View source for Krylov-Bogolyubov method of averaging

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k0559401.png" /> from the free terms of the equations, up to a given degree of accuracy in terms of a small parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k0559402.png" />; 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:

<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/k/k055/k055940/k0559403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k0559404.png" /> is the time and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k0559405.png" /> is a small positive parameter. The fundamental assumptions adopted in regard to this system are that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k0559406.png" /> is a smooth function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k0559407.png" /> and that this function is in a sense  "recurrent"  in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k0559408.png" />, implying the existence of the average

<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/k/k055/k055940/k0559409.png" /></td> </tr></table>

e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594010.png" /> might be a periodic or almost-periodic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594011.png" />.

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594012.png" />-th approximation to the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594013.png" /> of the system (1) is, according to the method, defined by

<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/k/k055/k055940/k05594014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594015.png" /> is the solution of the  "averaged"  equation

<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/k/k055/k055940/k05594016.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594018.png" />, are functions chosen so that the expression (2) should satisfy equation (1) up to quantities of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594019.png" /> and so that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594020.png" /> should satisfy the same recurrence conditions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594021.png" /> as the free term of equation (1). The determination of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594022.png" /> is elementary; the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594023.png" /> are found by averaging the right-hand side of equation (1) in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594024.png" /> has been replaced by (2). In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594025.png" /> is a periodic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594026.png" />, with

<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/k/k055/k055940/k05594027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594028.png" /> is determined from (3) by

<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/k/k055/k055940/k05594029.png" /></td> </tr></table>

and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594030.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594031.png" />) are determined by analogous formulas using the relation

<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/k/k055/k055940/k05594032.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/k/k055/k055940/k05594033.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/k/k055/k055940/k05594034.png" /></td> </tr></table>

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

<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/k/k055/k055940/k05594035.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594036.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594038.png" /> is a constant independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594039.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594040.png" />; 2) one proves the existence of a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594041.png" /> of the system (1) which lies in a sufficiently small neighbourhood of the equilibrium position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594043.png" />, of the averaged 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/k/k055/k055940/k05594044.png" /></td> </tr></table>

and shows that this solution is stable and periodic or almost-periodic; 3) one proves the existence of an integral manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594045.png" />:

<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/k/k055/k055940/k05594046.png" /></td> </tr></table>

of the system (1), in a neighbourhood of a periodic trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594049.png" />, of the averaged 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/k/k055/k055940/k05594050.png" /></td> </tr></table>

and investigates the behaviour of the solutions of (1) that lie in the neighbourhood of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055940/k05594051.png" />.

====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

<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/k/k055/k055940/k05594052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

(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>