How to go about studying chaos theory/dynamical systems/fluid dynamics in grad school with a physics background? [closed]

I'm turning to you as I find myself in need of help as to how to best go about studying something related to chaos theory/dynamical systems/fluid dynamics in postgraduate school.

I'm currently in my third year of physics and my first question would be what courses would be especially pertinent for me to choose as my electives? Whenever I had the chance thus far, I've taken maths courses as my electives, and I'm also taking the "honors" version of maths courses offered at my university, which are targeted at Maths Honors majors and Mathematical Physics majors. Not only does maths interest me a great deal, I now figure a good mathematical background is essential if I want to go on into aforementioned fields. But what maths courses would be especially relevant if I want to pursue the aforementioned fields?

Thus far I've taken the following mathematics courses: Honors Calculus I and II, Honors Advanced Calculus I and II, Honors Linear Algebra I and II, Introduction to Group Theory, Ordinary Differential Equations and Honors Complex Analysis. I'm taking Linear Algebra III and PDE's next term, and I also heard topology is necessary if I want to go into the fields I mentioned. Is this true? What else would I need?

Note that I realize I probably wouldn't be able to study these fields from a pure maths perspective due to my background, but I'm hoping I'd be able to from a mixed maths/physics one.

Any help here would be greatly appreciated.

edit: For anyone reading this years after the creation of the thread, I actually switched to math in my last year of undergrad, am now doing a Ph.D. in Pure Mathematics, and after taking the required courses hope to start with research this Fall (2016). I'm still keen on looking at dynamical systems, but it's probably going to be ergodic theory/symbolic dynamics/complex dynamics.

Solution 1:

There are several things you could study.

Topology is definitely a must, especially when you start talking about bifurcations, and it sounds like you have a chance to take that. That could be a great first start, especially as an undergrad.

In the long view, since you're coming from a physics background, you might want to learn Hamiltonian mechanics and start looking at things like integrable systems, perturbation theory, and the stability of the solar system -- which is something you'll probably do at some point anyways. (Besides, this is historically how the subject got started.) You may get some of that in an advanced undergraduate class on classical mechanics (they should at least touch on Lagrangian mechanics), but probably not much. An excellent segue is V.I. Arnold's Mathematical Methods of Classical Mechanics. Another very good book, and heavy on the math, is Goldstein's Classical Mechanics. Both of these are considered graduate-level texts but you sound like you can handle them.

About the math involved: if you haven't had a class on real analysis yet, you might want to think about that. If you have, measure theory might be the next logical step. Differential geometry could also help, although that might be a bit ambitious, and besides, Arnold develops it as he goes. If you don't get this as an undergrad, it may not be a big deal, but you should certainly think about all of these things in grad school.

Some of the big guns you might aim to understand are the KAM theorem and things like the Poincare-Bendixon theorem. If I remember, Strogatz gives a great discussion of the latter.

As soon as you start talking about strange attractors, the possibility of fractals comes in, and then topology and measure theory will be helpful, so that you can talk about things like the fractal dimension of your basin of attraction. One of my favorite examples of this is the magnetic pendulum which has wonderfully strange properties.

If it's offered, you could consider taking a course on Statistical Mechanics, where you would hopefully get introduced to the ergodic hypothesis. In this context, you might want to look at things like the Poincare recurrence theorem, applied to, say, a gas in a box. This could give you a feeling for how measure-theoretic questions and ergodicity can enter into dynamical systems. You might also look at Arnold's Ergodic Problems of Classical Mechanics.

In general, Scholarpedia is an excellent source on all of these topics: it's like wikipedia on math-crack. They have a whole section devoted to dynamical systems, with articles written by people who made significant contributions to the subject.

Solution 2:

Here are some items for you to think about and review.

Nonlinear Dynamics:

Undergraduate courses in Calculus and / or Differential Equations, Dynamics, Linear Algebra, Mechanical Vibration

Texts:

Steven Strogatz, Nonlinear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering, Westview Press, 1st edition (Edit: as of 2014 there is now a 2nd edition containing additional applications.)

Holger Kantz and Thomas Schreiber, Nonlinear Time Series Analysis, Second Edition, Cambridge University Press.

An Introduction to Dynamical Systems, D. K. Arrowsmith, C. M. Place

Introduction to Applied Nonlinear Dynamical Systems and Chaos, Stephen Wiggins

Scratchpad Wiki

Suggested Optional Texts:

(1) Julien C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, 2003,

(2) Lawrence N. Virgin, Introduction to Experimental Nonlinear Dynamics: A Case Study in Mechanical Vibration, Cambridge University Press, Cambridge, UK, 2000, and

(3) Chaos in Ecology: Experimental Nonlinear Dynamics, Academic Press, Elsevier Science, 2003.

Review Nonlinear Dynamical Systems and Chaos Review MediaWiki Nonlinear Dynamical Systems and Chaos. See the reading materials listed to give you an idea of the prerequisites for you to consider.

Fluid Dynamics Classes in Dynamics, Calculus III, Differential equations

Reference books (Math oriented and there are many others)

Theoretical hydrodynamics by L.M. Milne-Thomson

An introduction to theoretical fluid mechanics by S. Childress

Fluid Mechanics by Kundu and Cohen

Fundamental Mechanics of Fluids by I. G. Currie

A mathematical introduction to fluid mechanics by Chorin and Marsden

Fluid dynamics for physicists by T. E. Faber

Physical fluid dynamics by D. J. Tritton

Regards