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[[Lyapunov stability|Lyapunov stability]], uniform with respect to the initial time. A solution  $  x _ {0} ( t) $, 
$  t \in \mathbf R  ^ {+} $, 
of a system of differential equations

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

is called uniformly stable if for every  $  \epsilon > 0 $
there is a  $  \delta > 0 $
such that for every  $  t _ {0} \in \mathbf R  ^ {+} $
and every solution  $  x ( t) $
of the system satisfying the inequality

$$ 
| x ( t _ {0} ) - x _ {0} ( t _ {0} ) |  <  \delta ,
$$

the inequality

$$ 
| x ( t) - x _ {0} ( t) |  <  \epsilon
$$

holds for all  $  t \geq  t _ {0} $.

A Lyapunov-stable fixed point of an autonomous system of differential equations  $  \dot{x} = f ( x) $, 
$  x \in \mathbf R  ^ {n} $, 
is uniformly stable, but, in general, a Lyapunov-stable solution need not be uniformly stable. For example, the solution  $  x ( t) = 0 $, 
$  t \in \mathbf R  ^ {+} $, 
of the equation

$$ \tag{1 }
\dot{x}  =  [ \sin   \mathop{\rm ln}  ( 1 + t) - \alpha ] x
$$

is stable for each  $  \alpha \in ( 1/ \sqrt 2 , 1) $
but is not uniformly stable for such  $  \alpha $.

Suppose one is given a linear system of differential equations

$$ \tag{2 }
\dot{x}  =  A ( t) x,\ \ 
x \in \mathbf R  ^ {n} ,
$$

where  $  A ( \cdot ) $
is a mapping  $  \mathbf R  ^ {+} \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $
that is summable on each interval.

In order that the solution  $  x = 0 $
of (2) be uniformly stable, it is necessary that the upper singular exponent  $  \Omega  ^ {0} ( A) $
of (2) be less than or equal to zero (cf. also [[Singular exponents|Singular exponents]]). For example, in the case of equation (1), the upper singular exponent  $  \Omega  ^ {0} ( A) = 1 - \alpha $, 
and the [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]  $  \lambda _ {1} ( A) = ( 1/ \sqrt 2 ) - \alpha $. 
For the existence of a  $  \delta > 0 $
such that the solution  $  x = 0 $
of any system

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

that satisfies the conditions of the existence and uniqueness theorem for the solution of the Cauchy problem as well as the condition

$$ 
| g ( t, x) |  <  \delta | x |
$$

be uniformly stable, it is necessary and sufficient that the upper singular exponent  $  \Omega  ^ {0} ( A) $
be less than zero.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Persidskii,   "On stability of motion in a first approximation"  ''Mat. Sb.'' , '''40''' :  3  (1933)  pp. 284–293  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.P. Demidovich,   "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.L. Daletskii,   M.G. Krein,   "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR></table>

====Comments====
The upper singular exponent is also called the Bohl exponent, cf. also [[Singular exponents|Singular exponents]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Rouché,   "Stability theory by Liapunov's direct method" , Springer  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.K. Hale,   "Ordinary differential equations" , Wiley  (1969)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.A. Coppel,   "Stability and asymptotic behavior of differential equations" , D.C. Heath  (1965)</TD></TR></table>