Monodromy group

of an ordinary linear differential equation or system of equations

The group of $( n \times n )$- matrices associated with the $n$- th order system

$$\tag{* } \dot{x} = A ( t) x ,$$

defined as follows. Let the matrix $A ( t)$ be holomorphic in a domain $G \subset \mathbf C$, let $t _ {0} \in G$ and let $X ( t)$ be the fundamental matrix of the system (*) given in a small neighbourhood of $t _ {0}$. If $\gamma \subset G$ is a closed curve with initial point $t _ {0}$, then by analytic continuation along $\gamma$, $X ( t) \rightarrow X ( t) C _ \gamma$, where $C _ \gamma$ is a constant $( n \times n )$- matrix. If two curves $\gamma _ {1} , \gamma _ {2}$ are homotopic in $G$, then $C _ {\gamma _ {2} } = C _ {\gamma _ {1} }$; if $\gamma = \gamma _ {1} \gamma _ {2}$, then $C _ \gamma = C _ {\gamma _ {1} } C _ {\gamma _ {2} }$. The mapping $\gamma \rightarrow C _ \gamma$ is a homomorphism of the fundamental group of $G$:

$$\pi _ {1} ( G , t _ {0} ) \rightarrow \mathop{\rm GL} ( n , \mathbf C ) ,$$

where $\mathop{\rm GL} ( n , \mathbf C )$ is the group of $( n \times n )$- matrices with complex entries; the image of this homomorphism is called the monodromy group $M ( t _ {0} , G )$ of (*). In this connection,

$$M ( t _ {1} , G ) = T ^ {-} 1 M ( t _ {0} , G ) T ,$$

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

References

[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)

Comments

Cf. also Monodromy matrix and Monodromy operator. If $\gamma ( s)$ is a closed differentiable curve in $G$ with initial point $t _ {0}$, then $Y ( s) = X ( \gamma ( s) )$ satisfies a matrix equation $\dot{Y} ( s) = \dot \gamma ( s) A ( \gamma ( s) ) Y ( s)$ and $C _ \gamma$ is the monodromy matrix of this system of linear differential equations with periodic coefficients.