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A method in which the right-hand side of a system of differential equations

$$ \tag{1 }

\frac{dx}{dt}
  =  f( t, x)
$$

is represented in the form

$$ 
f( t, x)  =  f _ {0} ( t, x) +
\epsilon g( t, x),\ \ 
\epsilon = 1 ,\ \ 
g= f - f _ {0} ,
$$

where  $  f _ {0} $
is the principal part (in some sense) of the vector function  $  f $, 
and  $  g $
is the totality of second-order terms. The decomposition of  $  f $
into  $  f _ {0} $
and  $  g $
is usually determined by the physical or analytical nature of the problem described by the system (1). Besides this system, the system with a parameter,

$$ \tag{2 }

\frac{dx _  \epsilon  }{dt}
  = \ 
f _ {0} ( t, x _  \epsilon  )+
\epsilon g ( t, x _  \epsilon  ),
$$

is also considered; if  $  \epsilon = 0 $, 
this system becomes the degenerate system

$$ \tag{3 }

\frac{dx _ {0} }{dt}
  = \ 
f _ {0} ( t, x _ {0} ).
$$

If  $  f( t, x) $
and  $  g( t, x) $
are holomorphic in a neighbourhood of a point  $  ( \tau , \xi ) $, 
the system (2) has the solution  $  x _  \epsilon  ( t;  \tau , \xi ) $, 
$  {x _  \epsilon  } ( \tau ;  \tau , \xi ) = \xi $
for values of  $  \epsilon $
which are, in modulus, sufficiently small. This solution can be represented in a neighbourhood of the initial values  $  ( \tau , \xi ) $
as a power series in  $  \epsilon $:

$$ \tag{4 }
x _  \epsilon  ( t;  \tau , \xi )  = \ 
x _ {0} ( t;  \tau , \xi )+
\epsilon \phi _ {1} ( t ;  \tau , \xi ) + \dots +
$$

$$ 
+ 
\epsilon  ^ {n} \phi _ {n} ( t;  \tau , \xi ) + \dots
,\  \phi _ {k} ( \tau ;  \tau , \xi )  =  0
$$

(in certain cases non-zero initial values may also be specified for  $  \phi _ {k} $). 
If the series (4) converges for  $  \epsilon = 1 $, 
it supplies the solution of the system (1) with initial values  $  ( \tau , \xi ) $. 
For an effective construction of the coefficients  $  \phi _ {n} $
it is sufficient to have the general solution of system (3) and a partial solution  $  z( t;  \tau , 0) $
of an arbitrary system

$$ 

\frac{dz}{dt}
  =  f _ {0} ( t, z) + h( t),
$$

where  $  h( t) $
is holomorphic in a neighbourhood of  $  t = \tau $.

In particular, all  $  \phi _ {n} $
can be successively determined by quadratures if  $  {f _ {0} } ( t, x) = Ax $, 
where  $  A $
is a constant matrix.

The method of parameter introduction is very extensively employed in the theory of non-linear oscillations [[#References|[3]]] for the construction of periodic solutions of the system (1). (See also [[Small parameter, method of the|Small parameter, method of the]].) The method was employed by P. Painlevé to classify second-order differential equations whose solutions have no moving critical singular points (cf. [[Painlevé equation|Painlevé equation]]). The following theorem is true: Systems with fixed critical points can only be constituted by systems (1) which, after the introduction of a suitable parameter  $  \epsilon $, 
have systems without moving critical singular points as the degenerate systems (3). The parameter-introduction method is widely employed to construct new classes of essentially non-linear differential systems (1) without moving critical singular points, and in the study of systems belonging to these classes (cf. [[Singular point|Singular point]] of a differential equation).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,   "New methods of celestial mechanics" , '''1–3''' , NASA  (1967)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.M. Lyapunov,   "Stability of motion" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Bogolyubov,   Yu.A. Mitropol'skii,   "Asymptotic methods in the theory of non-linear oscillations" , Hindushtan Publ. Comp. , Delhi  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Golubev,   "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.P. Erugin,   "The analytic theory and problems of the real theory of differential equations with the first method and with methods of the analytic theory"  ''Differential Equations N.Y.'' , '''3''' :  11  (1967)  pp. 943–966  ''Differentsial'nye Uravneniya'' , '''3''' :  11  (1967)  pp. 1821–1863</TD></TR></table>

====Comments====
There exists no equivalent in the Western literature to the terminology parameter-introduction method. Systems of the structure (2) arise naturally in two ways:

The system (1) is non-linear and one wishes to study  "small solutions"  by a transformation  $  X( t) = \epsilon x _  \epsilon  ( t) $. 
Here  $  f _ {0} ( t, x _  \epsilon  ) $
is the linearization. Alternatively, (2) can be a perturbation of (3), including some effects that are neglected in (3) (for example, damping). In both cases  $  \epsilon $
is small. In mathematical terms, what is described is simply an iteration procedure. Convergence up to  $  \epsilon = 1 $
is sometimes observed, but should be considered exceptional.