## Are the asymptotics of Fourier coefficients to periodic solutions of ODE known?

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The Van der Pol equation, given by

$$x'' + x = g x' (1 - x^2),$$

has periodic solutions $x(t)$, with the period $T(g)$ depending on the parameter. Thus, one can expand $x(t)$ as a Fourier series with coefficients $a_n(g)$ also depending on $g$.

Question: Can one find an asymptotic formula for $a_n(g)$, as $g \to\infty$?

For example, the asymptotic formula for the period is well known:

$$T(g) \sim g [ (3 - \log 4) + O(g^{-4/3}) ].$$

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I believe it should be $x''+x=gx'(1-x^2)$. At least, that is what one usually means by the Van der Pol equation.   – fedja Nov 3 2009 at 2:33
Yes. Thanks for noticing. – Ricardo Nov 9 2009 at 1:37

## 2 Answers

This sounds like a homework problem :) By which means are we allowed to derived the form of $a_n(g)$ ? What is the asymptotic formula to be used for?

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No, not a homework problem. I'm working (with some collegues) on improved perturbation theory methods (for large parameters) that use the asymptotics of whatever one wants to calculate.  The method works well for calculating the period of Van der Pol solutions, for instance, or energy levels of nonharmonic oscillators (or even wave functions).  It works OK for the actual solutions of Van der Pol, but the problem is that the asymptotics are not known (at least to us).  To answer your first question: you can use whatever takes to calculate the asymptotics of $a_n(g)$. The second: I don't know. – Ricardo Nov 11 2009 at 1:32

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I am quite sure I found a perturbative solution to this during graduate school, what you are asking for is the Fourier representation of that. Are you familiar with the Method of Dominant Balance ?

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Perturbative solutions to the Van der Pol are known. However, we are interested in large values of the parameter $g$, where those methods don't work.  – Ricardo Nov 11 2009 at 9:00
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