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The operation of calculating the mean values of functions forming part of the structure of differential equations which describe periodic, almost-periodic, and, generally, oscillating processes. The operation of averaging can be considered as a type of smoothing operator. Averaging methods first came into use in celestial mechanics in the study of planetary motion around the Sun. Later, they propagated to a wide variety of areas: to the theory of non-linear oscillations, to physics, to the theory of automatic control, to astrodynamics, and to others. Averaging methods have often provided approximate solutions for the initial equations. The most typical classes of differential equations for which averaging methods are used are the following.
 
The operation of calculating the mean values of functions forming part of the structure of differential equations which describe periodic, almost-periodic, and, generally, oscillating processes. The operation of averaging can be considered as a type of smoothing operator. Averaging methods first came into use in celestial mechanics in the study of planetary motion around the Sun. Later, they propagated to a wide variety of areas: to the theory of non-linear oscillations, to physics, to the theory of automatic control, to astrodynamics, and to others. Averaging methods have often provided approximate solutions for the initial equations. The most typical classes of differential equations for which averaging methods are used are the following.
  
 
1) Standard systems in the sense of N.N. Bogolyubov
 
1) Standard systems in the sense of N.N. Bogolyubov
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a0142401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a0142402.png" /> are vectors, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a0142403.png" /> is the time, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a0142404.png" /> is a small positive parameter.
+
\frac{dx}{dt}
 +
= \  \mu X(x,\  t,\  \mu ),
 +
$$
  
2) Multi-frequency autonomous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a0142405.png" />-periodic systems
+
where  $  x,\  X $
 +
are vectors,  $  t $
 +
is the time, and  $  \mu $
 +
is a small positive parameter.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a0142406.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
2) Multi-frequency autonomous  $  2 \pi $-periodic systems
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a0142407.png" /> are vectors,
+
$$ \tag{2}
 +
\left . {
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a0142408.png" /></td> </tr></table>
+
\frac{dx}{dt}
 +
= \  \mu X(X,\  y), \atop
 +
\frac{dy}{dt}
 +
= \  \omega (x) + \mu Y(x,\  y),}\right \}
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a0142409.png" /> is the frequency vector.
+
where  $  x,\  y,\  X,\  Y $
 +
are vectors,
 +
 
 +
$$
 +
X(x,\  y + (2 \pi )) \  \equiv \  X(x,\  y),\ \
 +
Y(x,\  y + (2 \pi )) \  \equiv \  Y(x,\  y),
 +
$$
 +
 
 +
and $  \omega (x) $
 +
is the frequency vector.
  
 
3) Multi-frequency non-autonomous systems
 
3) Multi-frequency non-autonomous systems
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3}
 +
\left . {
 +
 
 +
\frac{dx}{dt}
 +
= \  \mu X(x,\  y,\  t), \atop
 +
\frac{dy}{dt}
 +
= \  \omega (x,\  y,\  t) + \mu Y(x,\  y,\  t).}\right \}
 +
$$
  
 
Instead of the systems (1)–(3),  "simpler"  averaged systems of a first approximation are considered:
 
Instead of the systems (1)–(3),  "simpler"  averaged systems of a first approximation are considered:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1prm)</td></tr></table>
+
$$ \tag{1'}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2prm)</td></tr></table>
+
\frac{d \mathbf x}{dt}
 +
= \  \mu \mathbf X _ {0} ( \mathbf x ) ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3prm)</td></tr></table>
+
$$ \tag{2'}
 +
\left . {
 +
\frac{d \mathbf x}{dt}
 +
= \  \mu \mathbf X _ {1} (
 +
\mathbf x ), \atop
 +
\frac{d \mathbf y}{dt}
 +
= \  \omega ( \mathbf x );} \right \}
 +
$$
 +
 
 +
$$ \tag{3'}
 +
\left . {
 +
\frac{d \mathbf x}{dt}
 +
= \  \mu
 +
\mathbf X _ {2} ( \mathbf x ,\  x _ {0} ,\  y _ {0} ,\  t), \atop
 +
\frac{d
 +
\mathbf y}{dt}
 +
= \  \omega ( \mathbf x ,\  \mathbf y ,\  t),} \right \}
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4}
 +
\mathbf X _ {0} ( \mathbf x ) \  = \  \lim\limits _ {T \rightarrow \infty} \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
\frac{1}{T}
 +
\int\limits _ { 0 } ^ T X( \mathbf x ,\  t,\  0) \  dt,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424016.png" /></td> </tr></table>
+
$$ \tag{5}
 +
\mathbf X _ {1} ( \mathbf x )\  =
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
\frac{1}{(2 \pi ) ^ n}
 +
\int\limits _ { 0 } ^ {2 \pi} \dots \int\limits _ { 0 } ^ {2 \pi} X( \mathbf x ,\  y _ {1} \dots y _ {n} ) \  dy _ {1} \dots dy _ {n} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424018.png" /></td> </tr></table>
+
$$ \tag{6}
 +
\mathbf X _ {2} ( \mathbf x ,\  x _ {0} ,\  y _ {0} ,\  t _ {0} )\  =  
 +
\lim\limits _ {T \rightarrow \infty} \ 
 +
\frac{1}{T}
 +
\int\limits _ {t _ 0} ^ {t _ {0} + T} X( \mathbf x ,\  \phi (x _ {0} ,\  y _ {0} ,\  t _ {0} ,\  t),\  t) \  dt.
 +
$$
  
