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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, 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.

Martin Sleziak
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Ryker
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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 06 '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 06 '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 06 '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 06 '13 at 18:43
  • @Alex Unfortunately there's no such course offered at my university, and that's quite disappointing. There's not even anything close either in the physics or maths department. In the maths department, there's a graduate level course on fluid dynamics, and various engineering departments offer that, as well, but nothing on chaos or nonlinear dynamics in general. – Ryker Jan 06 '13 at 19:56
  • @Amzoti Thanks for the suggestion, and I see I have the prerequisites for the undergraduate level course on chaos, at least. As for the graduate level course they offer, however, there are unfortunately none listed. – Ryker Jan 06 '13 at 20:03
  • @gnometorule The reason I was asking about topology is because I was thinking of taking it this term, and the professor teaching it told me it's a prerequisite for chaos theory, and that it can range from helpful to a prerequisite (depending on the particular context) for dynamical systems, as well. So I don't know what to think now. I just don't want to get rejected for grad school due to a lack of prereqs for that particular area. Also, thanks for the suggestion in regards to the book to learn from, I see they use it at MIT, as well. And I'll keep your girlfriend's recommendation in mind :) – Ryker Jan 06 '13 at 20:08
  • @Ryker: by all means, take topology! As said, its terminology is used in almost every last graduate level mathematics topic; and it is to me the most esthetic field of math. So this will be time well-spent, and probably very enjoyable. I just see what you already have taken and will take, and you certainly could read up on chaos/DS with what you know already. – gnometorule Jan 06 '13 at 20:15
  • @gnometorule Ah, I see. Yeah, I'm quite interested in it, as well, but the only thing holding me back from taking it is that I know the course is going to be extremely difficult, and I'm not sure if it'll fit into the context of all other ones I'm taking, leaving me enough time to do well in all of them. But I figured if it's necessary for the fields I want to go into, I'd kind of have to take it. – Ryker Jan 06 '13 at 20:18
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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 06 '13 at 20:19

2 Answers2

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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.

AndrewG
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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 08 '13 at 04:03
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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 08 '13 at 04:41
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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 08 '13 at 05:13
  • Here's probably a better example. – AndrewG Jan 08 '13 at 05:20
  • Yeah, I figured that course would be a good option, and now you kind of made me feel bad for not taking it this term :) I still have the option, as the term just started, but since it's a 400-level course, I took a 300-level Linear Algebra III course instead (I figured that could also be useful and fun, but I don't know), as I wasn't sure I could handle both that and Topology (in addition to other presumable tough physics courses) at the same time. It's good to know for next year, though, and hopefully the time slot won't clash with any of my required courses. – Ryker Jan 08 '13 at 18:33
  • BTW, this is the book used in that course. – Ryker Jan 08 '13 at 18:44
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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 08 '13 at 19:47
  • Thanks! In case I couldn't take this course next year, though, would this be something that would limit me when applying for grad school? Like I said, I technically could still drop that Linear Algebra III course and take this instead, but, again, I'm worried about whether I could handle that and topology (although, come to think of it, next year will probably be even tougher with that second real analysis course and the rest of the courses all being 400-level). – Ryker Jan 08 '13 at 20:08
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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 08 '13 at 20:15
  • Yeah, that's what I'm worried about, having to choose between hopefully better grades and fitting in that extra intermediate DE course. And as for linear and abstract algebra, is there any relevance to the fields I want to study there, as well, or not really? For example, this linear algebra course covers Hermitian and unitary matrices, spectral thm., Jordan canonical form, Cayley-Hamilton thm., bilinear forms, positive definiteness, Sylvester’s Law of inertia, geometric lattices, numerical methods, and application to discrete system evolution, matrix exponentials and differential equations. – Ryker Jan 08 '13 at 20:32
  • Wow, that's a serious linear algebra course! Between that and topology you will be working very hard! Yes, it's relevant. At the graduate level most of this stuff starts to connect in some way or another, so it's hard to point to a single topic and say "no, you don't need that." If you're really worried about time constraints you can always think about taking a few classes over the summer, or even staying an extra semester to take stuff you enjoy. If you can, though, I would try to take that DE course alongside classical mechanics. Having them side-by-side would surely give you extra insight. – AndrewG Jan 08 '13 at 20:42
  • I'm taking classical mechanics next year in the first term, while the DE course is offered in the second one. So I would either take it before or after that. Well, I'm hoping that linear algebra course actually wouldn't be that demanding, since it's quite a large class compared to other upper level courses (55 vs. ~ 15), and the instructor said it's mostly computational. But this might also mean I won't learn as much. As for other suggestions, I have to take a full course load due to my scholarship, and I'd rather not stay an extra term already paying international fees. – Ryker Jan 08 '13 at 20:54
  • In that case, given your enthusiasm for the subject, you might consider swapping either topology or linear algebra 3 for the DE course now and trying to fit in the other(s) later. It really is up to you. The DE course does sound like the closest thing you're going to find to what you're interested in and might be the most immediately enjoyable. Either way, if you go to grad school for this, you will be covering all of these subjects in detail then. – AndrewG Jan 08 '13 at 21:06
  • Well, Andrew, I decided to enroll into that DE course this term, and will thus not be taking the linear algebra course mentioned. I might take it next year, although I might also just not take it, have a lighter last term of undergrad, save some tuition fee money and learn those things on my own if I need to. – Ryker Jan 11 '13 at 03:24
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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

J W
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Amzoti
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    +1 for Strogatz. That book is so intuitive and wonderful. – AndrewG Jan 07 '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 07 '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 08 '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 08 '13 at 21:15
  • You should get a "Reference Expert" badge! (gold, at that!) +1 – amWhy May 09 '13 at 00:20