Customer Reviews

Chaos

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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.

This book presents brilliantly the foundations to Dynamical Systems and Chaos. You need to have some Linear Algebra, Calculus and Multivariable Calculus and Differential Equations knowledge. Full of exercises, computer experiments and Challenges. I think that the text looses some substance due to the lack of presenting more or all the solutions to the Exercises. They should be solved detailed in a Solutions Manual. Don't try to e-mail the authors for more solutions, they will not get them to you. This point is the only pitty in a text that is a great companion through chaotic dynamics. Also Very Brilliant for me at this Level are: Strogatz-Nonlinear Dynamics and Chaos, Kaplan-Understanding Nonlinear Dynamics, Gulick-Encounters with Chaos, Hilborn-Chaos and Nonlinear Dynamics, Devaney-An Introduction to Chaotic Dynamical Systems and A First Course to Chaotic Dynamics, Holmgren-A First Course in Discrete Dynamical Systems. More sofisticated maths but not too far away are: Schuster-Deterministic Chaos(graduate) and Ott-Chaos in Dynamical Systems (graduate).

This book is a must-own for anyone interested in nonlinear dynamics and chaos -- I also highly recommend the "Nonlinear Dynamics and Chaos" text by Strogatz.

I especially like the numerous diagrams that clarify everything so well in this book. In addition, the writing includes just the right amount of informal discussion to truly explain the material without retreating into jargon.

A favorite moment in the book is a "challenge" exercise that explains the famous "Period Three Implies Chaos" result: the reader is gently guided through 10 steps resulting in a proof of Sharkovskii's Theorem, a more general result that includes the Period 3 thing as a special case.

Buy it! Simply phenomenal.

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!

It was about the mid 1990's, still assimilating the big hype caused by the eventual and much-publicized proof by Andrew Wiles of Fermat's Last Theorem, when my curiosity (bolstered more by having seen a movie such as The Jurassic Park!) finally led me to taking a first college course on Chaos and Fractals at a California State school. At that time, the funny, surcastic, and somewhat sloppy foreign professor (who happened to be a country-mate of mine, for better or worse), had chosen the brand-new text "Fractals Everywhere" by Michael F. Barnsely for teaching our mid-size class consisting mainly of senior and first-year graduate students in math and sciences. I recall the discussion starting out by covering the basics about the metric spaces and sequences, and I having a head-start over many others coming fresh on the heels of a heavy-duty general topology course just in the previous semester (so for example I could show off to others on the first instruction day what it meant for two metrics to be equivalent). Still, I admit the semester went by without many of us really absorbing the nuts and bolts of the subject, for example why exactly topological transitivity was needed for chaos in an Iterated Function System, and why exactly some known fractals had the given fractional dimensions (eventhough we could compute them). However the students were generally happy to have scratched the surface of this vast, engaging subject, and for the time being it seemed about enough exposure for most of us. Consequently for me, during the several ensuing years in the late 90's the subject leapt mostly into the background, but nearly a decade later since I first took the college course, somehow it came back to the foreground in the company of several other applied subjects such as control, game theory, and information/coding theory.

Now looking back, I find Barnsley's text a very good choice having gone through at the time, but the title by Alligood, Sauer, and Yorke (as a recommendation by a college professor at a different school who had taught his students from it) seemed like a more well-balanced introduction to the area of dynamical systems. In fact I also recall at the time there was a discussion as to whether yet another text by Robert Devaney would have made for a better first course. The aforementioned professor duely noted that Devaney only dealt with the discrete dynamical systems, while A/S/Y treated both the discrete and continuous, hence making the choice of the latter a more suitable one. In any event, the rundown of the topics discussed in the 13 chapters of A/S/Y 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. There are answers and solutions to the selected exercises, as well as extensive references at the back, making up an ideal setting for self-study. The level and style of exposition is targeted towards an advanced undergraduate student who is into applied math or engineering fields. Therefore the authors emphasize concepts and applications instead of getting bogged down in too much mathematical rigor or heavy use of the abstract machinery (which is of course needed for a thorough treatment of the subject at an advanced level; there are in fact several newer titles which all occupy this niche). Notationally and stylistically also, A/S/Y is very accessible and attractive. All in all, an excellent first excursion/introduction to one of the most fascinating areas of applied math, whether for classroom use, or for self-study.

[Review updated and reposted on 08/08/08]

This book is both simple enough to understand, and sophisticated enough to provide further understanding. If you are an second or third year undergraduate planning on graduate work, this book is a great way to catch up on advanced mathematics.