 
The formulas (4)–(6) express the most widespread averaging methods.
 
The formulas (4)–(6) express the most widespread averaging methods.
  
Formula (6) expresses the scheme of averaging  "along the generating solution" . In the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424019.png" />, the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424020.png" /> is initially substituted by the generating solution of the system
+
Formula (6) expresses the scheme of averaging  "along the generating solution" . In the function $  X(x,\  y,\  t) $,
 +
the vector $  y $
 +
is initially substituted by the generating solution of the system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424021.png" /></td> </tr></table>
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424022.png" /></td> </tr></table>
+
\frac{dx}{dt}
 +
= 0,
 +
$$
 +
 
 +
$$
 +
 
 +
\frac{dy}{dt}
 +
= \  \omega (x,\  y,\  t),
 +
$$
  
 
after which the integral average (6) is calculated.
 
after which the integral average (6) is calculated.
  
The principal question which arises when the systems (1)–(3) are changed is the construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424023.png" />-estimates for the norms
+
The principal question which arises when the systems (1)–(3) are changed is the construction of $  \epsilon $-
 +
estimates for the norms
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424024.png" /></td> </tr></table>
+
$$
 +
\| x(t,\  \mu ) - \mathbf x (t,\  \mu ) \| ,\ \
 +
\| y(t,\  \mu ) - \mathbf y (t,\  \mu ) \|
 +
$$
  
on the largest possible interval (of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424025.png" />) of time, if
+
on the largest possible interval (of order $  1/ \mu $)  
 +
of time, if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014240/a01424026.png" /></td> </tr></table>
+
$$
 +
x(0,\  \mu ) \  = \  \mathbf x (0,\  \mu ),\ \
 +
y(0,\  \mu ) \  = \  \mathbf y (0,\  \mu ).
 +
$$
  
 
This is the essence of the problem of averaging methods. For systems (1), this problem of averaging methods was proposed by Bogolyubov, whose results formed the basis of the modern algorithmic theory of ordinary differential equations.
 
This is the essence of the problem of averaging methods. For systems (1), this problem of averaging methods was proposed by Bogolyubov, whose results formed the basis of the modern algorithmic theory of ordinary differential equations.
Line 63: Line 139:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Bogolyubov,  Yu.A. Mitropol'skii,  "Asymptotic methods in the theory of non-linear oscillations" , Gordon &amp; Breach , Delhi  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  V.M. Volosov,  B.I. Morgunov,  "Averaging methods in the theory of non-linear oscillatory systems" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.A. Grebenikov,  Yu.A. Ryabov,  "Constructive methods of analysis of non-linear systems" , Moscow  (1979)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Bogolyubov,  Yu.A. Mitropol'skii,  "Asymptotic methods in the theory of non-linear oscillations" , Gordon &amp; Breach , Delhi  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  V.M. Volosov,  B.I. Morgunov,  "Averaging methods in the theory of non-linear oscillatory systems" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.A. Grebenikov,  Yu.A. Ryabov,  "Constructive methods of analysis of non-linear systems" , Moscow  (1979)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 03:42, 21 March 2022


The operation of calculating the mean values of functions forming part of the structure of differential equations which describe periodic, almost-periodic, and, generally, oscillating processes. The operation of averaging can be considered as a type of smoothing operator. Averaging methods first came into use in celestial mechanics in the study of planetary motion around the Sun. Later, they propagated to a wide variety of areas: to the theory of non-linear oscillations, to physics, to the theory of automatic control, to astrodynamics, and to others. Averaging methods have often provided approximate solutions for the initial equations. The most typical classes of differential equations for which averaging methods are used are the following.

1) Standard systems in the sense of N.N. Bogolyubov

$$ \tag{1} \frac{dx}{dt} \ = \ \mu X(x,\ t,\ \mu ), $$

where $ x,\ X $ are vectors, $ t $ is the time, and $ \mu $ is a small positive parameter.

