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> "there is a huge difference between an equation that doesn't have a solution and one whose solution we can't find." Of particular note, with some classes of
by lotharbot 11y ago
> "there is a huge difference between an equation that doesn't have a solution and one whose solution we can't find."
Of particular note, with some classes of differential equations, we can't write an explicit solution, but we can characterize the solutions very well. For example, we might be able to describe the solution in terms of its long-term behavior ("it approaches such-and-such point" or "it has a stable orbit around such-and-such path".)
See also: http://en.wikipedia.org/wiki/Dynamical_system http://en.wikipedia.org/wiki/Dynamical_system
- JadeNB 11y agoI have long wanted to, but have never actually got to, teach a differential-equations course from Hubbard & West (http://www.amazon.com/Differential-Equations-Dynamical-Approach-Mathematics/dp/0387972862 http://www.amazon.com/Differential-Equations-Dynamical-Appro...). I thought that there was an article in Math. Mag. about their approach via "fences and funnels", but MathSciNet only knows about a 1995 CMS conference proceedings: http://www.ams.org/mathscinet-getitem?mr=1483923 http://www.ams.org/mathscinet-getitem?mr=1483923 . EDIT: The article I wanted (which has the same title as the proceedings article above) is actually in College Math. J.; it's http://www.jstor.org/stable/2687507 http://www.jstor.org/stable/2687507 . Another, I think, slightly gentler approach along similar lines is Blanchard, Devaney, & Hall (http://math.bu.edu/odes http://math.bu.edu/odes). The pedigree for this latter is very good; Devaney gave the (as far as I know) first rigorous mathematical definition of chaos. I have taught out of this one a number of times, and very much enjoy it.
- lotharbot 11y agoAt a more advanced level: my favorite graduate course was taught out of http://www.amazon.com/Introduction-Applied-Nonlinear-Dynamical-Mathematics/dp/0387001778/ref=la_B004LQHLZ6_1_1?s=books&ie=UTF8&qid=1433370754&sr=1-1 http://www.amazon.com/Introduction-Applied-Nonlinear-Dynamic... .
- JadeNB 11y agoAnother one that I meant to mention in my previous post is Strogatz (http://www.stevenstrogatz.com/books/nonlinear-dynamics-and-chaos-with-applications-to-physics-biology-chemistry-and-engineering http://www.stevenstrogatz.com/books/nonlinear-dynamics-and-c...). Do you know how Wiggins compares?
- lotharbot 11y agoI have the first edition of Strogatz' book. It's fairly accessible to students who have taken calculus and been exposed to ODEs, and can be worked through in about a semester. Great for a 200 or 300 level course. Wiggins is a much more advanced, and much more thorough, book. It's appropriate for 400 level at a minimum, and probably more accessible to graduate students. I wouldn't even attempt it without having taken ODEs and Linear Algebra, and probably some real analysis or PDEs or another high level course just for exposure to mathematical rigor (and I might be leaving out other prereqs; I can't presently locate my copy of the book.) It's tremendously well presented for a book at that level, and could be enough material for a three-quarter or two-semester sequence. It's advanced enough that someone working on a related Masters or Doctoral thesis would likely refer to it regularly.