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''of a linear homogeneous system of ordinary differential equations''
 
''of a linear homogeneous system of ordinary differential equations''
  
A [[Fundamental system of solutions|fundamental system of solutions]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067530/n0675301.png" /> such that any other fundamental system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067530/n0675302.png" /> satisfies the inequality
+
A [[Fundamental system of solutions|fundamental system of solutions]] $  x _ {1} ( t) \dots x _ {n} ( t) $
 +
such that any other fundamental system $  \widehat{x}  _ {1} ( t) \dots \widehat{x}  _ {n} ( t) $
 +
satisfies the inequality
  
<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/n/n067/n067530/n0675303.png" /></td> </tr></table>
+
$$
 +
\sum _ { i= } 1 ^ { n }
 +
\lambda _ {\widehat{x}  _ {i ( t) }  }  \geq  \
 +
\sum _ { i= } 1 ^ { n }
 +
\lambda _ {x _ {i ( t) }  } ;
 +
$$
  
 
here
 
here
  
<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/n/n067/n067530/n0675304.png" /></td> </tr></table>
+
$$
 +
\lambda _ {y ( t) }  = \
 +
\overline{\lim\limits}\; _
 +
{t \rightarrow + \infty } 
 +
\frac{1}{t}
 +
  \mathop{\rm log}  | y ( t) |
 +
$$
  
is the [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]] of a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067530/n0675305.png" />. Normal fundamental systems of solutions were introduced by A.M. Lyapunov [[#References|[1]]], who proved that they exist for every linear system
+
is the [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]] of a solution $  y ( t) $.  
 +
Normal fundamental systems of solutions were introduced by A.M. Lyapunov [[#References|[1]]], who proved that they exist for every linear 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/n/n067/n067530/n0675306.png" /></td> </tr></table>
+
$$
 +
\dot{x}  = A ( t) x ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067530/n0675307.png" /> is a mapping
+
where $  A ( \cdot ) $
 +
is a mapping
  
<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/n/n067/n067530/n0675308.png" /></td> </tr></table>
+
$$
 +
\mathbf R  ^ {+}  \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) \ \
 +
( \textrm{ or }  \mathbf R  ^ {+}  \rightarrow  \mathop{\rm Hom} ( \mathbf C  ^ {n} ,\
 +
\mathbf C  ^ {n} ) )
 +
$$
  
 
that is summable on every segment and satisfies the additional condition
 
that is summable on every segment and satisfies the additional condition
  
<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/n/n067/n067530/n0675309.png" /></td> </tr></table>
+
$$
 +
\overline{\lim\limits}\; _ {t \rightarrow \infty } \
 +
 
 +
\frac{1}{t}
 +
\int\limits _ { 0 } ^ { t }
 +
\| A ( \tau ) \|  dt  < + \infty .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,  "Collected works" , '''1–5''' , Moscow-Leningrad  (1956)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,  "Collected works" , '''1–5''' , Moscow-Leningrad  (1956)  (In Russian)</TD></TR></table>

Revision as of 08:03, 6 June 2020


of a linear homogeneous system of ordinary differential equations

A fundamental system of solutions $ x _ {1} ( t) \dots x _ {n} ( t) $ such that any other fundamental system $ \widehat{x} _ {1} ( t) \dots \widehat{x} _ {n} ( t) $ satisfies the inequality

$$ \sum _ { i= } 1 ^ { n } \lambda _ {\widehat{x} _ {i ( t) } } \geq \ \sum _ { i= } 1 ^ { n } \lambda _ {x _ {i ( t) } } ; $$

here

$$ \lambda _ {y ( t) } = \ \overline{\lim\limits}\; _ {t \rightarrow + \infty } \frac{1}{t} \mathop{\rm log} | y ( t) | $$

is the Lyapunov characteristic exponent of a solution $ y ( t) $. Normal fundamental systems of solutions were introduced by A.M. Lyapunov [1], who proved that they exist for every linear system

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

where $ A ( \cdot ) $ is a mapping

$$ \mathbf R ^ {+} \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) \ \ ( \textrm{ or } \mathbf R ^ {+} \rightarrow \mathop{\rm Hom} ( \mathbf C ^ {n} ,\ \mathbf C ^ {n} ) ) $$

that is summable on every segment and satisfies the additional condition

$$ \overline{\lim\limits}\; _ {t \rightarrow \infty } \ \frac{1}{t} \int\limits _ { 0 } ^ { t } \| A ( \tau ) \| dt < + \infty . $$

References

[1] A.M. Lyapunov, "Collected works" , 1–5 , Moscow-Leningrad (1956) (In Russian)
How to Cite This Entry:
Normal fundamental system of solutions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_fundamental_system_of_solutions&oldid=48014
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article