Bifurcation

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.

As a postgrad student searching for textbooks to cover all those things I learnt about dynamical systems in my undergrad studies, this one was one of my first choices. It is easy to read, which is surprising, considering it's a maths textbook. It is systematic in its treatment of dynamical systems. The important things are easy to find, since the index is very comprehensive and the layout of the page makes formulae stand out. The many diagrams help those of us who think about dynamics geometrically. All in all, a great book, which I use on a regular basis as a reference.

This is an unusual book in the Springer-Verlag series in one aspect: Unlike other Springer-Verlag books, the mathematical symbols and notations in this one are *not* weird unusual made-up symbols! I know this may not be something tenured professors think about, but for the rest of us who are still learning the craft, so to speak, it is important to be able to switch from source to source without looking up the authors' use of unusual symbols all the time.

This is an excellent and comprehensive book on bifurcation theory for a graduate-level course on this subject. It is also a very useful resource for research on nonlinear bifurcations. I strongly recommend this book.

As it promises in its title, this book is concerned with developing a practising knowledge of bifurcation analysis. The book starts with a very simple introduction, and takes off from there. The content is lively and straightforward. Concepts are introduced in plain language, without the technical complexity of formal mathematical jargon. Few theorems are proved, so the concepts remain in focus. Computational techniques are an important concern, and their problems. All in all, an entertaining and at once informative book to learn using a hands-on-approach. It manages to cover a fairly large amount of material, for a thorough introduction. Useful for people who are primarily interested in applications.

I am one of the authors of this book. I am sorry to say that this book does not exist. The true title of the existing book is " Topics in Bifurcation Theory and Applications". same authors.

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.

I have used Iooss and Joseph for over 20 years now, starting with the first edition back in the early 80s.

The book is for the more advanced student, one who has a bacic working knowledge of real and functional analysis. Unfortunately, these days, few engineering and physical science students have such a background. Hence, the book would be better if it contained some supporting basic material on mathematical analysis.

The first edition contained numerous "typos". While much improved, the second edition still contains too many "typo" errors.

Overall, the book is a good source of information that should be consulted by anyone interested in bifurcation theory. The book contains material (like the bifurcation of forced T-periodic solutions) not normally included in an elementary treatment of bifurcations.

John Stensby, Professor
Electrical and Computer Engineering
University of Alabama in Huntsville
Huntsville, AL

The book consists solely of exercises and hints for every exercise, which he curiously calls "answers". This book is perfect if you are looking to review geometrically-tinged algebraic structures like matrix groups, symmetry groups, and wallpaper groups. There is also some basic pure algebra in here. I don't think this book would work all that well for a student new to algebra, although someone with some backgroud in algebra can definitely get something out of the geometric chapters.

This is an exellent book on stability and bifurcation theory, from an applied math perspective. A reader could just skim and pick up a broad outline but would be better off working though at least some of the messy details to make sure that (s)he is really following the thread of the argument.

The first two chapters give a introduction to bifurcation and is easy to read and understand for a chemical engineer.
Explanation of the theory is nicely represented bij graphs.
It always keeps in mind that it is for chemical engineers and therefore does not go into detail on the numerical analysis.
The latter 2 chapter I did not use.
But overal a good book for bifurcation and modelling for chemical engineers.

hypertext novel - excellent text read and enjoy