Dynamics book reviews

This is a list of very brief book reviews regarding dynamical systems books as well as background material. These reviews are very informal, are intended for students, are from my personal viewpoint, and thus may not carry much validity. If you have questions or comments, please let me know (I am easy to contact). The reviews will be broken up into the following categories and sub-categories: dynamics (numerical, undergrad, grad, and monograph), numerical analysis as applied to dynamics, and background books. I hope I don't offend anyone, if I did, I am sorry, just email me and I'll try to fix it. This is also far from being exhaustive, so if I omitted something important, I am sorry, it is out of ignorance --- so please let me know so I can correct the omission. Dynamics: books to add: katok and has. intro book., geometric methods by de mello, one d dynamics book by collet etc..., random motions book by moser. kaneko's coupled map lattice book. new bbook by viana..., jost's dynamics book.

Numerical dynamics:

Kuznetsov - "Elements of applied bifurcation theory" It has a lot of hard core numerical stuff in it as well as a boat load of dynamics theory. It is definitely a grad book, but can really be used by two audiences, applied crowds doing applied, computational bifurcation theory in their modeling and analytic bifurcation theory folks. I talk about this book more in a later section.

Sprott - "Chaos and time-series analysis" Clint has quite a bit of time-series analysis in the book; the entire book is from the computational, scientific prospective. I think it is the best book from a computational perspective for applied dynamics. It can definitely be used for advanced undergrads and beginning grad students depending on the field and amount of background. I use it as a reference frequently as it has all of the standard, important cases treated completely and because it is rather exhaustive in it's content.

Kantz and Schreiber "Nonlinear time-series analysis"

Undergraduate (beginning and advanced):

Strogatz - "Nonlinear dynamics" This is my favorite elementary book. It is geared for early undergrads and I think it is one of the best places to learn dynamics out of, however, there is a huge gap between this and the "graduate" books. So, jumping from Strogatz to, say, Wiggins, Brin, or even Devaney is a non-trivial, somewhat large step.

Sprott - "Chaos and time-series analysis" I really like this book of course; I read through it quite a bit when he was writing it. I think it is the best book for advanced physics, non-math types. It has a huge amount of numerical simulations type stuff, lots of exercises, etc. It is the book to use for advanced undergrads to beginning grads to use for computational studies. It is missing some things which I hope will get fixed in a future edition, like the LCE spectrum, but that is easily supplemented.

Devaney - "A first course in chaotic dynamical systems: theory and experiment" (also known as baby-Devaney) I haven't ever spent much time with this book. It seems quite nice, and I like it, given the little time I've spent with it. I remember it being pretty elementary, sort of a good first dynamics book from the mathematics perspective. It has a bunch of numerical content, which is great I think. I know many folks who love this book.

Robinson - "An introduction to dynamical systems" (baby-Robinson) This book is relatively new and I haven't looked at it yet.

Hale and Kocak - "Dynamics" This is probably a better undergraduate math book than Strogatz in some ways, at least for those who want to move on to Wiggins or the like and is maybe a graduate book for physics or science-like folks. It is a pretty unique book in my experience. I like it and think it is interesting, but maybe a little out of the mainstream thinking. I wouldn't hesitate to teach from it however, but it would require some qualifications sometimes. It basically moves from dimension 1 to dimension 1.5 to dimension 2, etc, covering various dynamics properties that occur while adding dimensions. It is quite a unique book.

Ott - "Chaos in dynamical systems" I have looked through it but not super carefully. It appears to be a good engineering style book. It says it is for graduate students; I wouldn't teach math graduate students from it but I might use it for engineers and other scientists. Math folks would probably hate it, physics undergrads might like it. There aren't many books like it from a computational perspective so it fills a necessary gap.

Altigood, Sauer, and Yorke - "" I think this is a great book from the standard Maryland-style dynamics perspective. Many topics are handled very well and in an intuitive way. I think there is a big gap between this and the graduate math books I have used (this is a re-occuring theme). I think this is geared more for physics types, but could be used for either undergraduate math or physics folks.

Hilborn - "" This is definitely a physics style book with a great reference section (although not as good as Clint's). I think the prose is a little dry, but this is a matter of taste. Engineers and experimental physicists might like it the best as it is very practical. I do like it and it was the first book I learned out of.

