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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, Introdcuction 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.

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Have you tried taking a course on nonlinear dynamics and chaos? Usually such courses are offered as cross listed for math/physics/engineering. –  Alex R. Jan 6 '13 at 17:49
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I would recommend looking into the courses at MIT Opencourseware and looking to see which books they use and the pre-requisites for the areas of chaos, dynamical systems and fluid dynamics. There are also many wonderful books in each of these areas. –  Amzoti Jan 6 '13 at 17:59
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(Point-set) topology is just something you should be familiar with if you plan on doing graduate level mathematics, but except for basic terminology, it's not that important for what you plan on focusing on. With your background (already taken/plan to take), you seem well-prepared to go straight to any class offered at your school, found online, or just self-studying a book on dynamical systems/chaos. For DS, I personally like an old classic by Hirsch/Smale, which also referenced physics frequently. –  gnometorule Jan 6 '13 at 18:37
    
I hear my gf recommends Strogatz, Nonlinear dynamics and chaos (and that you should apply to the Theoretical Neuroscience program at Columbia :) ). –  gnometorule Jan 6 '13 at 18:43
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@Ryker: Munkres: Topology is almost certainly the best first graduate/advanced undergraduate level intro text book. Its first chapter alone on necessary set theory is worth the price of admission; its clearly written (I self-studied it); with great exercises for which someone posted a complete set of solutions (if you get stuck) on dbfin, which were mostly elegant and (except for very few errors I noticed) correct. –  gnometorule Jan 6 '13 at 20:19

2 Answers 2

up vote 4 down vote accepted

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.

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Whoa, thanks for that! In regards to your suggestion, I've decided to take general topology this year, and next year I have two courses on real variables (so I guess that's real analysis), as well as classical mechanics that I think touches on Hamiltonian mechanics, lined up. The maths department also offers a course whose description includes existence and uniqueness theorems, systems of equations and their stability, perturbation theory and phase plane analysis? Would this be useful? Oh, and how useful would, say, ring theory be? –  Ryker Jan 8 '13 at 4:03
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That last course you describe sounds like exactly what you are after. It would complement an introductory book like Strogatz perfectly (which is probably one of the best places to start anyways.) One of the very first things you learn is how to linearize small systems and then look for limit cycles, etc. This is precisely phase plane analysis. I would take it first chance you get, you will probably love it. The other courses also sound good. But ring theory is more in the realm of abstract algebra. While algebra will probably be helpful in the long run, it's probably not as vital right away. –  AndrewG Jan 8 '13 at 4:41
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I would check the book listing for that math course and compare the table of contents to those of the books people have recommended here. There will probably be a lot of overlap. Stability (read: bifurcations), phase-plane methods, and perturbation theory are some of the very beginning topics in dynamical systems and chaos theory. You really couldn't ask for a better first acquaintance. You might google for fixed points, limit-cycles, and examples of simple systems analyzed by these methods to get a feel for what you would be doing. An example –  AndrewG Jan 8 '13 at 5:13
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That looks like a good book, although maybe more DE and less dynamical systems. Still lots of good stuff. Lyapunov stability and Lienard systems (especially the van der Pol oscillator) jump out at me, and it looks like they do plenty of phase-plane stuff and have Poincare-Bendixon in the index. If you manage to get all these classes you should be very well prepared! –  AndrewG Jan 8 '13 at 19:47
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I doubt it would limit you. More important than specific classes are overall grades, GRE, and letters. Topology is a big step mathematically, and it's probably wise to devote a lot of time to that. It also may not be immediately obvious how things like compactness matter, so there may be little cross-over at first. On the other hand, complementing that DE course with one/some of the books people have recommended, and a solid course on classical mechanics, would make a very good intro to dynamics. (e.g. the pendulum can be understood in many ways.) You sound like you have a good plan lined up. –  AndrewG Jan 8 '13 at 20:15

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:

"Nonlinear Dynamics and Chaos---With applications to Physics, Biology, Chemistry and Engineering", by Steven Strogatz.

Holger Kantz and Thomas Schreober, 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 trhere 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

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+1 for Strogatz. That book is so intuitive and wonderful. –  AndrewG Jan 7 '13 at 17:17
    
@AndrewGibson, thanks. There are a few other excellent books in this area, but I need to be home and dig them up and will likely add them as I think they will be helpful to the OP. Regards –  Amzoti Jan 7 '13 at 17:30
    
@Amzoti, thanks for this extensive list! I was looking at Chaos: An Introduction to Dynamical Systems by Alligood et. al book in our online library over the holidays, but haven't really started reading it. I noticed it's published by Springer, which I thought had a good standing as far as textbooks are concerned, but what would your thoughts on this particular book be? –  Ryker Jan 8 '13 at 20:44
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@Ryker: As an introductory text, it looks just fine and seems to be well received. I looked through the index and it looks useful. There are some other excellent texts in this area and I will add them (also Springer books) that are just beautifully done texts. I would certainly recommend that you learn a Computer Algebra System - CAS and make sure to experiment with each system and really explore how changing parameters, initial conditions and the like affect the system. Regards –  Amzoti Jan 8 '13 at 21:15
    
You should get a "Reference Expert" badge! (gold, at that!) +1 –  amWhy May 9 '13 at 0:20

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