There are no mandatory exercises. That does not mean it's a good idea to skip the exercises, however. The exercise web page is password protected using a password that has been announced at the lectures.
2008-06-05: Today's exam problem with solution (in Norwegian) can be found on the old exams page. Several student asked, in different ways, how much is to be expected on problem 2. I answered something like, “as much as you can do without getting into trouble with available time”. Awfully vague, I agree. Selecting what is important to talk about is part of the problem. An answer to problem 2 may look quite different from my answer and still receive a good score.
D. W. Jordan & P. Smith: Non-linear ordinary differential equations: An introduction to dynamical systems, Fourth Edition,^{*} Oxford University Press 2007, ISBN-13 978-0-19-920824-1.
Ch. 1, 2, 3, 8, 10, 11, 12.1–4, 13.1 and Appendix A. Except 1.6, 3.3, 3.5, uniform stability i 8.3–8.4, all details concerning topographical systems in Ch. 10 (i.e., most of 10.2–10.4). I don't understand Theorem 10.10 (p. 351 (362 in the third edition)), so skip that.
You will note that my notes overlap quite a bit with the book, in which case you may choose which text to read. (But note that the statement of the Poincaré–Bendixson theorem in the book is wrong.)