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.
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.
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.
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.