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The mapping  $  f $
in the system of ordinary differential equations

$$ \tag{1 }
\dot{x}  =  A ( t) x + f ( x, t).
$$

A perturbation is usually assumed to be small in some sense, for example

$$ \tag{2 }

\frac{| f ( x, t) | }{| x | }
  \rightarrow  0 \ \ 
\textrm{ if }  | x | \rightarrow 0.
$$

The solution  $  \phi ( t) $
of the perturbed system (1) and the solution  $  \Psi ( t) $
of the linear system

$$ \tag{3 }
\dot{y}  =  A ( t) y
$$

with the same initial value  $  y _ {0} $
at  $  t = t _ {0} $, 
are connected by the relation

$$ 
\phi ( t)  =  \Psi ( t) \left ( y _ {0} +
\int\limits _ {t _ {0} } ^ { t } 
\Psi  ^ {-} 1 ( \tau ) f ( \phi ( \tau ), \tau )  d \tau \right ) ,
$$

known as the formula of variation of constants, where  $  \Psi ( t) $
is the [[Fundamental matrix|fundamental matrix]] of the linear system (3).

It was shown by A.M. Lyapunov [[#References|[1]]] that the trivial solution of the system (1) is asymptotically stable (cf. [[Asymptotically-stable solution|Asymptotically-stable solution]]) if relation (2) is valid uniformly in  $  t $, 
the matrix  $  A( t) $
is constant and all real parts of the eigen values of  $  A( t) $
are negative; if only one such real part is positive, the trivial solution is not stable.

The study of the periodic solution  $  \phi $
of the system  $  \dot{x} = P( x, t) $, 
describing an oscillating process, reduces in the general case by the transformation  $  x = \phi ( t) + y $
to the study of a perturbed linear system, the right-hand side of which is periodic in  $  t $[[#References|[3]]].

====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">  L.S. Pontryagin,   "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR></table>

====Comments====
Results of this type are usually called Poincaré–Lyapunov theorems. There are several extensions, described, e.g., in [[#References|[a1]]]. A recent tutorial text containing these matters is [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Roseau,   "Vibrations non linéaires et théorie de la stabilité" , Springer  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Verhulst,   "Nonlinear differential equations and dynamical systems" , Springer  (1989)</TD></TR></table>