Are the asymptotics of Fourier coefficients to periodic solutions of ODE known? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:59:54Z http://mathoverflow.net/feeds/question/3471 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3471/are-the-asymptotics-of-fourier-coefficients-to-periodic-solutions-of-ode-known Are the asymptotics of Fourier coefficients to periodic solutions of ODE known? Ricardo 2009-10-30T21:03:07Z 2009-11-25T06:43:26Z <p>The Van der Pol equation, given by</p> <p>$$x'' + x = g x' (1 - x^2),$$</p> <p>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 $g$.</p> <p>Question: Can one find an asymptotic formula for $a_n(g)$, as $g \to\infty$?</p> <p>For example, the asymptotic formula for the period is well known: </p> <p>$$T(g) \sim g [ (3 - \log 4) + O(g^{-4/3}) ].$$</p> http://mathoverflow.net/questions/3471/are-the-asymptotics-of-fourier-coefficients-to-periodic-solutions-of-ode-known/4826#4826 Answer by Duke Leto for Are the asymptotics of Fourier coefficients to periodic solutions of ODE known? Duke Leto 2009-11-10T07:27:34Z 2009-11-10T07:27:34Z <p>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?</p> http://mathoverflow.net/questions/3471/are-the-asymptotics-of-fourier-coefficients-to-periodic-solutions-of-ode-known/4987#4987 Answer by Duke Leto for Are the asymptotics of Fourier coefficients to periodic solutions of ODE known? Duke Leto 2009-11-11T06:14:10Z 2009-11-11T06:14:10Z <p>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 ?</p>