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A method in which the right-hand side of a system of differential equations
 
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>
+
$$ \tag{1 }
 +
 
 +
\frac{dx}{dt}
 +
  = f( t, x)
 +
$$
  
 
is represented in the form
 
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>
+
$$
 +
f( t, x)  = f _ {0} ( t, x) +
 +
\epsilon g( t, x),\ \
 +
\epsilon = 1 ,\ \
 +
g= f - f _ {0} ,
 +
$$
 +
 
 +
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  ),
 +
$$
  
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,
+
is also considered; if  $  \epsilon = 0 $,  
 +
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/p0714809.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{3 }
  
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
+
\frac{dx _ {0} }{dt}
 +
  = \
 +
f _ {0} ( t, x _ {0} ).
 +
$$
  
<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  $  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 $:
  
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" />:
+
$$ \tag{4 }
 +
x _  \epsilon  ( t;  \tau , \xi ) = \
 +
x _ {0} ( t;  \tau , \xi )+
 +
\epsilon \phi _ {1} ( t ;  \tau , \xi ) + \dots +
 +
$$
  
<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>
+
$$
 +
+
 +
\epsilon  ^ {n} \phi _ {n} ( t; \tau , \xi ) + \dots
 +
,\  \phi _ {k} ( \tau ; \tau , \xi )  = 0
 +
$$
  
<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  $  \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
  
(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>
+
\frac{dz}{dt}
 +
  = f _ {0} ( t, z) + h( t),
 +
$$
  
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" />.
+
where $  h( t) $
 +
is holomorphic in a neighbourhood of $  t = \tau $.
  
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.
+
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 [[#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).
+
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 $  \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|Singular point]] of a differential equation).
  
 
====References====
 
====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>
 
<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====
 
====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:
 
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.
+
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.

Latest revision as of 08:05, 6 June 2020


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.

How to Cite This Entry:
Parameter-introduction method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parameter-introduction_method&oldid=15914
This article was adapted from an original article by Yu.S. Bogdanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article