Hirsh, Smale, and Devaney - "Differential equations, dynamical systems and an introduction to chaos" I have seen it but haven't spent much time with it. Some portions of it are out of the old ODE and linear algebra book of Hirsh and Smale which seems between graduate and undergraduate for an ODE book. I know that Moe and Smale just signed off on this version, so Devaney must have done most of the new writing and updating. It seems, and I have heard, that this book is much easier and more elementary than the old ODE book, and that this is an undergraduate book and the Hirsh-Smale book is for hardcore grad students, but I do not know.

Graduate:

Hirsch and Smale - "Differential equations, dynamical systems and linear algebra" I like this book a lot. It fits in with Arnold's "baby" ODE book. It is maybe a bit more applied and less encyclopedic than Arnold's book. It is a good book for those working on applications who want some actual rigor, but many say this book is pretty hardcore and definitely a graduate math book. I think advanced undergrads who have already had some ODE and linear algebra stuff would get a lot out of this book. I am not sure if it is a graduate book or not, but I wouldn't use it for any less than advanced undergraduates and would use it for graduate students, probably with some more modern supplements.

Peterson - "Ergodic theory" This is maybe my favorite classic ergodic theory book. It has a clear presentation with background. It contains the standard results, e.g., Birkoff ergodic theorem, with very pleasant proofs. The introduction is particularly useful as it gives a wonderful context to the field and how the topics are related.

Devaney - "An introduction to chaotic dynamical systems" This is a nice math text. It presents low dimensional cases, has some real examples such as the DA diffeo and the solenoid. This book is not so bad for applied folks either. I would love to teach from this book. It is a great beginning math grad book or an advanced undergrad book. I think it has the right perspective, it avoids too much complication and nutty abstract geometry, but has a lot of practical geometry in it. Basically, it strips away all the unnecessary formalism and presents the important examples in the most simple context, which is ideal for a first go at mathematical dynamics for real.

Hartman - "Ordinary differential equations" THE standard classical text for years for those who don't care about numerics or maybe modern global analysis. It contains standard results a la the averaging method, existence and uniqueness, etc. It is the book that taught many really, really good math folks (e.g., Charles Pugh).

Brin and Stuck - "Introduction to dynamical systems" This is quickly becoming my favorite ergodic theory and dynamics book. I have heard it tears up math grad students and spits them out, but I think if taught well, it would make a good grad math book. It is hardish as an independent study book. It has supposedly the best treatment of absolute continuity which doesn't exist anywhere but in Anosov's book which is apparently unreadable (I have never tried) and in a book by Pesin which is another seemingly good place to learn absolute continuity from. I would say, for people who already have a good idea of how the arguments go, this book is pretty ideal as it is compact, to the point, and clear. This of course makes it not super friendly to newcomers who are not math folks or to those who do not have much background. It is by Brin (need I say more?).

Nemytski and Stepanov - "Qualitative theory of ordinary differential equations" I really like this book; it is a Dover collection of two old Russian books. It is a good place to learn some of the older notation which is also sometimes more akin to explicit matrix algebra and computation. That all said, it isn't a particularly easy book. I have spent a lot of time with it, but it isn't a book I recommend to most folks because they will have a fit. Moreover, it is old, and is, of course, somewhat out of date (of course, in some ways this makes it useful, e.g., the notation is less abstract). It is Russian old school, and reminds me of an old nuclear theory book I learned a bunch of quantum mechanics out of somehow. I still use this book quite a bit and am still learning from it. I have used this book quite a bit.

Robinson - "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos" This is one of the standard, modern dynamical systems books for math folks. It is self contained, doesn't assume too much geometry, and is quite explicit, and hence sometimes a bit tedious. Those who would rather have too much detail rather than too little will like it. It is good book and I have learned from it.

