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

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 '09 at 2:33
Yes. Thanks for noticing. – Ricardo Nov 9 '09 at 1:37
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? – Duke Leto Nov 10 '09 at 7:27
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 '09 at 1:32

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 '09 at 9:00