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- Thread starter dwsmith
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- Jan 26, 2012

- 890

What have you tried?$$

\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 might help?

CB

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- #3

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.What have you tried?

What might help?

CB

- Jan 26, 2012

- 890

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

http://www.ingelec.uns.edu.ar/asnl/Materiales/Cap04Extras/VanDerPol/BuonomoSIAM.pdf

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

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So I would integrateThe method for finding the asymptotic form for the period is complicated but elementary (another singular perturbation series problem), Google will if you are careful will turn up links which show how it is found, In particular see:

http://www.ingelec.uns.edu.ar/asnl/Materiales/Cap04Extras/VanDerPol/BuonomoSIAM.pdf

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

$$

\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

$$

\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

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.

- Jan 26, 2012

- 890

Sorry, that makes no sense, please provide context.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?

CB

- Jan 26, 2012

- 890

From the nature of the Google hits you do realise that the question you have asked is a research level problem don't you?

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.

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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- #10

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

- Jan 26, 2012

- 890

See my first post in this thread where the second order approximation for the angular frequency from 4.7 of Buonomo in your notation is given.I don't see how to use it for a specified $\mu$ to get the period though.

CB

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- #12

When $\mu = 0$, shouldn't the period be $2\pi$? Since the other $\mu$ values are close to circular, shouldn't they be around $2\pi$ as well?See my first post in this thread where the second order approximation for the angular frequency from 4.7 of Buonomo in your notation is given.

CB

Using that formula for $\omega$, doesn't seem like it will produce that result.

- Jan 26, 2012

- 890

Go back and read the next line.When $\mu = 0$, shouldn't the period be $2\pi$? Since the other $\mu$ values are close to circular, shouldn't they be around $2\pi$ as well?

Using that formula for $\omega$, doesn't seem like it will produce that result.

Also revise the relationship between frequency \(f\), angular frequency \(\omega\) and period \(\tau\) for a periodic signal/function.

CB

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