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''of an ordinary linear differential equation or system of equations''
 
''of an ordinary linear differential equation or system of equations''
  
The group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m0646601.png" />-matrices associated with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m0646602.png" />-th order system
+
The group of $  ( n \times n ) $-
 +
matrices associated with the $  n $-
 +
th order system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m0646603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\dot{x}  = A ( t) x ,
 +
$$
  
defined as follows. Let the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m0646604.png" /> be holomorphic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m0646605.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m0646606.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m0646607.png" /> be the [[Fundamental matrix|fundamental matrix]] of the system (*) given in a small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m0646608.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m0646609.png" /> is a closed curve with initial point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466010.png" />, then by analytic continuation along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466013.png" /> is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466014.png" />-matrix. If two curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466015.png" /> are homotopic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466017.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466019.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466020.png" /> is a homomorphism of the [[Fundamental group|fundamental group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466021.png" />:
+
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|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|fundamental group]] of $  G $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466022.png" /></td> </tr></table>
+
$$
 +
\pi _ {1} ( G , t _ {0} )  \rightarrow  \mathop{\rm GL} ( n , \mathbf C ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466023.png" /> is the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466024.png" />-matrices with complex entries; the image of this homomorphism is called the monodromy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466025.png" /> of (*). In this connection,
+
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,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466026.png" /></td> </tr></table>
+
$$
 +
M ( t _ {1} , G )  = T  ^ {-} 1 M ( t _ {0} , G ) T ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466027.png" /> is a constant matrix. The monodromy group has been computed for the equations of Euler and Papperitz (see [[#References|[1]]], [[#References|[2]]]).
+
where $  T $
 +
is a constant matrix. The monodromy group has been computed for the equations of Euler and Papperitz (see [[#References|[1]]], [[#References|[2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Golubev,  "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Ince,  "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Golubev,  "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Ince,  "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Cf. also [[Monodromy matrix|Monodromy matrix]] and [[Monodromy operator|Monodromy operator]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466028.png" /> is a closed differentiable curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466029.png" /> with initial point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466031.png" /> satisfies a matrix equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064660/m06466033.png" /> is the monodromy matrix of this system of linear differential equations with periodic coefficients.
+
Cf. also [[Monodromy matrix|Monodromy matrix]] and [[Monodromy operator|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.

Latest revision as of 08:01, 6 June 2020


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.

How to Cite This Entry:
Monodromy group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monodromy_group&oldid=47884
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article