Golibinsky and Gulliman - "Stable mapping and their singularities" This book is a bifurcation theory book from the ODE, singularity theory, perspective. It has a great treatment of bifurcation theory using k-jets, and converses Thom transversality theory extremely well. For the topics it covers, it covers them very well, but sometimes in an intimidating style. This book includes ALL the details of proofs. For some folks, this is heaven. I struggle with books like this sometimes, and often prefer books like Arnold's Geometric theory of ODEs book. However, I am a dyslexic moron and crazy overloads of notation are a nightmare for me; I am better off struggling though the proofs myself. Anyway, I definitely own this book, it is a must have for those trying to understand bifurcation theory.

de Mello - "One dimensional dynamics" This is a special book that deals one dimensional discrete time maps only. It contains invariant measure stuff, and many results for maps of the interval (e.g., the logistic map/real quadratic family). If you want a book on one-dimensional dynamics, this is a rockstar book.

Shub - "Global stability of dynamical systems" This book is somewhat dry (it seems ok to admit this because theses are words out of Shub's mouth), but nevertheless is a nice intro into global analysis. It contains a nice proof of some of the stable manifold theorems and a lot of stuff that can't be found anywhere else. This is a book many people (including me) own and is a standard reference. It is what I used to sort out the proof of the stable manifold theorem.

Perko - "Ordinary differential equations" I think this is the current standard ordinary differential equations graduate math book. It is nice and mellow --- kind and gentle compared to the more old school books. Some of the normal form treatments should be supplemented with Wiggins or Kuznetsov. I took a class using it, but it was a bit on the soft side.

Arrowsmith and Place "An introduction to dynamical systems" This is a great dynamics book. It has everything, is kind of applied, but has the right examples, e.g., the circle map, toral automorphism, etc. I have come to really like this book even though it is not particularly formal. There is a nice, explicit treatment of normal forms. If you could only have one dynamics book, this is one of the candidates for the one to have.

Arnold (baby) "Ordinary differential equations" This is a hardcore, nice geometric ordinary differential equations book. I use it as a reference. If I could pick one book to teach physicists a year of ODEs, this would be the book, but they would scream about it the whole time. The book can be kinda a hard sometimes. It is in the Russian style; nice practical examples mixed with abstract as hell concepts. Physics book equivalents might be the Landau book on mechanics which I also love.

Hale - "Ordinary differential equations" This is an old school book much like Hartman's book. I have not spent much time with it, don't own a copy, but many folks love it. It is one of the old standards and I wish I had a copy.

Ruelle - "Elements of differentiable dynamics and bifurcation theory" This book is nice, but pretty dense and without much explanation. Most theorems are very general --- formulated on Banach spaces which is useful sometimes but not for others (sometimes it is nice to have coordinates). I think it is a nice reference, but hard to learn from.

Sinai - "Ergodic theory" This is Sinai's intro ergodic theory book. It is not too mathematical and is much more physically orientated than many books. It gives a nice sketch of the field. I like it a lot but haven't spent a ton of time with it. I would, however, highly recommend it.

Sinai - "Topics in ergodic theory" This is the follow-up to Sinai's ergodic theory book. I like it a lot, it is Russian but applied. I think it is a good ergodic theory book for a physicist --- but it does have some math in it (of course).

Nitecki - "Differentiable dynamics" This is a great old global analysis book. It gives a nice picture (or snapshot if you will) of global analysis/dynamics Berkeley style in the 70's. It is good for history and contains explanations of papers that do not exist anywhere else and thus is very useful. Many things in this book are "known" by folks in the field but are more part of the oral tradition and do not show up in many textbooks anymore. It has a special place. If you do not have access to someone who can fill in the oral history of the field, I would HIGHLY recommend this book.

Walters - "Ergodic theory" This is another standard ergodic theory book much like Peterson. I like it about as much. It is in the same mold with slightly different treatments and slightly different contents. I like this book and have learned stuff (=proofs) out of it without too much trouble. I also use it as a reference.

Wiggins - "Introduction to applied nonlinear dynamical systems and chaos" This is one of THE standard "applied" books. (Many might argue it is not very applied, however folks like Katok might argue this is not a mathematics book.) I like it a lot and have spent a huge number of hours with it. It is a book that you need some geometry and differential topology and hardcore ODE book to really completely understand, but without those things it is nevertheless extremely useful. It is more wordy and G&H, has more global bifurcation theory, and in some ways is less applied. It has a nice way of introducing normal form theory for those who want to move on to Kuznetsov's book.

