Derivation of van der pol oscillator

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Homework Help Overview

The discussion revolves around deriving the van der Pol oscillator equation from a characteristic equation related to a circuit containing a diode. The original poster expresses confusion about fitting a polynomial of degree three as mentioned in a referenced paper, particularly regarding the manipulation of terms to achieve the desired form.

Discussion Character

  • Exploratory, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • The original poster considers whether to plug the polynomial into the characteristic equation and manipulate it, while others suggest that the polynomial fitting process might be straightforward. There are discussions about how certain terms behave when derivatives are applied and the implications of setting specific variables to simplify the equation.

Discussion Status

Participants are actively exploring different approaches to manipulate the characteristic equation, with some providing insights into how terms might cancel or transform. There is a recognition of the challenges in eliminating certain terms, particularly the cubic term, indicating a productive exchange of ideas without a clear consensus on the next steps.

Contextual Notes

Participants are working under the constraints of the original paper's methodology and are grappling with the specifics of polynomial fitting and term manipulation as outlined in the text. The original poster's goal is to derive the van der Pol equation, which adds a layer of complexity to the discussion.

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Homework Statement


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?
 
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He is just filling it in I think.

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)

It looks like it is going to look as is stated. Have you tried it and doesn't it work?
 
barefeet said:
He is just filling it in I think.

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)

It looks like it is going to look as is stated. Have you tried it and doesn't it work?
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.
 
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.
 
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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