5 comments about An Introduction to General Systems Thinking (Wiley Series on Systems Engineering & Analysis).

1. In computing, a timeless classic is anything that is worth reading for any reason other than to obtain a historical context after five years. If that still holds true after twenty five years, then it is truly an extraordinary piece of work. That label applies to this book. It is not about computing per se, but about how humans think about things and how "facts" are relative to time, our personal experience and environmental context.
   Human thinking is a complex operation and that is the point of this book. The problems and examples presented are not those in computing, but problems in how we think about the world and how that world can be different from person to person. In many ways, Weinberg anticipates the development of the science of chaos, where small changes lead to disproportionate large changes. His example of the "small" change of a single character is a classic. A man was considering the purchase of a piece of real estate, but when told the cost was fourteen million dollars, sent the response by telegram, "No, price too high." However, somehow a character was dropped, so the agent received the message, "No price too high", purchased the property and so a classic error was invented.
   Weinberg uses science and mathematics as the genesis point for most of his examples. The laws of thermodynamics, chance and simulations in state spaces are used to demonstrate the points. As someone with a wide background in science, I found his examples of how scientific thought gives us an anchor but yet alters over time excellent learning material. Thought problems are included at the end of each chapter and they cover many different areas. Some involve mathematics, others science and many could be the point of a vigorous philosophical debate. Together they form the best collection of thought experiments and points of contention that I have ever seen gathered together in one location.
   This is a book that is a true classic, not in computing but in the broad area of scholarship. It is partly about the philosophy and mechanisms of science; partly about designing things so they work but mostly it is about how humans view the world and create things that match that view. This book will still be worth reading for a long time to come and it is on my list of top ten computing books of the year.



2. Weinberg distills the essence from von Bertalanffy's classic and manages to present it in a very accessible fashion. The book has been out of print for quite a while so it is great to see a new edition. The message and information contained in here, although originally published in 1975, is now more relevant than ever.

   Weinberg covers many aspects of systems theory beginning with the main stumbling block with the present scientific paradigm: the idea that the universe is mechanistic. His treatment is much more general than Robert Rosen's in "Life Itself" but still conveys why the mechanistic notion is flawed.

   He then outlines the general systems theory approach before leading into the idea that a system is simply a way of looking at the world. He then outlines the principle of indifference. This leads straight into two sections outlining various aspects of making observations. Finally he discusses behaviour and then some general systems questions.

   Throughout the book he uses many examples from disparate fields in conjunction with questions for further research. It is great to see someone who doesn't preach systems but actually uses the ideas.

   Definitely a must-read as we decided how to solve the myriad of issues before us.




3. I was searching for an alternative to the out-of-print book 'Quality Software Management, vol.1: Systems Thinking', written by the same Author.
   Having read few Sofware Management series books from the same Author, which I rate at the very top of my list, I was biased on very high expectations. Surprisingly, I have found it being quite verbose and in the end, I couldn't get too much inspiration out of it.



4. This book is excellent. I first read this book in graduate school in 1976, and I continue to find Weinberg's ideas useful. It was outstanding then, and it has held up with time.



5. Weinberg's book will not teach you how to be a systems thinker. It will, however, provide a stimulating discussion and thoughtful examination of an alternative approach to problem analysis and solution. The book is not so much about how the systems approach works or how it can be applied to complex problems as it is an invitation to his readers to explore their perceptions of what they think they know versus what they really do know. Throughout the book, Weinberg follows the strategy of leading the reader through a series of logical discussions designed to bring them face to face with their biases and misconceptions about systems vs. reductionist thinking. In doing so, Weinberg exposes the shortcomings of the reductionist approach to problem solving by demonstrating to his readers that the real solutions to some familiar and apparently simple problems are very complex.
   
   Through his examples, Weinberg shows that by viewing a system holistically within its environment, we may be able to discern patterns of behavior/actions and recognize interactions, interrelationships, and interdependencies among the components that will be missed in a reductionist approach. From that view, we can better understand the system and, perhaps, better predict how it will evolve over time. The success of his approach is demonstrated by the fact that people are still reading and quoting his book 25+ years after it was written.
   
   One facet of this book which I found beneficial may be a drawback for some readers. Weinberg wrote from the viewpoint of a computer programmer and a scientist. A person not versed in either field might have difficulty understanding his examples.

No comments about Linear Systems Theory.

5 comments about Principles of Plasma Discharges and Materials Processing , 2nd Edition.

