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Archived Nonlinear Dynamics and Chaos, Strogatz: 2.1.5

  1. Jun 3, 2013 #1
    1. The problem statement, all variables and given/known data
    a) Find a mechanical system that is approximately governed by [itex]\dot{x}=sin(x)[/itex]
    b) Using your physical intuition, explain why it now becomes obvious that x*=0 is an unstable fixed point and x*=[itex]\pi[/itex] is stable.

    2. Relevant equations

    [itex]\dot{x}=sin(x)[/itex] (?)

    3. The attempt at a solution
    I'm thinking a pendulum can be used as a mechanical system that varies with sinθ, but I'm not sure how to solidify my answer.

    Could it possibly be an inverted pendulum in a very viscous medium?
     
    Last edited: Jun 3, 2013
  2. jcsd
  3. Feb 5, 2016 #2

    haruspex

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    Consider a particle sliding over terrain y=f(x). Assuming conservation of energy (and some convenient total) what is the PE at x? What does that yield for f?
     
  4. Feb 6, 2016 #3

    epenguin

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    Question of understanding what they want. I think all you need to do is draw a diagram of the given function (familiar!) extending a little further on both sides than the points mentioned; xdot is the ordinate, but more important, with little horizontal arrows show which way x Is moving on each side of the named points, and you will soon see what they mean about stability/instability. Just explain this in your own words.
     
    Last edited: Feb 7, 2016
  5. Jan 5, 2017 #4
    Immerse your pendulum in honey and evolve your equations. When you finish, you're gonna get a second order differencial equation. At this point, you gotta be audacious and destroy the second order term. Voilà.
     
  6. Jan 5, 2017 #5

    haruspex

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    The OP never came back, and this is years old.
    But for what it's worth, I believe my suggestion in post #2 gives a very easy model, no approximations needed.
     
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