Dynamical-Systems

It's now been exactly a decade since the time of a college course on chaos and fractals, when the book "Fractals Everywhere" by Michael F. Barnsely was used as the text for the class. Back then, riding high on the heels of a good semester of point set topology, the first half of the class for me was just a breeze as the instructor found himself having to spend the first few weeks teaching basic toplogy topics to most applied math students, before moving on to dealing with the actual geometrical chaos topics such as Hausdorff dimension and the iterated collage theorem. For a few ensuing years however, the subject of dynamical systems theory had to leap into the background as I was persuing my studies in physics subjects such as relativity theory, quantum mechanics, some other branches of mathematics, and finally moving towards specializing in both differential and algebraic geometries. However, rather recently I have regained my interest in applied analysis-related subjects such as dynamical systems and chaos, control theory, ergodic theory, symplectic geometry, and information theory. The book by Alligood, Sauer, and Yorke, was a somewhat recent recommendation by an enthusiastic college professor who had already used it to teach his classes and research students from it.

I recall at the time there was a discussion as to whether Robert Devaney's book would have made up for a better first course, but he correctly mentioned that Devaney only deals with the discrete dynamical systems, while ASY treats both discrete and continuous systems. For those of you curious to know, some of the topics discussed in the 13 chapters of ASY include: one and two dimensional maps, fixed points, iterations, sinks, sources, saddles, Lyapunov exponents, chaotic orbits, conjugacy, fractals and their dimension, chaotic attractors, measure, Lotka-Volterra models, Poincare-Bendixson theorem, Lorentz and Roessler attractors, stable manifolds and crises, homoclinic and heteroclinic points, bifurcations, and cascades, to name a few.There are also answers and solutions to the selected exercises, as well as extensive references at the back of the book, making up an ideal setting for self-study. The level and style of exposition is targeted towards an advanced undergraduate or beginning graduate student who is into applied math and/or engineering fields. The authors justly tend to emphasize concepts and applications, instead of getting bogged down in too much mathematical rigor or heavy use of the abstract machinery. All in all, an excellent first excursion/introduction to one of the most fascinating areas of applied mathematics, whether for classroom use, or merely for self-study.

When I purchased this book three years ago, I had only a rudimentary understanding of dynamical systems. Thankfully, all that was needed to get me started was some intermediate calculus and some basic college-level linear algebra. Since I had been doing both from the time I was a sophmore in high school, I had no trouble getting comfortable with it. The authors present dynamical systems in an easy-to-read style with tests that appear at the end of each chapter after you've had time to catch on.

If you're seriously thinking about getting started in dynamical systems, get this book!

I was enrolled in a course at GMU in which the draft version of this text was used. The math was not as difficult as some of the graduate texts, therefore it serves as a good intoduction for someone with as little as 2 years of undergraduate math. The challenges at the end of each chapter are more difficult than the regular problems, but they are meant to be. Many of the systems can be modeled on a spreadsheet. If you have any interest in Chaos, this book will only strengthen it.

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.

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...

An excellent text that is written in a very understandable and careful style. It gives the readers a good grasp of the fundamentals by emphasizing main ideas instead of harping on technical definitions. The bibliography at the end of the book is also a good source for readers who want to delve further into the technical literature.

This book is highly recommended to all those who aspire to enjoy the world of chaos and fractals. It is written by one of the masters of discrete dynamics and difference equations. I was sold on Devaney's book until I saw this book. Moreover, I am not fully satisfied with the explanations and the writing style in the book by Alligood, Sauer and Yorke. At last we have some genuine and realistic applications to discrete dynamical systems. The best bargain of all is the chapter on Iterated function systems; the book provides a delightful and very readable , yet mathematically honest, account. Almost all the results in the book are proved completely and clearly. I enjoyed the relaxed and lucid writing style of this author. Keep them coming and way to go Elaydi!

Discrete chaos is an excellent text. It can serve both as an introductory text in discrete-time dynamical systems (chaos theory) and as a resource for more advanced work. The book provides an accessible introduction to discrete-time dynamical systems with many interesting applications.

