the theory of twist maps and various extensions of the classic Poincaré-Birkhoff fixed point theorem. ... Read more

Product Description
This textbook presents a systematic study of the qualitative and geometric theory of nonlinear differential equations and dynamical systems.Although the main topic of the book is the local and global behavior of nonlinear systems and their bifurcations, a thorough treatment of linear systems is given at the beginning of the text. All the material necessary for a clear understanding of the qualitative behavior of dynamical systems is contained in this textbook, including an outline of the proof and examples illustrating the proof of the Hartman-Grobman theorem, the use of the Poincare map in the theory of limit cycles, the theory of rotated vector fields and its use in the study of limit cycles and homoclinic loops, and a description of the behavior and termination of one-parameter families of limit cycles.In addition to minor corrections and updates throughout, this new edition includes materials on higher order Melnikov theory and the bifurcation of limit cycles for planar systems of differential equations, including new sections on Francoise's algorithm for higher order Melnikov functions and on the finite codimension bifurcations that occur in the class of bounded quadratic systems. ... Read more

Customer Reviews (2)

Very good graduate ODEs text, with some flaws
Perko's book is one of the best books that gives an advanced introduction to dynamical systems from the point of differential equations.Many other good books tread the same ground, without emphasizing the connection to ODEs.Perko's text is particularly strong in several respects.First, the dynamical systems it considers are almost always expressed in terms of underlying differential equations.Second, it gives proofs or outlines of proofs of most major theorems used in this field.Third, it covers the most important topics, including: local theory of hyperbolic equilibria, invariant manifolds, Hamiltonian systems, flows on R^2, stability theory, and elementary bifurcations.Also reviewed are the results from linear systems theory, in a particularly well-written and easy to follow introductory chapter.Another great feature of this book is its solid coverage of center manifold theory, which is an important and somewhat difficult topic.

There are a couple of problems with this book.The proofs to some of the major theorems are occasionally abstruse or poorly derived.Perko seems to bend over backwards to give analytical proofs, when algebraic or topological proofs might be easier.Many of the problems reuse the same elementary example equations.This is OK insofar as it allows the reader to see how different techniques can be used to analyze the same systems, but it limits the reader's exposure to the full variety of interesting dynamical systems that can arise in practice.The author also tends to emphasize polynomial vector fields, which is a potential limitation.Occasionally the problems are significantly more difficult than the examples worked in the text.

Overall, Perko's text is a very solid introduction to advanced ODEs and continuous dynamics.It is especially well-suited for scientists and engineers who want a readable introduction to the qualitative theory of ODEs.

A Book on Advanced Dynamical Systems
This book is a useful textbook for advanced courses on differential equations and dynamical systems for senior undergraduate students or first year graduate students.

The book presents a systematic study of thequalitative and geometric theory of nonlinear differential equations anddynamical systems.

The book has a sketch of the proof of theHartman-Grobman Theorem which was useful for my second undergraduate courseon dynamical systems and nonlinear differential equations.

I liked thebook and I am quite sure it will become a classic textbook on this veryuseful branch of Math that has so many old and new applications in Physics,Economics and Finance. ... Read more

Product 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. ... Read more

Customer Reviews (2)

Great expectations, unmet
The author is a very good mathematician (and a grandfather of Google) so I was expecting a short and lucid introduction to dynamical system. Imagine my sadness when I found the book barely comprehensible. Apparently, the author learned his writing skills in Russian in the 60s, where paper was scarce, and any sort of explanation was viewed a waste thereof. If you actually want to understand dynamics, Katok/HasselblattIntroduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications) is vastly superior.

Great Survey Book
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. ... Read more

Product Description
Written by one of the most respected mathematicians in the field, this book conveys the essential mathematical ideas in dynamical systems using a combination of theory and computer experimentation. This introductory look at dynamical systems includes investigating the rates of approach to attracting and indifferent fixed points to the discovery of Feigenbaum's constant; exploring the window structure in the orbit diagram; and understanding the periods of the bulbs in the Mandelbrot set. ... Read more

Customer Reviews (4)

There Are Better Choices
This is a fine text, and I was able to follow it fairly easily.However, it is rather dated (1992) and there have been improvements in the subject in several areas.I found Steven Strogatz's "Nonlinear Dynamics And Chaos" (2001) a significantly better book for both content and readability.

