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A method used in non-linear oscillation theory to study oscillatory processes; it is based on an averaging principle, that is, the exact differential equation of the motion is replaced by an averaged equation.

Long before the work of N.M. Krylov and N.N. Bogolyubov, various averaging schemes (Gauss, Fatou, Delone–Hill, etc.) were widely applied in celestial mechanics. These two authors have the credit of working out a general algorithm, known as the Krylov–Bogolyubov method of averaging, and proving that the solutions of the averaged system approximate those of the exact one (see [[#References|[1]]], [[#References|[2]]]). The rigorous theory of the method, with a comprehensive explanation of the essence of the general averaging principle, is due to N.N. Bogolyubov (see [[#References|[3]]], [[#References|[4]]]), who showed that the averaging method is related to the existence of a certain transformation of variables which enables one to eliminate the time  $  t $
from the free terms of the equations, up to a given degree of accuracy in terms of a small parameter  $  \epsilon $; 
he also established the asymptotic nature of the approximations that the method yield, and established a relationship between the solutions of the exact and the averaged equations over an infinite time interval. These results were later extended by Yu.A. Mitropol'skii and others (see [[#References|[5]]]–[[#References|[8]]]); they are used in the study of non-linear oscillations.

The standard form of the system of equations for which the Krylov–Bogolyubov method of averaging has been developed is:

$$ \tag{1 }
{
\frac{dx}{dt}
 }  = \ 
\epsilon X ( t, x),\ \ 
x \in \mathbf R  ^ {n} ,
$$

where  $  t $
is the time and  $  \epsilon $
is a small positive parameter. The fundamental assumptions adopted in regard to this system are that  $  X $
is a smooth function of  $  t, x $
and that this function is in a sense  "recurrent"  in  $  t $, 
implying the existence of the average

$$ 
\lim\limits _ {T \rightarrow \infty }
 {
\frac{1}{T}
 }
\int\limits _ { 0 } ^ { T } 
X ( t, x)  dt  = \ 
X _ {0} ( x),
$$

e.g.  $  X $
might be a periodic or almost-periodic function of  $  t $.

The  $  m $-
th approximation to the solution  $  x = x ( t) $
of the system (1) is, according to the method, defined by

$$ \tag{2 }
x  =  \xi + \epsilon F _ {1} ( t, \xi ) + \dots +
\epsilon  ^ {m} F _ {m} ( t, \xi ),
$$

where  $  \xi = \xi ( t) $
is the solution of the  "averaged"  equation

$$ 

\frac{d \xi }{dt }
  = \ 
\epsilon X _ {0} ( \xi ) +
\epsilon  ^ {2} P _ {2} ( \xi ) + \dots +
\epsilon  ^ {m} P _ {m} ( \xi ),
$$

$  F _ {1} , F _ {j} , P _ {j} $, 
$  j = 2 \dots m $, 
are functions chosen so that the expression (2) should satisfy equation (1) up to quantities of order  $  \epsilon ^ {m + 1 } $
and so that the functions  $  F _ {j} $
should satisfy the same recurrence conditions in  $  t $
as the free term of equation (1). The determination of the functions  $  F _ {j} $
is elementary; the functions  $  P _ {j} $
are found by averaging the right-hand side of equation (1) in which  $  x $
has been replaced by (2). In particular, if  $  X $
is a periodic function of  $  t $, 
with

$$ \tag{3 }
X ( t, x)  = \ 
X ( t + 2 \pi , x)  = \ 
\sum _ {- \infty < k < \infty }
 X _ {k} ( x) e  ^ {ikt} ,
$$

the function  $  F _ {1} $
is determined from (3) by

$$ 
F _ {1} ( t, \xi )  = \ 
\sum _ {k \neq 0 }

\frac{X _ {k} ( \xi ) }{ik }

e  ^ {ikt} ,
$$

and the functions  $  F _ {m} , P _ {m} $(
$  m \geq  2 $) 
are determined by analogous formulas using the relation

$$ 
X ( t, \xi + \epsilon F _ {1} ( t, \xi ) + \dots +
\epsilon ^ {m - 1 } F _ {m - 1 }  ( t, \xi )) =
$$

$$ 
= \ 
X ( t, \xi ) + \epsilon 
\frac{\partial  X ( t, \xi ) }{\partial  x }
 F _ {1} ( t, \xi ) + \dots +
$$

$$ 
+ 
\epsilon ^ {m - 1 } 
\frac{\partial  X ( t, \xi ) }{\partial  x }
 F _ {m - 1 }  ( t, \xi )
$$

