View source for Almost-reducible linear system

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''of ordinary differential equations''

A system

$$ \tag{* }
\dot{x}  =  A (t) x ,\ \ 
x \in \mathbf R  ^ {n} ,
$$

$$ 
A ( \cdot ) :  \mathbf R  \rightarrow   \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) ,
$$

having the following property: There exist a system  $  \dot{y} = B y $, 
$  y \in \mathbf R  ^ {n} $, 
with constant coefficients and, for every  $  \epsilon > 0 $, 
a [[Lyapunov transformation|Lyapunov transformation]]  $  L _  \epsilon  (t) $
such that by the change of variables  $  x = L _  \epsilon  (t) y $, 
the system (*) is transformed into the system

$$ 
\dot{y}  =  ( B + C _  \epsilon  (t) ) y ,
$$

where

$$ 
\sup _ {t \in \mathbf R } \ 
\| C _  \epsilon  (t) \|
 <  \epsilon .
$$

Every [[Reducible linear system|reducible linear system]] is almost reducible.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.A. Izobov,   "Linear systems of ordinary differential equations"  ''J. Soviet Math.'' , '''5''' :  1  (1976)  pp. 46–96  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''12'''  (1974)  pp. 71–146</TD></TR></table>