Has soul - but no cigar
The content of the book is great, only a little short and condensed. I wish more examples were available. Also I wish sometimes author would approach a topic from different angles, instead of leaving us behind after plowing through something...

Very poor print quality and layout, the colored pages are FALLING OUT from the book. Definitions are not always apparent and it is difficult to discern important parts in the book. Graphs are "grayscale" and are also of very poor quality.
I guess this is all due to the book being old, and it is printed on-demand, as shows in the font/typeface - which looks like it's been "xeroxed".

Nice begginers text
This text is a great begginners guide to chaotic systems, it provides very clear explanations and proofs as well as some examples to help you along.

Excellent book.Explains concepts clearly.
I went from knowing absolutlely nothing about dynamical systems to being able to look at a point on the Mandelbrot Set and visualize what the corresponding Julia Set looks like.Ever wonder why weather cannot be predicted accurately??Read this book... ... Read more

Product Description
The 3n+1 function T is defined by T(n)=n/2 for n even, and T(n)=(3n+1)/2 for n odd. The famous 3n+1 conjecture, which remains open, states that, for any starting number n>0, iterated application of T to n eventually produces 1. After a survey of theorems concerning the 3n+1 problem, the main focus of the book are 3n+1 predecessor sets. These are analyzed using, e.g., elementary number theory, combinatorics, asymptotic analysis, and abstract measure theory. The book is written for any mathematician interested in the 3n+1 problem, and in the wealth of mathematical ideas employed to attack it. ... Read more

Product 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. ... Read more

Customer Reviews (4)

Great, advanced intro to dynamical systems
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
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
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
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. ... Read more

Product Description
Both fractal geometry and dynamical systems have a long history of development and have provided fertile ground for many great mathematicians and much deep and important mathematics. These two areas interact with each other and with the theory of chaos in a fundamental way: many dynamical systems (even some very simple ones) produce fractal sets, which are in turn a source of irregular ``chaotic'' motions in the system. This book is an introduction to these two fields, with an emphasis on the relationship between them. The first half of the book introduces some of the key ideas in fractal geometry and dimension theory--Cantor sets, Hausdorff dimension, box dimension--using dynamical notions whenever possible, particularly one-dimensional Markov maps and symbolic dynamics. Various techniques for computing Hausdorff dimension are shown, leading to a discussion of Bernoulli and Markov measures and of the relationship between dimension, entropy, and Lyapunov exponents. In the second half of the book some examples of dynamical systems are considered and various phenomena of chaotic behaviour are discussed, including bifurcations, hyperbolicity, attractors, horseshoes, and intermittent and persistent chaos. These phenomena are naturally revealed in the course of our study of two real models from science--the FitzHugh-Nagumo model and the Lorenz system of differential equations. This book is accessible to undergraduate students and requires only standard knowledge in calculus, linear algebra, and differential equations. Elements of point set topology and measure theory are introduced as needed. This book is a result of the MASS course in analysis at Penn State University in the fall semester of 2008. ... Read more

Customer Reviews (1)

Not aparticularly great book
This book is not supposed to teach you chaos/fractals from scratch. The book is too shabby for self-study. It has a collection of lectures that are not very neatly organized. The only reason I purchased this book was because I took a class by the author. If you want to learn Chaos and Fractals, there are better books available. ... Read more

Product Description
This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces; the authors have included a self-contained treatment of that subject. ... Read more

Customer Reviews (6)

A valuable source
This book (the original version) has all the basics to introduce the future differential equations/dynamical systems researchers into the field. Written by authorities in the field (Hirsch and Smale,) this text offers a wide variety of topics, including linear systems, local and global stability theory for non-linear systems, and applications to physics and biology. As an added treat, the inclusion of basic linear algebra and operator theory makes this a rather self-contained work. The dedicated reader will not be disappointed - the material is well organised with sufficient level of detail, illustration, and exercises.

