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,
|||V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)|
|||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.
Monodromy group. M.V. Fedoryuk (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Monodromy_group&oldid=15529