# Parameter-introduction method

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