The validity of the averaging method is established as follows. 1) One proves an estimate of the type

$$ 
\| \mathbf x ( t) - \xi ( t) \|
 \leq   \eta ( \epsilon ),\ \ 
t \in \left [ 0, {
\frac{L} \epsilon 
 } \right ] ,
$$

where  $  \eta ( \epsilon ) \rightarrow 0 $
as  $  \epsilon \rightarrow 0 $, 
$  L $
is a constant independent of  $  \epsilon $, 
and  $  x ( 0) = \xi ( 0) $; 
2) one proves the existence of a solution  $  x = x _ {0} ( t) $
of the system (1) which lies in a sufficiently small neighbourhood of the equilibrium position  $  \xi = \xi _ {0} $, 
$  X _ {0} ( \xi _ {0} ) = 0 $, 
of the averaged system:

$$ 
\sup _ {t \in (- \infty , \infty ) }
 \| x ( t) - \xi _ {0} \|  \leq   \eta ( \epsilon ),
$$

and shows that this solution is stable and periodic or almost-periodic; 3) one proves the existence of an integral manifold  $  \tau $:

$$ 
x  =  f ( t, \phi , \epsilon ),\ \ 
f ( t, \phi + 2 \pi , \epsilon )  = \ 
f ( t, \phi , \epsilon ),
$$

of the system (1), in a neighbourhood of a periodic trajectory  $  \xi = \xi _ {0} ( \phi ) $, 
$  \phi = \epsilon \nu $, 
$  \nu = \textrm{ const } $, 
of the averaged system:

$$ 
\sup _ {t, \phi \in (- \infty , \infty ) }
 \| f ( t, \phi , \epsilon ) - \xi _ {0} ( \phi ) \|
 \leq   \eta ( \epsilon ),
$$

and investigates the behaviour of the solutions of (1) that lie in the neighbourhood of the manifold  $  \tau $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.M. Krylov,   N.N. Bogolyubov,   "Méthodes approchées de la mécanique non-linéaire dans leurs application à l'Aeetude de la perturbation des mouvements périodiques de divers phénomènes de résonance s'y rapportant" , Kiev  (1935)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.M. Krylov,   N.N. Bogolyubov,   "Introduction to non-linear mechanics" , Princeton Univ. Press  (1947)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Bogolyubov,   "On certain statistical methods in mathematical physics" , Kiev  (1945)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.N. Bogolyubov,   ''Sb. Tr. Inst. Stroitel. Mekh. Akad. Nauk SSSR'' , '''14'''  (1950)  pp. 9–34</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.N. Bogolyubov,   Yu.A. Mitropol'skii,   "Asymptotic methods in the theory of nonlinear oscillations" , Gordon &amp; Breach , Delhi  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Yu.A. Mitropol'skii,   "An averaging method in non-linear mechanics" , Kiev  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  Yu.A. Mitropol'skii,   "Nonstationary processes in non-linear oscillatory systems" , ''ATIC-270579 F-9085/V'' , Qir Techn. Intell. Transl.  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.M. Volosov,  , ''Mechanics in the USSR during 50 years'' , '''1''' , Moscow  (1968)  pp. 115–135  (In Russian)</TD></TR></table>

====Comments====
A generalization of the Krylov–Bogolyubov method to equations of the form

$$ \tag{a1 }
\dot{x}  =  A x + \epsilon X ( x , t , \epsilon )
$$

(instead of (1)) can be found in [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.K. Hale,   "Oscillations in nonlinear systems" , McGraw-Hill  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.H. Nayfeh,   "Perturbation methods" , Wiley  (1973)  pp. Sect. 5.2, 5.4</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.M. Volosov,   "Averaging in systems of ordinary differential equations"  ''Russian Math. Surveys'' , '''7'''  (1962)  pp. 1–26  ''Uspekhi Mat. Nauk'' , '''17''' :  6  (1962)  pp. 3–126</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Yu.A. Mitropolski,   "Problems of the asymptotic theory of nonstationary vibrations" , D. Davey  (1965)  (Translated from Russian)</TD></TR></table>