Guckenheimer and Holmes(G&H) "Nonlinear oscillations and bifurcations of vector fields" This book is one of the first and one of the best. In some ways it is a more elegant version of Wiggins, but from a different perspective and with a somewhat different set of topics. I love this book and have nearly worn mine out. Some applied folks have the complaint that it assumes to much abstract math, but it is nevertheless, very useful. It was the first "real" book I learned out of.

Advanced (and mellow monographs):

Wiggins - "Global bifurcations" This is a nice survey of global bifurcation theory. Also, it is the only big treatment of global bifurcation theory I know of besides Palis and Takens, which considers homoclinic bifurcations and wild hyperbolic sets in particular. I like this book a lot. It extends his intro book very well.

Kuznetsov - "Elements of applied bifurcation theory" This is the best all around bifurcation theory book. There is hardcore numerical stuff in it, some global stuff, really good normal form treatments, all the latest advancements with respect to codim 2 and 3 bifurcations. From this book, all that exist are really difficult monographs a la lecture notes in math. I LOVE this book. It has elementary stuff in it but it also has problems and issues that are up to current research. It would make a great book for a grad math course in bifurcation theory.

Katok and Hasselblatt - "Introduction to the modern theory of dynamical systems" This is the mother of all dynamical systems books and could be titled "Katok's view of dynamics." It is a reference book as far as I'm concerned, but the most encyclopedic, complete, and accessible (although it isn't particularly accessible). I think that Katok has realized it is a reference and in the second edition is gearing it more toward that audience. This is a hardcore math book, and it is distinctly from Katok's perspective of what is interesting, what is important, and what approaches are the right way of dealing with stuff. However, even if you are not in love with Katok's perspective, it handles an insane number of circumstances, and it often the first place to look for information regarding a topic. I would say it is pretty valuable, but hard to use at first. If you are a math person, this is probably one of the books you don't leave home without. My copy is pretty worn at this point (and I am not a math person as far as Katok would be concerned) although I still struggle with it because I have many holes in my mathematics training.

Chow and Hale - "Bifurcation theory" It has been a long time since I have read this book. It is one of the most advanced bifurcation theory books from the ODE perspective. I learned Sard's theorem out of it as an extra project for an analysis class. This book and be sort of hard, and I would only get a copy if it has topics you need. Of course, if this is the case, it is probably the only book out there.

Barreira and Pesin - "Smooth ergodic theory" This is THE book to learn Lyapunov exponents out of by one of THE Lyapunov exponent masters. I love this book. It is written well which is refreshing. It has a presentation of absolute continuity I haven't read yet, but I have heard is good. Of course it is a monograph, so it isn't the most accessible book on the planet, but for what it is it is great. Some differential topology is assumed, as well as measure theory and geometry, but it isn't so bad. The notation is great. This thing rocks. I would use it to supplement a any variety of dynamics courses because it is rather understandable given the topic.

Pesin - "Lectures on partial hyperbolicity" I love this book too. It explains stuff really well, the notation is great, the writing is great. Again, it is written to bring folks right up to the edge of where people are working as of when it was published. Where many of the chapters end there are groups of folks currently working on the extensions. It is a great presentation and is as up to date as it gets (circa 2004).

Arnold - "Geometric methods in the theory of ordinary differential equations" This is one of my favorite books. Some people hate it --- those are the same people who hate Landau's mechanics book. It is very Russian, the proofs are outlines, a few pages can take days to work through, but I really like the presentation, the topics, the level, etc. This book comes with me very often. It is one of the all time classics. If you want a book that has more of the details filled in, then I would suggest Golibinsky and Gulliman' book.

Palis and Takens "Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations" This is the other global bifurcation book I know reasonably well. I really really like this book. It has the best treatment of Newhouse's wild hyperbolic sets work (according to Newhouse anyway) and basically just goes through dynamics near homoclinic tangencies very well. If you are going to learn global bifurcation theory, I think this is the right book to do it out of, and it is written pretty well, so that is a bonus. For what it is, it is KKKKKK for what it is.

Abraham and Robbin - "Transversal mappings and flows" This is one of the classics and is cited everywhere. I haven't used it too much, but when I have I have been quite happy with it. The title says it all; if you want to learn transversality, then this is your book.

Mane "Ergodic theory and dynamical systems" This is a wonderful book that contains a bunch of mistakes or typos. Mane rocked, so you get his perspective via this book which is extremely valuable. There is an ok version of absolute continuity, but again some mistakes. If you want a version of the Brazilian way and how the Brazilian school thinks at least as is related to Mane, this is a good book. I own it and cherish it.

