[SOLVED] period of a limit cycle

[dwsmith]

$$
\begin{cases}
x'=.05\left[y-\left(\frac{1}{3}x^3 - x\right)\right]\\
y'=-\frac{1}{.05}x
\end{cases}
$$
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?

 

[CaptainBlack]

$$
\begin{cases}
x'=.05\left[y-\left(\frac{1}{3}x^3 - x\right)\right]\\
y'=-\frac{1}{.05}x
\end{cases}
$$
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

 

[dwsmith]

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.

 

[CaptainBlack]

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), 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

 

[dwsmith]

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:

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

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?

 

[dwsmith]

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?

 

[dwsmith]

http://www.iaeng.org/publication/IMECS2011/IMECS2011_pp1539-1544.pdf

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.

 

[CaptainBlack]

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

 

[CaptainBlack]

http://www.iaeng.org/publication/IMECS2011/IMECS2011_pp1539-1544.pdf

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

 

[dwsmith]

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.

 

[CaptainBlack]

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

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

 

[dwsmith]

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

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.

 

[CaptainBlack]

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.

Go back and read the next line.

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

CB