The book is divided into six sections. Section 1 and Section 2 introduce essential concepts for describing and analyzing chaotic discrete-time systems. Section 3 and Section 5 focus on chaos in one dimension and two dimensions, respectively. Section 4 describes the stability of two-dimensional systems. Fractals are formulated and analyzed in Section 6. Section 7 is devoted to the study of The Julia and Mandelbrot Sets.

All chapters end with excellent exercises. The book also has answers to the exercises. The book is designed to be independent of any particular computing environment.

"Discrete Chaos" resembles in many ways the highly successful 1989 introductory text by Robert Devaney on chaotic dyanamical systems. However, a great deal has happened since Devaney's text appeared, and Discrete Chaos includes some of the new finds (e.g., the new results on stability of maps on the real line and the author's own work on the converse of Sharkovsky's Theorem). Discrete Chaos also adds a new element unique to it: The author's perspective as a successful researcher and a talented expositor in the area of difference equations. This is important since it balances the attention between the familiar topological/algebraic ideas and the more purely analytical results. Equally important perhaps, is the author's talent for writing introductory level books (he also wrote the very readable "An Introduction to Difference Equations"). Discrete Chaos contains many interesting examples and helpful exercises, although the proofs of the more technical results are not given.

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.

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...

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

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.

Chaos has been a fascinating and fruitful field for scientists in the last twenty years. It has immensely deepened our understanding of nature. The ideas of basins of attraction, Julia sets, stability in chaotic systems and others have been discovered and been found quite valid.

During much of this, engineers have probably glanced at it with interest. But until recently, engineering systems did not deal with it or invoke even the presence or possibility of chaotic behaviour. Which is why this book is promising. It moves chaos firmly into the engineering sphere.

Most of the discussion is on dynamical systems, with remarks on the impact on traditional control systems theory of the presence of chaos. The mathematical treatment is fairly sophisticated, and engineering readers should probably already be quite well versed in understanding dynamical systems.

An excellent introduction to chaos for the mathematically inclined, this work offers a brief tour through the concepts and theorms behind the field of chaos and nonlinear dynamics. Having a college mathematical background is decidedly helpful in getting use out of it, but having a mathematical mindset is absolutely essential.

Very clear and intuitive upper-undergrad/graduate level text. I highly recomend it.

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.

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 read 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.

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.

This book has been a continuing source of information and guidance for 18 years now. Students and researchers in many different fields have used this book due to its breadth and detail of coverage. The book does require a fairly advanced mathematical background, but the authors do include a glossary for the reader lacking this.

Chapter one is an overview of differential equations and dynamical systems. All the concepts needed for a study of such systems are discussed in great detail and also very informally, stressing instead the understanding of the concepts, and not merely their definition. Some of the proofs of the main results, such as the Hartman-Grobman and the stable manifold theorems, are omitted however.

This is followed in Chapter 2 by a very intuitive discussion of the van der Pols equation, Duffings equation, the Lorenz equations, and the bouncing ball. Numerical calculations are effectively employed to illustrate some of the main properties of the systems modeled by these equations.

A taste of bifurcation theory follows in Chapter 3. Center manifolds are defined and many examples are given, but the proof of the center manifold theorem is omitted unfortunately. Normal forms and Hopf bifurcations are treated in detail.

Averaging methods are discussed in Chapter 4, with part of the averaging theorem proved using a version of Gronwall's lemma. Several interesting examples of averaging are given, along with a discussion of to what extent the bifurcation properties of the averaged equations carry over to the original equations. Most importantly, this chapter discusses the Melnikov function, so very important in the study of small perturbations of dynamical systems with a hyperbolic fixed point. A full proof that simple zeros of the Melnikov function imply the transversal intersection of the stable and unstable manifolds is given.

