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.