Monodromy group

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of an ordinary linear differential equation or system of equations

The group of -matrices associated with the -th order system


defined as follows. Let the matrix be holomorphic in a domain , let and let be the fundamental matrix of the system (*) given in a small neighbourhood of . If is a closed curve with initial point , then by analytic continuation along , , where is a constant -matrix. If two curves are homotopic in , then ; if , then . The mapping is a homomorphism of the fundamental group of :

where is the group of -matrices with complex entries; the image of this homomorphism is called the monodromy group of (*). In this connection,

where is a constant matrix. The monodromy group has been computed for the equations of Euler and Papperitz (see [1], [2]).


[1] V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)
[2] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)


Cf. also Monodromy matrix and Monodromy operator. If is a closed differentiable curve in with initial point , then satisfies a matrix equation and is the monodromy matrix of this system of linear differential equations with periodic coefficients.

How to Cite This Entry:
Monodromy group. M.V. Fedoryuk (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098