1. This book provides a theoretical overview of plasma's, including coverage of ECR applications. This was very valuable for Hitachi Etchers. The theory is presented at an undergraduate level and assumes the reader has knowledge of vector analysis. Highly recommended for any Etch Process Engineer in the Semiconductor Industry.



2. Lieberman covers many of the standard processing devices and much of the physics needed to model them effectively.

   The presentation is clear and extremely useful both as a reference and as a tutorial. A must-have book for anyone interested in plasma processing.




3. This book provides an excellent introduction and overview of plasma discharges applied to semiconductor manufacturing. It is well-organized, clearly-written and full of useful examples and exercises. And unlike many books on plasma physics, it is not overly-mathematical and contains many useful physical insights. I strongly recommend this book for anyone wanting to review the field of plasma processing.



4. As a praciticing process engineer my opinion is this book lacks insight. Typical text book written a professor in the academic community with no practical experience. This book spends far to much time deriving equations and not discussing the basics concepts. The author makes a half hearted attempt to relate the first 14 chapters to the real worl, in a short and inadequate Chapter 15. This is the first mail book order book i took the time to return. In all honesty this book is not worth the $90.00. Spend your money elsewhere



5. One of the most practical and comprehensive resources on plasma phyics and engineering. The book is much easier to understand and more in depth than most other books on the subject, except for maybe chen (who takes more of a physics approach, whereas lieberman takes more of an engineering approach) This book is a must have for anyone working with or studying plasmas.

4 comments about The Essence of Chaos (The Jessie and John Danz Lecture Series).

1. Edward Lorenz takes a complicated topic and makes it accessible for all people, regardless of prior knowledge of chaos theory. He provides interesting and easy to follow examples of chaos, fractals and complexity. The illustrations are helpful and he includes a glossary of terms to aid the beginning chaos enthusiasts to quickly become familiar with the terminology. Mr. Lorenz gives a brief history of chaos and explains how it is used in the study of mathematics, meteorology, economics, music, and other fields. The book is very interesting and is highly recommended for those who would like to acquaint themselves with the exciting world of chaos.



2. Lorenz has done it again. This is a terrific inside look at chaos by the man who made Gleick's book possible. And it had a few interesting new ideas too--who would have thought there was a different way to present fourth-order Runge-Kutta? Who would have thought Runge-Kutta could convert a phase-space circle to a nice-looking fractal attractor? A good book for the air plane.



3. My first intro to chaos was Gleick's book *Chaos: Making a New Science* which focused on the history of the discovery of chaos. Although this was fascinating - and a good read for those just learning about dynamical systems, strange attractors, and the like - Lorenz's *Essence of Chaos* was much more satisfying. Lorenz analyzes specific chaotic functions, gives you the math (equations are in the appendix) and generally accomplishes what the title suggests - that is, exploring the essence of chaos. I highly recommend it for anyone interested in this deeply fascinating subject.



4. Lorenz did a great job when he wrote this book!
   The very first time when I heard of chaos theory was year ago while watching some old documentary about Nostadamus. In film was mentioned chaos theory and said that acceptance of it by many people could change whole look to life and so on. Movie left to me questions - what is that theory, what it's standing for.
   Finaly my interest lead me to this book and it clearly showed me what kind of staff is that chaos theory! That was and is really intriguing!
   Book is well written. There was of course some places that wasn't easy to understand. I myself have studied high math,encountered differential equations but anyway had some difficulties. That's why not 5 stars to book - it's really not for absolutely everyone although almost close to it. I couldn't stop it reading, I was done in two days.
   This book encouraged me for further reading.

5 comments about Chaos.

1. 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).



2. 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.



3. 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!




4. 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.



5. 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]

2 comments about Automating with STEP 7 in STL and SCL: Programmable Controllers SIMATIC S7-300/400.

1. A must for the person using Siemens PLC's. This is the book for the beginner or for the experienced Siemens S7 Programmer. Book cover the things you need to know or reference when programming.



2. "Automating with S7...STL/SCL" is a reference book for any EXPERT programmer using Siemens S7 PLCs. Hans Berger, the author, is the "father" of S7 PLC family (in a technical way). If you want to reach the EXPERT level, YOU MUST HAVE THIS BOOK!!! On the other side, if you know you will stay to ladder logic level at least 95% of the time (or, you "know and use / program" ten types of PLCs, ten types of drives, etc) then, YOU DON'T NEED THIS BOOK!

No comments about Signals and Systems (The Oxford Series in Electrical and Computer Engineering).

