Dynamical Systems for undergraduate students Hello,
I'm searching for a good subject of Dynamical Systems theory in which I can propose a theme for a undergraduate research opportunity program. As I'm a undergraduate student, I have had only be exposed to subjects such as linear and abstract algebra, real analysis (calculus, basic topology of $\mathbb{R}^{n}$, metric spaces, and a "informal" view of measure theory). 
I have been looking at subjects such as unidimensional dynamics and circle homeomorphisms, and reading An Introduction to Chaotic Dynamical Systems by R. Devaney.
-Thanks.
 A: I would recommend reading through Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering by Steven Strogatz. He does a great job of motivating the applications of the field to various branches of science with a plethora of exercises. I wouldn't say it's entirely rigorous at times, but it covers a huge amount of material which should give you plenty to think about if you want to do a research topic. Out of all the books I've seen on the subject, his is probably the only elementary one that tackles renormalization techniques and universality, involving Feigenbaum's constant. 
A: Another worthy one:
Lasota, Andrzej; Mackey, Michael C.: Chaos, fractals, and noise. Stochastic aspects of dynamics. Second edition. Applied Mathematical Sciences, 97. Springer-Verlag, New York, 1994. xiv+472 pp. ISBN: 0-387-94049-9 
http://books.google.com/books?id=2xsb6iveF9QC&dq=inauthor:Lasota&source=gbs_navlinks_s
Despite the word "stochastic" in the title, the book  treats "deterministic" systems as well. It starts from quite a low point and develops techniques and problems along the way. With some background in metric spaces and exposure to measure theory, you should have little problem following along. There are routine exercises, numerical experiments as well as difficult or open problems throughout the book.
A: I see that you have already received suggestions about books, so I am not going to suggest any books, but I will suggest a topic, if that's what you are looking for. I think the notion of entropy and its role in dynamical systems could potentially be a good research theme. You can find out about different notions of entropy, their connections and their importance in dynamical systems.
A: Although the book is now 18 years old, you might benefit from looking at the bibliography in Stephen H. Kellert's In the Wake of Chaos: Unpredictable Order in Dynamical Systems (Chicago: The University of Chicago Press, 1993). The technical as well as the philosophical papers cited therein might suggest a topic for your project.
