On the OP's request I make my comment an answer, a slightly more elaborated version of it follows below:
It's a very broad subject, involving many different things. I'd recommend to plunge right in and try to figure out what you like and what seems appealing to you. Much of it can be understood without much previous knowledge. A bit of real and complex analysis, linear algebra and ordinary differential equations seem indispensable, however. Whatever you know about geometry (differential geometry, topology), analysis (measure and integration, functional analysis) and physics can only help.
Dynamical systems are a very broad subject, involving and connecting so many different things that virtually everything can help, ranging from combinatorics to measure and probability theory, number theory and differential geometry, physics and computational mathematics, you name it.
Thus, it is very hard to recommend anything specific, especially since you don't give much background info. As I said and as lhf pointed out, you should definitely have a good knowledge of real analysis and ordinary differential equations, because that's the initial motivation (besides physics, of course). As you probably know, linear algebra and complex analysis help the understanding of these topics very much. Everything beyond seems optional but only helpful.
My advice would be: look around, ask more advanced students or your tutors what they'd recommend as books or preparatory lectures. Take some topics of your liking, try to develop your skills and intuition, and you'll see what you need for further progress and understanding.
I myself learned a lot from lectures by E. Zehnder on the subject, who recently (finally!) published this nice book based on these lectures. At least the introductory chapters should be accessible without too much prior knowledge beyond a good course in analysis. Another book I liked was Hasselblatt-Katok's bible [there are several ones, don't start with the handbook :), I mean Introduction to the modern theory of dynamical systems, but I'm sure that the first course is even more appropriate, even if I haven't read it].
But I should say, these recommendations are my own preferences and might be completely orthogonal to what's offered at your university.