2) Multi-frequency autonomous $ 2 \pi $-periodic systems

$$ \tag{2} \left . { \frac{dx}{dt} \ = \ \mu X(X,\ y), \atop \frac{dy}{dt} \ = \ \omega (x) + \mu Y(x,\ y),}\right \} $$

where $ x,\ y,\ X,\ Y $ are vectors,

$$ X(x,\ y + (2 \pi )) \ \equiv \ X(x,\ y),\ \ Y(x,\ y + (2 \pi )) \ \equiv \ Y(x,\ y), $$

and $ \omega (x) $ is the frequency vector.

3) Multi-frequency non-autonomous systems

$$ \tag{3} \left . { \frac{dx}{dt} \ = \ \mu X(x,\ y,\ t), \atop \frac{dy}{dt} \ = \ \omega (x,\ y,\ t) + \mu Y(x,\ y,\ t).}\right \} $$

Instead of the systems (1)–(3), "simpler" averaged systems of a first approximation are considered:

$$ \tag{1'} \frac{d \mathbf x}{dt} \ = \ \mu \mathbf X _ {0} ( \mathbf x ) ; $$

$$ \tag{2'} \left . { \frac{d \mathbf x}{dt} \ = \ \mu \mathbf X _ {1} ( \mathbf x ), \atop \frac{d \mathbf y}{dt} \ = \ \omega ( \mathbf x );} \right \} $$

$$ \tag{3'} \left . { \frac{d \mathbf x}{dt} \ = \ \mu \mathbf X _ {2} ( \mathbf x ,\ x _ {0} ,\ y _ {0} ,\ t), \atop \frac{d \mathbf y}{dt} \ = \ \omega ( \mathbf x ,\ \mathbf y ,\ t),} \right \} $$

where

$$ \tag{4} \mathbf X _ {0} ( \mathbf x ) \ = \ \lim\limits _ {T \rightarrow \infty} \ \frac{1}{T} \int\limits _ { 0 } ^ T X( \mathbf x ,\ t,\ 0) \ dt, $$

$$ \tag{5} \mathbf X _ {1} ( \mathbf x )\ = \frac{1}{(2 \pi ) ^ n} \int\limits _ { 0 } ^ {2 \pi} \dots \int\limits _ { 0 } ^ {2 \pi} X( \mathbf x ,\ y _ {1} \dots y _ {n} ) \ dy _ {1} \dots dy _ {n} , $$

$$ \tag{6} \mathbf X _ {2} ( \mathbf x ,\ x _ {0} ,\ y _ {0} ,\ t _ {0} )\ = \lim\limits _ {T \rightarrow \infty} \ \frac{1}{T} \int\limits _ {t _ 0} ^ {t _ {0} + T} X( \mathbf x ,\ \phi (x _ {0} ,\ y _ {0} ,\ t _ {0} ,\ t),\ t) \ dt. $$

The formulas (4)–(6) express the most widespread averaging methods.

Formula (6) expresses the scheme of averaging "along the generating solution" . In the function $ X(x,\ y,\ t) $, the vector $ y $ is initially substituted by the generating solution of the system

$$ \frac{dx}{dt} \ = \ 0, $$

$$ \frac{dy}{dt} \ = \ \omega (x,\ y,\ t), $$

after which the integral average (6) is calculated.

The principal question which arises when the systems (1)–(3) are changed is the construction of $ \epsilon $- estimates for the norms

$$ \| x(t,\ \mu ) - \mathbf x (t,\ \mu ) \| ,\ \ \| y(t,\ \mu ) - \mathbf y (t,\ \mu ) \| $$

on the largest possible interval (of order $ 1/ \mu $) of time, if

$$ x(0,\ \mu ) \ = \ \mathbf x (0,\ \mu ),\ \ y(0,\ \mu ) \ = \ \mathbf y (0,\ \mu ). $$

This is the essence of the problem of averaging methods. For systems (1), this problem of averaging methods was proposed by Bogolyubov, whose results formed the basis of the modern algorithmic theory of ordinary differential equations.

References

[1] N.N. Bogolyubov, Yu.A. Mitropol'skii, "Asymptotic methods in the theory of non-linear oscillations" , Gordon & Breach , Delhi (1961) (Translated from Russian)
[2] Yu.A. Mitropol'skii, "An averaging method in non-linear mechanics" , Kiev (1971) (In Russian)
[3] V.M. Volosov, B.I. Morgunov, "Averaging methods in the theory of non-linear oscillatory systems" , Moscow (1971) (In Russian)
[4] E.A. Grebenikov, Yu.A. Ryabov, "Constructive methods of analysis of non-linear systems" , Moscow (1979) (In Russian)

Comments

A recent up-to-date treatment of averaging matters is given in [a1].

References

[a1] J.A. Sanders, F. Verhulst, "Averaging methods in nonlinear dynamical systems" , Springer (1985)
How to Cite This Entry:
Averaging. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Averaging&oldid=12772
This article was adapted from an original article by E.A. Grebenikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article