Books for M531-532

For the course M531 - M532, I will not adopt a book as the textbook for the course. However, there are two books (Chicone, and Walter) that match parts of the course closely, and I will follow, for most of the material, one of these two, with neither book available for the whole course, but some core material available in both.

These books will be in Reserve at the library for the time of the course, and other material may be placed there at the appropriate time. Find below some more details about the books and judge for yourself if you want to buy either or both. Older editions (if available) will be just as good as the latest one.

  1. Carmen Chicone: Ordinary Differential Equations with Applications, Springer Texts in Applied Mathematics # 34 --- about 55 bucks at amazon
  2. Wolfgang Walter: Ordinary Differential Equations Springer Graduate Texts in Mathematics, # 182 --- about 52 bucks at amazon
    This is the translation of a German book, which sells for 25 Euros (approx $30-35), and offers used ones yet more cheaply. If any of you reads German and wants me to bring a copy for them in (without freight costs) contact me before I fly home.
  3. Lawrence Perko: Differential Equations and Dynamical Systems, Springer Texts in Applied Mathematics # 7 --- about 64 bucks at amazon

Advice on these books:

The book by Chicone is a very concise book, which I can warmly recommend for contents, aesthetics and focus. However it is rather steep, and you may find it difficult reading. We'll rely on a large subset of chapters 1-3, and maybe a bit of chapter 4. Further material would make a good selection towards a follow-up course. If you plan to specialize in Diff'Eq's (ordinary or partial), it is a good investment for the rest of your graduate and postgraduate carreer. However, if you are headed towards an MS and take the course merely as a prelim prereq, then this book is not your primary choice to buy (or at least, not to keep).
--- Chicone's book focuses on the qualitative theory of (nonlinear) ODEs (also called Dynamical Systems); to get into the subject, you'll first want a more leisurely introduction into linear systems of ODEs than what he provides, and we'll need some other linear material that he doesn't include (power series methods, and boundary value problems; Sturm-Liouville theory).

We'll use mainly the last chapters of the book by Walter, namely Ch. 5, the complex treatment of linear systems (power series methods), and Ch. 6, on boundary value problems, including Sturm-Liouville, and possibly Ch. 7 on Stability (stability is also included in Chicone).
--- Most of the material for which I'll follow Chicone is also contained in Walter's book. Those of you who find Chicone too steep may want to look into Walter. However, some of this material is served in a piecemeal way through Chapters 2-4, which makes it easy for learning step by step, but difficult to perceive the unity and coherence. So if you follow this part of my course through Walter's book, you'll find me jumping erratically through 3 chapters. (And if it appears erratic to you, you miss the philosopy behind the modern view on ODEs). One primary goal of the course is to teach you ODEs with the Dynamical Systems view in mind, and Walter's book does not support this part of the course philosophy.
--- If you need to review some advanced calculus material, Walter's appendices will be very helpful to you. (If you understand Chicone's calculus chapter (it's great!), then you don't need to review our advanced calculus course.) --- If you have already sold the incredibly boring (they all are) 200 level ODE textbook, but need to review some of that stuff, Walter has much of it (what's worth remembering anyways).

The book by Perko has a nice chapter on linear systems that does all the linear algebra you may need to learn or review as part of this course, and its 2nd chapter on stability (up to 2.11 at most) may be helpful reading for the course. It then diverges from our course, but contains some good stuff for a possible follow-up course.
--- Useful, but the M531-532 alone is not a reason to buy it.

Books for further reading:

The book by Herbert Amann (Ordinary Differential Equaltions) covers a reasonable subset of our course (and stuff beyond) at a level approximately appropriate for our level. (Steeper than Walter, not as steep as Chicone). It's also a translation from the German and may be available at a lower price in the original.

Hirsch, Smale (Differential Equations, Dynamical Systems, and Linear Algebra) is advanced undergraduate level. Nice chapter on Poincare-Bendixson, predator-prey models, and some material on Jordan normal form. Useful reading or browsing to begin with.

The book by Arrowsmith and Place (An Introduction to Dynamical Systems) is quite nice, but has virtually no intersection with our course. Once you're through with the basics (existence theory, flow) in our course, it's sensible independent reading or reference, if your studies move in this direction.
--- A different syllabus choice at this level could reasonably contain some of Arrowsmith's material in 532; and if I get to do a 631, it will contain some.

Katok, Hasselblatt (Introduction tho the Modern Theory of Dynamical Systems), is a very comprehensive book on non-dissipative dynamical systems. Good reference; contents beyond 531-532. The authors also have another textbook (A first course in dynamics: with a panorama of recent developments), at which I haven't had an opportunity to look at yet, but I would only expect good stuff from the authors. From the online-description, I conclude that it will have little intersection with our course subject, but may be useful for puruing complementary subjects at the level of our course.

Online Material on Prerequisites

I have the (loved or dreaded) habit of producing additional notes for UG courses. They will usually be written in a way that looks at the material from an advanced point of view, such as to have a second look *after* the lecture. You may find familiar material organized in a way that helps you blend it in the new, more advanced material.

If you forgot the linear theory from 231, I have this material posted for everybody who has use for it. --- Some of the 431 material (also posted) will be reused for 531 (since 431 is not a prereq for 531), but in the 531 course we actually prove the theorems. For an overview of part of the course, you may find the 431 notes useful. I won't produce similar notes for 500 level courses, because at this level, (text)books are not infested with the bore-plug-and-chug virus yet.