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Hi, 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. 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 $z$. Thanks, Victor |
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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 this paper, Floquet himself worked a bit on this generalization, deriving his theory for ODEs with singly periodic coefficients from the more general case). |
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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 monodromy, 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. |
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