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  1. MHB Master

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    #1
    $$
    \begin{cases}
    x'=.05\left[y-\left(\frac{1}{3}x^3 - x\right)\right]\\
    y'=-\frac{1}{.05}x
    \end{cases}
    $$
    So this a Van de Pol equation where $\mu = .05$. It is basically a circle at the origin with radius 2. How do I find the period?

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    #2
    Quote Originally Posted by dwsmith View Post
    $$
    \begin{cases}
    x'=.05\left[y-\left(\frac{1}{3}x^3 - x\right)\right]\\
    y'=-\frac{1}{.05}x
    \end{cases}
    $$
    So this a Van de Pol equation where $\mu = .05$. It is basically a circle at the origin with radius 2. How do I find the period?
    What have you tried?

    What might help?

    CB

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    #3 Thread Author
    Quote Originally Posted by CaptainBlack View Post
    What have you tried?

    What might help?

    CB
    I only know how to solve for large mu. I read the section in Strogatz book but it didn't tell me anything or I couldn't decipher the meaning.

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    #4
    Quote Originally Posted by dwsmith View Post
    I only know how to solve for large mu. I read the section in Strogatz book but it didn't tell me anything or I couldn't decipher the meaning.
    The method for finding the asymptotic form for the period is complicated but elementary (another singular perturbation series problem), will if you are careful will turn up links which show how it is found, In particular see:



    But to paraphrase and simplify equation 4.7 of the above paper, for small \(\mu\) we have the asymptotic approximation:
    \[\omega(\mu)\approx 1-\frac{\mu^2}{16}\] and the period \(\tau=\frac{2\pi}{\omega}\)

    The straight forward method to find the period is in fact to numerically integrate the equation and extract the period from a record of the time history of the path.

    CB
    Last edited by CaptainBlack; October 12th, 2012 at 15:20.

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    Quote Originally Posted by CaptainBlack View Post
    The method for finding the asymptotic form for the period is complicated but elementary (another singular perturbation series problem), will if you are careful will turn up links which show how it is found, In particular see:



    But to paraphrase and simplify equation 4.7 of the above paper, for small \(\mu\) we have the asymptotic approximation:
    \[\omega(\mu)\approx 1-\frac{\mu^2}{16}\] and the period \(\tau=\frac{2\pi}{\omega}\)

    The straight forward method to find the period is in fact to numerically integrate the equation and extract the period from a record of the time history of the path.

    CB
    So I would integrate
    $$
    \int\tau d\tau
    $$
    What would be the bounds? $[0,2\pi]$? What happens if $\mu$ is small but the limit cycle is no longer circular?

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    #6 Thread Author
    under the heading average equations for van del pol there was
    $$
    \int\frac{8dr}{r(4-r^2)}=\int dT
    $$
    Then they have
    $$
    x(t,\mu) = \frac{2}{\sqrt{1+3e^{-\mu t}}}\cos t +\mathcal{O}(\mu)
    $$
    Does plugging in $\mu$ here yield the period?
    If so, what would be t?

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    #7 Thread Author


    I read this paper on the part about two timing the van der pole equation. It presents a solution for small $\mu$ but I don't understand how to use it.

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    #8
    Quote Originally Posted by dwsmith View Post
    So I would integrate
    $$
    \int\tau d\tau
    $$
    What would be the bounds? $[0,2\pi]$? What happens if $\mu$ is small but the limit cycle is no longer circular?
    Sorry, that makes no sense, please provide context.

    CB

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    #9
    Quote Originally Posted by dwsmith View Post


    I read this paper on the part about two timing the van der pole equation. It presents a solution for small $\mu$ but I don't understand how to use it.
    From the nature of the Google hits you do realise that the question you have asked is a research level problem don't you?

    The SIAM J Appl Math paper by Buonomo that I gave a link to gives a relatively straight forward treatment of the problem and in equation 4.7 a direct answer to the question asked (which is skated over in the last paragraph of the link in your last post).

    CB

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    Quote Originally Posted by CaptainBlack View Post
    From the nature of the Google hits you do realise that the question you have asked is a research level problem don't you?

    The SIAM J Appl Math paper by Buonomo that I gave a link to gives a relatively straight forward treatment of the problem and in equation 4.7 a direct answer to the question asked.

    CB
    I don't see how to use it for a specified $\mu$ to get the period though.

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