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''of ordinary differential equations''

A system of the form

$$ \tag{1 }
\dot{x}  =  A( t) x,\  x \in \mathbf R  ^ {n}
$$

(where  $  A( \cdot ) $
is a mapping  $  \mathbf R  ^ {+} \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {m} ) $
that is summable on every interval and has the property that

$$ 
\lim\limits _ {t \rightarrow \infty }  
\frac{1}{t}
 \int\limits _ { 0 } ^ { t } 
 \mathop{\rm Tr}  A( \tau )  d \tau
$$

exists and is equal to  $  \sum _ {i= 1 }  ^ { n }  \lambda _ {i} ( A) $, 
where  $  \lambda _ {1} ( A) \geq  \dots \geq  \lambda _ {n} ( A) $
are the characteristic Lyapunov exponents (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]) of the system (1)).

For a triangular system

$$ 
\dot{u}   ^ {i}  =  \sum _ {j= i } ^ { n }  p _ {ij} ( t) u
 ^ {j} ,\  i= 1 \dots n,
$$

to be regular it is necessary and sufficient that the limits

$$ 
\lim\limits _ {t \rightarrow \infty }  
\frac{1}{t}
 \int\limits _ { 0 } ^ { t }  p _ {ii} ( \tau )  d \tau ,\  i= 1 \dots n,
$$

exist (Lyapunov's criterion). Every [[Reducible linear system|reducible linear system]] and every [[Almost-reducible linear system|almost-reducible linear system]] is regular.

The role of the concept of a regular linear system is clarified by the following theorem of Lyapunov. Let the system (1) be regular and let  $  k $
of its characteristic Lyapunov exponents be negative:

$$ 
0  >  \lambda _ {n-} k+ 1 ( A)  \geq  \dots \geq   \lambda _ {n} ( A).
$$

Then for every system

$$ \tag{2 }
\dot{x}  =  A( t) x + g( t, x),
$$

where  $  g( t, x) $
satisfies the following conditions:  $  g $
and  $  g _ {x}  ^  \prime  $
are continuous, and  $  g( t, 0)= 0 $, 
$  \sup _ {t \geq  0 }   \| g _ {x}  ^  \prime  ( t, x) \| = O( | x |  ^  \epsilon  ) $, 
where  $  \epsilon = \textrm{ const } > 0 $, 
there is a  $  k $-
dimensional manifold  $  V  ^ {k} \subset  \mathbf R  ^ {n} $
containing the point  $  x= 0 $, 
such that every solution  $  x( t) $
of (2) starting on  $  V  ^ {k} $(
i.e.  $  x( 0) \in V  ^ {k} $) 
exponentially decreases as  $  t \rightarrow \infty $; 
more precisely, for every  $  \delta > 0 $
there is a  $  C _  \delta  $
such that the inequality

$$ 
| x( t) |  \leq   C _  \delta  e ^ {[ \lambda _ {n-} k+ 1 ( A)+
\delta ] t } | x( 0) |
$$

is satisfied.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,   "Stability of motion" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.F. Bylov,   R.E. Vinograd,   D.M. Grobman,   V.V. Nemytskii,   "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[3]</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''' :  1  (1974)  pp. 71–146</TD></TR></table>