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Classical Looking for rigorous text on dynamical systems

  1. Feb 9, 2015 #1

    A_B

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    Hi,

    I'm looking for a modern rigorous text on (Hamiltonian) dynamical systems, perhaps with emphasis on perturbation theory. It should be in the same vein is Poincare's "methodes nouvelles", but modern.


    Thanks
     
  2. jcsd
  3. Feb 9, 2015 #2
    Y. Bar-Yam, "Dynamics of Complex Systems" (Addison - Wesley, 1997)
     
  4. Feb 9, 2015 #3

    A_B

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    Thanks for your suggestion, Joel.
    Having a quick browse, my first impression is that "Dynamics of Complex Systems" looks a lot like Strogatz's book. Very good, but not rigorous.
    What I'm looking for is a book that proves the theorems about convergence of Lindstedt series, birth of limit cylces under small perturbation, KAM theory, etc. etc.

    I have in the meantime found https://www.amazon.com/Nonlinear-Differential-Equations-Dynamical-Universitext/dp/3540609342 which is more the style I'm looking for.
    Though other suggestions are still very welcome.


    Thanks
     
  5. Feb 9, 2015 #4
  6. Feb 9, 2015 #5
    A_B,

    Congratulations on finding the text for which you were looking.

    Here is an idea from Norbert Wiener that you may find interesting. All that physicists really know are the results of measurements of experiments, and the measurements of initial conditions and of the resulting conditions are always statistically spread over some nonzero range of values. Wiener argues that in using differential equations to generate models we are trying to predict one mean from the other, and that it would be more useful to instead search for an operator that transforms the initial distribution into the resulting distribution.

    In the cases in which our models are nonlinear equations containing branch points, we are always left wondering how nature chooses the branch that it follows. Perhaps, it doesn't. Using Wieners idea we would find the operators that transform the initial distributions into multimodal distributions.

    I hope that this interests you,

    Joel
     
    Last edited: Feb 9, 2015
  7. Feb 17, 2015 #6

    vanhees71

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    2016 Award

    How about

    Arnold, Mathematical Methods of Classical Mechanics, Springer

    He's the "A" in "KAM theorem". For me it looks pretty rigorous, but I'm not a mathematician but a theoretical physicist.

    In any case, I'd recommend to read a physics text first, e.g., Goldstein, Landau-Lifshits vol. I, or my beloved Sommerfeld vol. I, because from mathematicians' physics books you usually don't get the theoretical-physics intuition for the subject ;-).
     
  8. Feb 17, 2015 #7

    A_B

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    Xiuh : Looks like a very interesting book, thanks for the suggestion!

    Joel : This idea does interest me very much. But my interest lies mainly in justifying the approximation of deterministic dynamical systems by stochastic processes. This is often done by estimating the measure of parts of phase space where the initial conditions lead to some special behaviour (eg arnold diffusion or scattering/capture by resonances). The stochastic approach becomes useful when these different parts of phase space leading to qualitatively different behaviour are very intertwined, so that when we represent an experimental initial condition by some block in phase space, the measure of the different parts of phase space can be thought of as probabilities for the one or the other behaviour.

    vanees71 : Thanks for your suggestion. I am already familiar with Arnold's book which covers the basics of hamiltonian perturbation theory in its final chapter.
     
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