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Math Help - Limit cycles

  1. #1
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    Limit cycles

    Whenever I say ... dot I mean the derivative of ...

    I have to show that the system of Ordinary Diff. Eqns

    x (dot) = y + x* f(r) / r
    y (dot) = -x + y * f(r)/r

    r=sqrt(x^2+y^2)

    has limit cycles and the limit cycles have radius which correspond to the zeros of f(r)

    Also I have to find the direction of motion on the limit cycles.


    My workings:

    I first found the nullclines.

    C(x): y + x* f(r) / r =0 and C(y): -x + y * f(r)/r =0 implies that x=y*f(r) / r

    Substituting into C(x) we get that the system has an equilibrium point at
    (y * f(r) / r, 0)

    Then we consider polar coordinates. Using r^2=x^2 + y^2

    Then differentiate to obtain that r dot = (f(r) / r^2 ) * (2x^2 + 2y^2)

    Now if 2x^2 + 2y^2 = 0 => r dot = 0

    if 2x^2 + 2y^2 > 0 => r dot >0


    Now I have to find C1 and C2 s.t they bound a region R that contains no equilibrium points and trap all trajectories so I can use Poincaré–Bendixson theorem to show that there exists a limit cycle.

    Any help would be greatly appreciated!

    Thank you!
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  2. #2
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    Re: Limit cycles

    What is f(r)?
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  3. #3
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    Re: Limit cycles

    Quote Originally Posted by Danny View Post
    What is f(r)?
    I am not given f(r) at this point, I am given f(r) at a later point, so I have to deduce generally first.
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  4. #4
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    Re: Limit cycles

    Ok, I figured them all out, thank you though
    Last edited by Darkprince; December 7th 2011 at 03:15 PM.
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  5. #5
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    Re: Limit cycles

    Quote Originally Posted by Darkprince View Post
    Using polar coordinates: x=rcosf, y=rsinf
    You don't want to do this. What you what is

    x = r \cos \theta,\;\;\;y = r \sin \theta.

    f(r) is a separate issue
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  6. #6
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    Re: Limit cycles

    Quote Originally Posted by Danny View Post
    You don't want to do this. What you what is

    x = r \cos \theta,\;\;\;y = r \sin \theta.

    f(r) is a separate issue

    What do you mean? When writing f I was meaning theta.
    Are my workings not correct?
    Last edited by Darkprince; December 6th 2011 at 04:24 PM.
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  7. #7
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    Re: Limit cycles

    Ok, I figured them all, thank you!
    Last edited by Darkprince; December 7th 2011 at 03:16 PM.
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