5 comments about An Introduction to Stochastic Modeling, Third Edition.

1. First, let me say that I found the content of this book to be, on the overall, wonderful and fairly well explained. Concepts are presented well and, unlike many other books on Stochastic Modeling, sigma algebra is avoided (this is a definant plus for making it into an undergrad or low-level grad textbook).

   That having been said, this book has some of the worst organization I have ever seen in a textbook. Every chapter is divided into sections and at the end of each section there are questions which are separated into "Exercises" and "Problems"; this in-and-of itself is not as much of a problem as that everything is numbered the same way.

   Therefore problem 5 in section 4 chapter 3 is numbered the same way (4.5) as exercise 5 in the same section and chapter is numbered the same way as exercise/problem 5 in the same section of any other chapter in the book. The only real difference between "Exercises" and "Problems" is that exercises tend to be answered in the back of the book.

   There are also other organizational difficulties in the text itself--such as that it is never entirely clear where the examples are in the text: there are several things which are labeled as examples (and are), however, over half of the examples in some chapters seem to be simply thrown into the text without any special indicator that they are examples of what is being discussed.

   While the content in this book is good, the organization is so wretched that I have to knock it down two stars.




2. I purchased this book to use as the text for a graduate level Stochastic Processes course that I am taking by independent study, and have had a large role in designing. I purchased the book, sight unseen, based on reviews that indicated there were many examples with solutions, wary that reviews also mentioned a lack of organization.
   
   The organization was worse than I could have anticipated, and is one of two major flaws that do not render the book unusable, but make it very unpleasant to work with.
   
   As has been mentioned, the outline numbering system makes chapters harder to follow, rather than easier, and it is difficult to distinguish the exercises with solutions from the problems with no solutions. This strange numbering system is carried out in the answer key portion, as well. When I read similar comments in reviews, I thought, how bad can it be? Creatively bad.
   
   The most problematic organizational point, however, is the fact that concepts are covered in homework problems before they are introduced in the text. Chapter 1, for example, contains problems that could only be done after reading Chapter 2. This juxtaposition of discussion and exercise is still taking place as I am about three fourths of the way through the book.
   
   The second issue with this text, besides the confusing organization, is the cumbersome use of notation with no key or explanation. Commonly, sections of text are only three or four pages long. They consist of, "Here is a formula. Now here is the proof," without any real explanation of what the formula is for, and perhaps, worse, no indication of what the variables stand for. The field of statistics is notorious for it's inconsistent use of symbology. Most texts address this by including a key of symbols. Not only has a key not been included in this text, but the symbology is most uncommon. It has taken me quite a bit of searching to decipher a number of symbols for which there were much more common alternatives.
   
   In it's favor, the exercises and problems in the book are good, appropriate, and even classic examples.
   
   With a strong enough background in probability, particularly Markov processes, or, with good instruction, this book is a decent source of exercises. But certainly there are better sources of exercises if we must look elsewhere for instruction.



3. Letting the other reviews critique the content, I would like to mention that I was unimpressed with the quality of the binding.
   
   Although I may be a unique occurence, the binding of my book was cheap and broke so that several pages came loose. For an $80 book I expect better durability than a paperback.



4. This book opens with a nose dive into Conditional probability. Unlike some other authors that devote a half their entire bloody book on review of probability, random variables, and conditional probability distributions, this book assumes a firm or atleast an introductory knowledge of the above. Ideally, a good probability book such as Hogg and Tanis would prove to be quite helpful as a supplementary reference.
   
   Markov Chains and Processes are introduced in the third chapter and the definition is lucid, complete with examples that are easy to comprehend. One of the examples that calculated the frequency with which an autoparts store must replenish its stock was absolutely brilliant and made things a lot easier to understand. The exercises are rather thorough, so if you are purchasing this book for a class and will be assigned homework assignments from the text, be prepared to devote atleast an hour on an average to each problem.
   
   The book is relatively easy to read, if you have a good background in random variables, and hence, i repeat, keep a book on introductory probability and statistics handy.



5. Since the third edition is out and I have not seen I must say upfront that my comments are on an earlier edition. This pair of authors have done an excellent job with three textbooks on stochastic processes. Unlike the other two books that give a rigorous treatment of stochastic processes this book is more applied with the emphasis on examples and models for real applied problems. Introductory probability and statistics is assumed by the authors and they deal with all the standard topics.
   
