# 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  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).