Is there a generalization of Floquet theory to elliptic functions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T11:48:33Z http://mathoverflow.net/feeds/question/41877 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41877/is-there-a-generalization-of-floquet-theory-to-elliptic-functions Is there a generalization of Floquet theory to elliptic functions? Victor Galitski 2010-10-12T06:04:30Z 2010-10-12T08:52:06Z <p>Hi, </p> <p>Consider a system of linear differential equations $${d f \over dz} = A(z) f,$$ where $A(z)$ is a matrix-function. If $z \in \mathbb{R}$ and the function is periodic $A(z) = A(z + T)$, Floquet theorem applies. </p> <p>I am curious to know if there exists a generalization of Floquet theorem to the case, where $z \in \mathbb{C}$ and $A(z)$ is a doubly-periodic elliptic function of $z$.</p> <p>Thanks,</p> <p>Victor</p> http://mathoverflow.net/questions/41877/is-there-a-generalization-of-floquet-theory-to-elliptic-functions/41881#41881 Answer by J. M. for Is there a generalization of Floquet theory to elliptic functions? J. M. 2010-10-12T07:24:15Z 2010-10-12T07:24:15Z <p>After some searching around, it looks to me that it was Picard who thought of generalizing Floquet theory to linear ODEs with doubly-periodic coefficients (though according to <a href="http://www.ams.org/proc/1998-126-04/S0002-9939-98-04668-1/S0002-9939-98-04668-1.pdf" rel="nofollow">this paper</a>, Floquet himself worked a bit on this generalization, deriving his theory for ODEs with singly periodic coefficients from the more general case).</p> http://mathoverflow.net/questions/41877/is-there-a-generalization-of-floquet-theory-to-elliptic-functions/41886#41886 Answer by Denis Serre for Is there a generalization of Floquet theory to elliptic functions? Denis Serre 2010-10-12T08:52:06Z 2010-10-12T08:52:06Z <p>Another, far reaching, aspect of Floquet theory in differential equations $$a_0(z)f^{(n)}(z)+\cdots+a_nf(z)=0$$ with holomorphic coefficients is Fuchs theory of <strong>monodromy</strong>, where the leading coefficient $a_0$ has a zeros at $z_0$. You cannot solve a Cauchy problem at $z_0$, but you can solve it in a pointed disk $D\setminus z_0$. When you follow a circle around $z_0$, the coefficients look periodic. </p>