
#1
October 12th, 2012,
00:05
$$
\begin{cases}
x'=.05\left[y\left(\frac{1}{3}x^3  x\right)\right]\\
y'=\frac{1}{.05}x
\end{cases}
$$
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?

October 12th, 2012 00:05
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#2
October 12th, 2012,
06:43
Originally Posted by
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?
What have you tried?
What might help?
CB

#3
October 12th, 2012,
13:46
Thread Author
Originally Posted by
CaptainBlack
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.

#4
October 12th, 2012,
15:15
Originally Posted by
dwsmith
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.

#5
October 12th, 2012,
15:39
Thread Author
Originally Posted by
CaptainBlack
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?

#6
October 12th, 2012,
17:56
Thread Author
under the heading average equations for van del pol there was
$$
\int\frac{8dr}{r(4r^2)}=\int dT
$$
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?

#7
October 12th, 2012,
19:35
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.

#8
October 12th, 2012,
23:44
Originally Posted by
dwsmith
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

#9
October 12th, 2012,
23:56
Originally Posted by
dwsmith
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

#10
October 12th, 2012,
23:58
Thread Author
Originally Posted by
CaptainBlack
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.