This is not a recipe book
I can see that this is not the book for you if you want to solve a particular differential equation.But in terms of understanding the field of dynamical systems, there is no rival.This book is a pleasure to read, for the first time I understood the importance and beauty of linear algebra.Academic Press says that this is their most successful mathematics text, and it is not hard to see why.I wish more texts were as clearly written and as beautiful to read.

A complete waste
This is not a book, it's a piece of trash!!! This so called book is a meaningless mess which wasn't even understandable for the person who had a PhD in math and was teaching our class. Do NOT bother with this nonsense. if you want to learn something just read Ordinary Differential Equations by V. I. Arnold.I would have given no star if I could!!!Just go with Arnold's and I'll be WAY better off.

Not for the average undergrad!
As a senior undergrad majoring in math and economics, this book is everything but an easy read. To all fellow undergrads who are not math superheroes (that should about 75% of us), if you happen to come across this book in an upcoming course description, it may be a good idea to look for alternative. Currently, I'm looking for another book that I may be able to use as a supplement to get me through this course with a passing grade. Up to this point in my math career, I have never come across a text as ungraspable as this one; this is unfortunate since it appears that there is a lot of knowledge and content on the pages.

Thorough and solid introduction
This is the book from which I was introduced to dynamical systems some twenty-odd years ago. It's a thorough introduction that presumes a basic knowledge of multivariate differential calculus but is pretty well self-contained as far as linear algebra is concerned. Rigorous but readable, it provides a foundational understanding of n-dimensional linear dynamical systems and their basic exponential solution.

But my opinions won't be as helpful to the Amazon math shopper as a simple listing of what's in the book. So here's the table of contents.

Chapter 1: First Examples

Chapter 2: Newton's Equation and Kepler's Law

Chapter 3: Linear Systems with Constant Coefficiants and Real Eigenvalues

Chapter 4: Linear Systems with Constant Coefficients and Complex Eigenvalues

Chapter 5: Linear Systems and Exponentials of Operators

Chapter 6: Linear Systems and Canonical Forms of Operators

Chapter 7: Contractions and Generic Properties of Operators

Chapter 8: Fundamental Theory

Chapter 9: Stability of Equilibria

Chapter 10: Differential Equations for Electric Circuits

Chapter 11: The Poincare-Bendixson Theorem

Chapter 12: Ecology

Chapter 13: Periodic Attractors

Chapter 14: Classical Mechanics

Chapter 15: Nonautonomous Equations and Differentiability of Flows

Chapter 16: Perturbation Theory and Structural Stability

Afterword

Appendix I: Elementary Facts

Appendix II: Polynomials

Appendix III: On Canonical Forms

Appendix IV: The Inverse Function Theorem

References

Answers to Selected Problems ... Read more

Product Description

Perturbation theory and in particular normal form theory has shown strong growth in recent decades. This book is a drastic revision of the first edition of the averaging book. The updated chapters represent new insights in averaging, in particular its relation with dynamical systems and the theory of normal forms. Also new are survey appendices on invariant manifolds. One of the most striking features of the book is the collection of examples, which range from the very simple to some that are elaborate, realistic, and of considerable practical importance. Most of them are presented in careful detail and are illustrated with illuminating diagrams.

Product Description
The area of hybrid dynamical systems (HDS) represents a difficult and exciting challenge to control engineers and is referred to as "the control theory of tomorrow" because of its future potential for solving problems. This relatively new discipline bridges control engineering, mathematics, and computer science. There is now an emerging literature on this topic describing a number of mathematical models, heuristic algorithms, and stability criteria. However, presently there is no systematic theory of HDS. "Hybrid Dynamical Systems" focuses on a comprehensive development of HDS theory and integrates results established by the authors.

The work is a self-contained informative text/reference, covering several theoretically interesting and practically significant problems concerning the use of switched controllers and examining the sensor scheduling problem. The emphasis is on classes of uncertain systems as models for HDS.

