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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.

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As a side note, your question reminded me of that famous example of excellent research done by an undergraduate: compmath.files.wordpress.com/2008/09/… (searching for knots in Lorenz system trajectories in phase space for high Rayleigh numbers done by Jonas Bergman while he was an undergraduate). I am not recommending this as a topic for your research, since I have no idea of later research done on that. That's why this is a comment, and not an answer. –  Harun Šiljak Sep 22 '11 at 19:21
    
I see that you're located in Brazil. I don't really know how undergraduate research works there, but in the US it's most often the case that a student approaches a professor first and then the professor assigns a problem/research area. Usually students pick advisors based on compatibility from courses taken or with a slight idea of what field the professor works in. I can't imagine an undergraduate research experience where the undergrad comes in having read several books and then suggests a topic to the advisor. But again, that's just my view from the US speaking. –  David White Sep 23 '11 at 0:45
    
Dear David, Yes, the same occur here in Brazil. But nothing forbids a student talk with a professor knowing something about the subject. –  Paulo Henrique Sep 23 '11 at 19:43
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4 Answers

up vote 7 down vote accepted

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.

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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.

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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.

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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.

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