Arnold (ed.) Dynamical systems V:

Peixoto (ed.) Dynamical systems: This is a conference proceedings from Brazil and it has a lot of really interesting and useful papers. If you can manage to get a copy, grab it.

"volume 14" This is the AMS "" volume 14 from 1968. There are many famous papers in this book and it is worth owning. There are many topics in this book that are part of the oral dynamics tradition at this point. This is where a bunch of really nice examples of the non-genercity of structural stability is presented (and really the first ones). This is a book I wish I had known about a lot earlier. The DA diffeo was in here, many of the standard examples showed up in here first (to my knowledge). It is an important book even today.

Pugh, Shub, Hirsch - "Invariant manifolds" Moe has told me that this book is unreadable --- so I have never tried very hard to read it. He recommends the papers in the AMS Bulletin instead. However, everyone cites this book, and I think you can download it from Shub's website. It is a classic kind of like Anosov's first book, invaluable but difficult.

Computation stuff:

Horn and Johnson - "Matrix analysis" This is my favorite linear algebra book. It is not built for computation specifically, but useful and to the point. If you don't own it, you should.

Robbin - "Matrix analysis" This book is great, and is from a very unique perspective. Joel got his PhD in logic, then made significant contributions to dynamics, and then moved on to other differential and algebraic topology like problems. Anyway, he learned linear algebra "for real" over the course of writing this book and thus, it contains a lot of info that would otherwise not be included, largely because he was writing it not for the perspective of someone who had worked in linear algebra for 40 years but as a mathematician who was sorting linear algebra out for himself. Anyway, I would highly recommend this book. It is also highly connected with Matlab, so it is really nice to learn from.

Golub and van Loan - "Matrix computation" This is the standard linear algebra computation book. I use it often, so does everybody else i know. It is extremely useful; if you don't own it, and you use a computer to conduct research, you should.

Higham "Accuracy and stability of numerical algorithms" This book could be called "honesty in numerical science" this is not a good book to learn out of but is perfect for the working numerical scientist. It is what is title says it is. It is a very good and comprehensive book, but it isn't easy.

Press et. al. "Numerical recipes" This is the first place anyone looks for anything numerical, it is good but has some flaws that math folks love to point out. It usually works however, and everybody who actually computes things seems to like this book.

Topology and geometry background:

Hirsch - "Differential topology" This is Moe's differential topology book and is my favorite of the few choices out there. I truly love my copy.

Lee - "Smooth manifolds"

Abraham, Marsden, and Ratiu - "Manifolds, tensor analysis, and applications" This is a fairly standard book that was meant to contain all a dynamics person needs to know about tensor analysis, geometry, topology, Lie algebras, etc. I like it and use it, but it does contain some mistakes, so be careful. Burns Geometry book... no proof of the whitney embedding theorem (find it in Moe's Book) milnor - diff top. jost brazilian @book{Lasota-Mackey, author = "Andrzej Lasota and Michael C. Mackey", title = "Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics", note = "First edition, {\em Probabilistic Properties of Deterministic Systems}, Cambridge University Press, 1985", series = "Applied Mathematical Sciences", volume = 97, address = "Berlin", publisher = "Springer-Verlag", year = 1994} @Book{jost_dsbook, author = {J. Jost}, ALTeditor = {}, title = {Dynamical systems: Examples of complex behavior}, publisher = {Springer-Verlag}, year = {2005}, OPTkey = {}, OPTvolume = {}, OPTnumber = {}, OPTseries = {}, OPTaddress = {}, OPTedition = {}, OPTmonth = {}, OPTnote = {}, OPTannote = {} } @Book{viana_book, author = {C. Bonatti and L. Diaz and M. Viana}, ALTeditor = {}, title = {Dynamics Beyond Uniform Hyperbolicity : A Global Geometric and Probabilistic Perspective}, publisher = {Springer-Verlag}, year = {2004}, OPTkey = {}, OPTvolume = {}, OPTnumber = {}, OPTseries = {}, OPTaddress = {}, OPTedition = {}, OPTmonth = {}, OPTnote = {}, OPTannote = {} }