Features and topics:

* Focuses on the design of robust HDS in a logical and clear manner

* Applies the hybrid control systems framework to two classical robust control problems: design of an optimal stable controller for a linear system and simultaneous stabilization of a collection of plants

* Presents a detailed treatment of stability and H-infinity control problems for a class of HDS

* Covers recent original results with complete mathematically rigorous proofs

Researchers and postgraduate students in control engineering, applied mathematics, and theoretical computer science will find this book covers the latest results on this important area of research. Advanced engineering practitioners and applied researchers working in areas of control engineering, signal processing, communications, and fault detection will find this book an up-to-date resource. ... Read more

Product Description
This introduction to dynamical systems theory treats both discrete dynamical systems and continuous systems. Driven by numerous examples from a broad range of disciplines and requiring only knowledge of ordinary differential equations, the text emphasizes applications and simulation utilizing MATLAB(r), Simulink(r), and the Symbolic Math toolbox.

Beginning with a tutorial guide to MATLAB(r), the text thereafter is divided into two main areas. In Part I, both real and complex discrete dynamical systems are considered, with examples presented from population dynamics, nonlinear optics, and materials science. Part II includes examples from mechanical systems, chemical kinetics, electric circuits, economics, population dynamics, epidemiology, and neural networks. Common themes such as bifurcation, bistability, chaos, fractals, instability, multistability, periodicity, and quasiperiodicity run through several chapters. Chaos control and multifractal theories are also included along with an example of chaos synchronization. Some material deals with cutting-edge published research articles and provides a useful resource for open problems in nonlinear dynamical systems.

Approximately 330 illustrations, over 300 examples, and exercises with solutions play a key role in the presentation.Additional applications and further links of interest are also available at the author's website.

The hands-on approach of "Dynamical Systems with Applications using MATLAB(r)" engages a wide audience of senior undergraduate and graduate students, applied mathematicians, engineers, and working scientists in various areas of the natural sciences.

Reviews of the author's published book "Dynamical Systems with Applications using Maple(r)":

"The text treats a remarkable spectrum of topics...and has a little for everyone. It can serve as an introduction to many of the topics of dynamical systems, and will help even the most jaded reader, such as this reviewer, enjoy some of the interactive aspects of studying dynamics using Maple(r)." - U.K. Nonlinear News

"...will provide a solid basis for both research and education in nonlinear dynamical systems." - The Maple Reporter ... Read more

Customer Reviews (1)

Review for Lynch's dynamical systems matlab edition text
I have been using the text for the last year or so. My familiarity with dynamical systems/non linear dynamics is over eight years now. I use this text as a reference for quick look-up on some of the more elementary techniques. Explanations are scarce and insufficient. Some minor inaccuracies are present, but can be easily disregarded if you are not particularly credulous/naive. You will have to consider Strogatz or J M T Thompson/H B Stewart texts if you want to understand what's happening physically. Or Nayfeh's body of texts on nonlinear physical systems for numerical techniques in simulating such systems or Phillip Holmes' and others for mathematical theory or geometric topology. But I love the book for its easily accessible presentation format for students and the succinctness of its prose. This book is certainly not for people from the mathematical side of dynamical systems, but great for undergrad or beginning grad level students in engineering or physics. This is mostly a cookbook, so don't expect brilliant flavors, just that you can put a meal on the table everyday. ... Read more

Product Description
This book is an introduction to the field of dynamical systems, in particular, to the special class of Hamiltonian systems. The authors aimed at keeping the requirements of mathematical techniques minimal but giving detailed proofs and many examples and illustrations from physics and celestial mechanics. After all, the celestial $N$-body problem is the origin of dynamical systems and gave rise in the past to many mathematical developments.Jürgen Moser (1928-1999) was a professor at the Courant Institute, New York, and then at ETH Zurich. He served as president of the International Mathematical Union and received many honors and prizes, among them the Wolf Prize in mathematics. Jürgen Moser is the author of several books, among them Stable and Random Motions in Dynamical Systems. Eduard Zehnder is a professor at ETH Zurich. He is coauthor with Helmut Hofer of the book Symplectic Invariants and Hamiltonian Dynamics.Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University. ... Read more

Product Description

This is the most thorough treatment of normal forms currently existing in book form. There is a substantial gap between elementary treatments in textbooks and advanced research papers on normal forms. This book develops all the necessary theory 'from scratch' in just the form that is needed for the application to normal forms, with as little unnecessary terminology as possible.