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Parameter-introduction method

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

(1)

is represented in the form

where is the principal part (in some sense) of the vector function , and is the totality of second-order terms. The decomposition of into and 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,

(2)

is also considered; if , this system becomes the degenerate system

(3)

If and are holomorphic in a neighbourhood of a point , the system (2) has the solution , for values of which are, in modulus, sufficiently small. This solution can be represented in a neighbourhood of the initial values as a power series in :

(4)

(in certain cases non-zero initial values may also be specified for ). If the series (4) converges for , it supplies the solution of the system (1) with initial values . For an effective construction of the coefficients it is sufficient to have the general solution of system (3) and a partial solution of an arbitrary system

where is holomorphic in a neighbourhood of .

In particular, all can be successively determined by quadratures if , where is a constant matrix.

The method of parameter introduction is very extensively employed in the theory of non-linear oscillations [3] for the construction of periodic solutions of the system (1). (See also 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). 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 , 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 of a differential equation).

References

[1] H. Poincaré, "New methods of celestial mechanics" , 1–3 , NASA (1967) (Translated from French)
[2] A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)
[3] N.N. Bogolyubov, Yu.A. Mitropol'skii, "Asymptotic methods in the theory of non-linear oscillations" , Hindushtan Publ. Comp. , Delhi (1961) (Translated from Russian)
[4] V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)
[5] 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


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 . Here 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 is small. In mathematical terms, what is described is simply an iteration procedure. Convergence up to is sometimes observed, but should be considered exceptional.

How to Cite This Entry:
Parameter-introduction method. Yu.S. Bogdanov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Parameter-introduction_method&oldid=15914
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098