Parameter-introduction method

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

$$ \tag{1 } \frac{dx}{dt} = f( t, x) $$

is represented in the form

$$ f( t, x) = f _ {0} ( t, x) + \epsilon g( t, x),\ \ \epsilon = 1 ,\ \ g= f - f _ {0} , $$

where $ f _ {0} $ is the principal part (in some sense) of the vector function $ f $, and $ g $ is the totality of second-order terms. The decomposition of $ f $ into $ f _ {0} $ and $ g $ 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,

$$ \tag{2 } \frac{dx _ \epsilon }{dt} = \ f _ {0} ( t, x _ \epsilon )+ \epsilon g ( t, x _ \epsilon ), $$

is also considered; if $ \epsilon = 0 $, this system becomes the degenerate system

$$ \tag{3 } \frac{dx _ {0} }{dt} = \ f _ {0} ( t, x _ {0} ). $$

If $ f( t, x) $ and $ g( t, x) $ are holomorphic in a neighbourhood of a point $ ( \tau , \xi ) $, the system (2) has the solution $ x _ \epsilon ( t; \tau , \xi ) $, $ {x _ \epsilon } ( \tau ; \tau , \xi ) = \xi $ for values of $ \epsilon $ which are, in modulus, sufficiently small. This solution can be represented in a neighbourhood of the initial values $ ( \tau , \xi ) $ as a power series in $ \epsilon $:

$$ \tag{4 } x _ \epsilon ( t; \tau , \xi ) = \ x _ {0} ( t; \tau , \xi )+ \epsilon \phi _ {1} ( t ; \tau , \xi ) + \dots + $$

$$ + \epsilon ^ {n} \phi _ {n} ( t; \tau , \xi ) + \dots ,\ \phi _ {k} ( \tau ; \tau , \xi ) = 0 $$

(in certain cases non-zero initial values may also be specified for $ \phi _ {k} $). If the series (4) converges for $ \epsilon = 1 $, it supplies the solution of the system (1) with initial values $ ( \tau , \xi ) $. For an effective construction of the coefficients $ \phi _ {n} $ it is sufficient to have the general solution of system (3) and a partial solution $ z( t; \tau , 0) $ of an arbitrary system

$$ \frac{dz}{dt} = f _ {0} ( t, z) + h( t), $$

where $ h( t) $ is holomorphic in a neighbourhood of $ t = \tau $.

In particular, all $ \phi _ {n} $ can be successively determined by quadratures if $ {f _ {0} } ( t, x) = Ax $, where $ A $ 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 $ \epsilon $, 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 $ X( t) = \epsilon x _ \epsilon ( t) $. Here $ f _ {0} ( t, x _ \epsilon ) $ 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 $ \epsilon $ is small. In mathematical terms, what is described is simply an iteration procedure. Convergence up to $ \epsilon = 1 $ is sometimes observed, but should be considered exceptional.