View source for Parameter-introduction method

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

<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/p/p071/p071480/p0714801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

is represented in 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/p/p071/p071480/p0714802.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p0714803.png" /> is the principal part (in some sense) of the vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p0714804.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p0714805.png" /> is the totality of second-order terms. The decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p0714806.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p0714807.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p0714808.png" /> 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,

<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/p/p071/p071480/p0714809.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

is also considered; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148010.png" />, this system becomes the degenerate 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/p/p071/p071480/p07148011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148013.png" /> are holomorphic in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148014.png" />, the system (2) has the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148016.png" /> for values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148017.png" /> which are, in modulus, sufficiently small. This solution can be represented in a neighbourhood of the initial values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148018.png" /> as a power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148019.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/p/p071/p071480/p07148020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</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/p/p071/p071480/p07148021.png" /></td> </tr></table>

(in certain cases non-zero initial values may also be specified for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148022.png" />). If the series (4) converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148023.png" />, it supplies the solution of the system (1) with initial values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148024.png" />. For an effective construction of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148025.png" /> it is sufficient to have the general solution of system (3) and a partial solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148026.png" /> of an arbitrary 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/p/p071/p071480/p07148027.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148028.png" /> is holomorphic in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148029.png" />.

In particular, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148030.png" /> can be successively determined by quadratures if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148032.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148033.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148034.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148035.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148036.png" /> is small. In mathematical terms, what is described is simply an iteration procedure. Convergence up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071480/p07148037.png" /> is sometimes observed, but should be considered exceptional.