# Preparatory Course of A Career in Dynamical Systems

I am sorry if this seems out of place. I am currently and undergrad student and I'm very much interested in Dynamical systems. However, I don't really know much about it. I would like to know what courses to take to prepare one's self in the area of Dynamical systems.

Thanks.

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Perhaps you could edit the title to make it more descriptive. The introductory Dynamical Systems course my school (University of Minnesota-Twin Cities) is only multivariable calculus and differential equations/linear algebra. –  Tyler May 30 '11 at 18:43
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. –  t.b. May 30 '11 at 18:44
@Theo: Can you make your comment an answer so I can accept it. Thanks. –  Joe May 30 '11 at 19:02

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.

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Thanks. :) –  Joe May 30 '11 at 19:28
@Joe: You're welcome. Good luck and have fun! –  t.b. May 30 '11 at 19:56

You should definitely take real analysis, differential equations, and linear algebra.

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Thanks. ;) –  Joe May 30 '11 at 19:28
@Joe, btw, the book that got my attention to the subject was Iterated Maps on the Interval as Dynamical Systems. –  lhf Apr 4 '12 at 2:09

You should take Real analysis and topology.

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Thanks:) –  Joe May 30 '11 at 19:28

Forgot to say that you need to know:

1. Calculus on manifolds.

2. Linear Algebra.

3. Measure theory (if I wanted to get involved with ergodic theory)

4. Functional analysis.

5. Basic Topology.

6. A good course in complex variable ...

There must be more prerequisites but can not remember now ...

To illustrate, take the example of a basic theorem of dynamical systems, The Stable Manifold Theorem, this theorem says roughly that: given a dynamical system $f: M \to M$ and a point in $M$ then the set of the points that will converge to $p$ over time by $f$ is an immersed submanifold of $M.$

Any of the ways to prove this theorem involving calculations in banch spaces and therefore functional analysis. So to make the courses Dynamical systems is good to have some baggage. Of course there are topics that do not have many pre-requisites will but as you get deeper, you will need more and more mathematical

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One very good book which does not require almost any prior knowledge is Holmgren's - "A First Course In Discrete Dynamical Systems". This is personaly my favourite mathematical book.

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As has been mentioned early, dynamical system is a subject that penetrates a great number of other mathematical areas. In order to preserve the spirit of generalization, I strong sugest that you have a good background on introductory number theory. One dimensional dynamics makes use of various results of theory of numbers. As an example, you can check out this presentation by Edson de Faria:

Slides

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