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Derivation of van der pol oscillator

  1. Jan 29, 2015 #1
    1. The problem statement, all variables and given/known data
    Given characteristic equation for a circuit containing a diode, I must figure out how to fit a polynomial to the curve so that the van der pol equation is obtained.

    The paper I am reading is here:

    http://www.unige.ch/math/hairer60/pres/pres_rentrop.pdf

    My doubts are located on page three where the author talks about fitting the polynomial of degree three. I am trying to fill in the details as it isn't quite clear what he is actually doing.

    Suggestions?

    Should I plug the polynomial into the characteristic equation and manipulate it? Or is there some other technique?
     
  2. jcsd
  3. Jan 30, 2015 #2
    He is just filling it in I think.

    [tex] C\ddot{U} + \left( \frac{RC}{L} + \frac{d}{du}\left(a_1U +a_2U^2 +a_3U^3\right)\right)\dot{U} + \frac{1}{L}\left(R(a_1U +a_2U^2 +a_3U^3) + U - U_{op}\right) [/tex]

    It looks like it is going to look as is stated. Have you tried it and doesn't it work?
     
  4. Jan 30, 2015 #3

    Yea, I tried plugging everything in like you just did, but I could not figure out how to get rid of the U^3 term.
     
  5. Jan 31, 2015 #4

    mfb

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    Staff: Mentor

    In the bracket it vanishes due to the derivative, but outside it does not.
    Setting y=U+d with the right constant d gets rid of the U term in the brackets, but then you still have those U^2 and U^3 outside.
     
  6. Feb 1, 2015 #5
    That's exactly what I don't want. I'm trying to figure out how they get to the van der pol equation.
     
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