Chapter 5 moves on to results of a more purely mathematical nature, where symbolic dynamics and the Smale horseshoe map are discussed. The proofs of the stable manifold theorem and the Palis lambda lemma are, however, omitted. Markov partitions and the shadowing lemma are discussed also but the latter is not proven. The authors do however give a proof of the Smale-Birkhoff homoclinic theorem. A purely mathematical overview of attractors is given along with measure-theoretic (ergodic) properties of dynamical systems.

The (local) bifurcation theory of Chapter 3 is extended to global bifurcations in the next chapter. A very detailed discussion of rotation numbers is given but the KAM theory is only briefly mentioned. The main emphasis is on 1-dimensional maps, the Lorentz system, and Silnikov theory. The authors give a very detailed treatment of wild hyperbolic sets.

The book ends with a discussion of bifurcations from equilibrium points that have multiple degeneracies. The discussion is more motivated from a physical standpont than the last few chapters. But some interesting mathematical constructions are employed, namely the role of k-jets, which have fascinating connections with algebraic goemetry, via the "blowing-up" techniques.

The concepts in the book have proven to have enduring value in the study of dynamical systems, and this book will no doubt continue to serve students and researchers in the years to come.

Guckenheimer is one of my favourite book in nonlinear science. Another absolute reference. This books deserved to be milestone in nonlinear dynamics.

This book has clearly withstood the test of time in over 15 years of continuous publication. On my bookcase, it stands among my most treasured and well-worn classics of fluid mechanics and differential equations--Hirsch and Smale, Birkhoff and Rota, Chandrasekhar, Bachelor, Lamb, Landau and Lifschitz... It changed many of the unquestioned assumptions of many fields besides my own. It redefined the terms of many scientific debates. And, it changed my life.

I obtained Guckenheimer and Holmes' classic when it first came out in 1983. It was so clear, concise and intellectually engaging that it inspired me to wonder whether the system of equations I was studying for my Ph.D. research at the time--the governing equations of thermal convection at infinite Prandtl number (which govern plate tectonics in the earth's mantle)--might have a chaotic solution. Guckenheimer and Holmes outlined a clear methodology to find out the answer.

My advisor at the University of Chicago thought not. Only steady solutions could be admitted in the absence of external forcing due to the lack of momentum transfer--this belief was widely held at the time, despite certain oscillatory solutions found by Fritz Busse (then at UCLA) and chaotic solutions found in certain limiting cases by Andrew Fowler at Oxford.

In despair, I left my studies at Chicago to work as a Unix sysadmin at my undergraduate alma mater --Cornell, where (unbeknownst to me when I took the job) John Guckenheimer had just relocated from UCSC. Delighted to find him there, I sat in on his courses. Later, with his help, I wrote a proposal to NASA to support the completion of my thesis--with him and Donald Turcotte serving as my advisors.

The 3-year fellowship was approved, and during this time I demonstrated and published that thermal convection at infinite Prandtl number--a condition that pervades many planetary interiors including our own--is indeed chaotic in the absence of external forcing.

Prior to this, planetary convection codes primarily looked for steady state solutions. Since, numerical analysts in the field have upgraded to time-dependent models. The source of chaos at infinite Prandtle number I identified--the heat advection term--is now widely accepted as the source of what is now called "Thermal Turbulence" in planetary interiors.

The defense at Chicago was quite an event. Since my new advisors were flown in from Ithaca, you might say my thesis--The Nonlinear Dynamics of Thermal Convection at Infinite Prandtl Number--passed with flying colors. Someone at Chicago might disagree, but his opinion is irrelevant.

Demonstrating the many possible solutions to a single set of equations and showing how the choice of solution depends very sensitively on the rather poorly-constrained initial conditions of the earth--does render mantle modeling itself rather superfluous and indeed, scientifically suspect. However, many important professors who stayed in the field nonetheless continue to run their time-dependent mantle convection codes, and never cease to wonder at the fact that they all get different results. It's rather amusing, really.

When all that too has passed away, the truths so beautifully put forth in Guckenheimer and Holmes will remain. Like I said, it's a classic. Furthermore, being number 42 in its series, it's got to be the answer to the ultimate question of life, the universe and everything. Was for me, anyway.