View source for Normal fundamental system of solutions

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''of a linear homogeneous system of ordinary differential equations''

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

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

here

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

is the [[Lyapunov characteristic exponent|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

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

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====
<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>