   However, I do have to agree with the other reviewers who criticize the organization. Karlin and Taylor are not known for great organizational skills. Sometimes that is a trait of a mathematical genius. Sam Karlin can certainly be put into that category. Although I never took a course from him at Stanford, the fact that he was a Stanford professor meant that I did learn a lot about him and see him on campus and hear an occasional lecture. Of course this was a long time ago in the mid 1970s when the first edition of the first course was out and the other two books had probably not yet been contemplated.
   
   In any case this would be a good reference for an applied statistician to have.

4 comments about Order Out of Chaos.

1. Prigogine argues persuasively that he has reconciled classical dynamics with the human conviction that the future cannot be predicted from a knowledge of initial conditions and differential equations alone. He draws the reader through his own intellectual odyssey from classical thermodynamics, through linear nonequilibrium thermodynamics, and finally to his holy grail of nonlinear nonequilibrium thermodynamics. I suspect he has identified the quantitative tools that will connect the Human Genome Project to a functional understanding of cell biology and physiology. Tools capable of dealing with complexity.
   If you are a scientist who has followed these disciplines from afar, and who has wished for a succinct summary that does not shrink from rigor, then acquire this book. You will chuckle at the constant barbs directed across the English Channel, and you will learn wonderful things about thermodynamics and thermokinetics. So few scientific books reveal the authors' insights. Instead, they teem with facts and formulas. Prigogine and Stengers have bedded physics with philosophy as if they were matchmakers for an illicit tryst. You will find yourself whispering, "Aha!" And you will, as I have, wear out your pen with underlining. I loved Carl Sagan's "Demon Haunted World", but Sagan was speaking to everyman. Prigogine and Stengers are speaking to scientists in fields outside their own. They believe they have seen the light, and they want you to see it too. Give them the chance to convince you. You will not be disappointed.



2. The whole problem with writing about a book, and especially this one, is that one has to cut a long story short. A story long enough to encompass a fair amount of scientific history - elaborated, if not referenced exhaustively. Not that it is meant to be. Prigogine's journey does not offer to take you by the hand for a guided tour of order, complexity and self-organisation. Rather, it keeps to the spirit of Toffler's introduction, (Was it coincidental that it was the other way round?!) where he talks about the wonderful art of scientific dissection. Order out of chaos, however, is a difficult read for the anyone who has been initiated into the scientific non-fiction. For those who expect the book to be a popular account of concepts in complexity and self-organisation, the intense style and the depth of detail can be exhausting. Like Penrose in the Emperor's New Mind, Prigogine's style is uncompromising. Toffler's introduction is fitting, if only in parts. The book does not offer explanations. Rather, Prigogine prefers to illumate his readers with his keen philosophical bent. It is here that the book triumphs. The effort that has gone into integrating the ideas in the book, the subtle nuances reflecting Prigogine's own views is truly commendable. But then, one should be fairly conversant with the loopholes that science finds itself in. The description of the behaviour of complex systems warrants some mention. The idea of switching between reality and mathematical description does not gel with the rest of the narrative in parts - specially when chemistry is the running example. Well, Prigogine wasn't writing the book with the intention of it being self-contained - and he makes no bones about it. That is the seed of inspiration, I suppose, for any writer, be it for the cause of science or for the sheer love for the written word. Prigogine has shown that philosophy is in some way inseparable from what many consider the scientist's playground. And we are glad that he has shared his views with us.



3. This work is one of the classics of the breakthrough period of chaos theory, complex systems, and self-organization theories. Mixing two modes and two cultures it stretches its bow between the nitty-gritty details of dissipative systems, and the history of the relations of the human and natural sciences, from the age of the emergence of thermodynamics to the present. The book has something now routinely filtered from discussion, the early critiques of the Newtonian mindset as it was starting to become dominant. The material on the history of the two cultures would seem to fall on deaf ears these days, and gives the book at depth not often seen in works of this type. Very much worth reading.



4. Prigogine describes his ideas of how order emerged from a ground of chaos and how the processes of entropy can lead a system open to its environment to evolve greater complexity. He also gives an exposition of the relevance of science to society. Prigogine's Nobel prize-winning models of dissipative structures are difficult to understand but persistent effort will reward the reader. His theories are as applicable to the evolution and expansion of consciousness as to the emergence of life on earth from a relatively simple environment.

4 comments about Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences Vol. 42).

1. For the moment it is "the" book on Dynamical Systems, through the world. Its first chapter is a good introduction on the mathematics needed to aboard the subject. The second introduces chaos, and the rest is for a good understanding of the newest and prolific science.



2. 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.




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



4. 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.