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Difference between revisions of "Equilibrium position"

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''of a system of ordinary differential equations
 
''of a system of ordinary 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/e/e036/e036010/e0360101.png" /></td> </tr></table>
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$$ \tag{* }
 +
\dot{x}  = \
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f ( t, x),\ \
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x \in \mathbf R  ^ {n}
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$$
  
 
''
 
''
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036010/e0360102.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036010/e0360103.png" /> is a solution of
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A point $  \xi \in \mathbf R  ^ {n} $
 +
such that $  x = \xi $
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is a solution of
  
(constant in time). The solution itself is also called an equilibrium position. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036010/e0360104.png" /> is an equilibrium position of
+
(constant in time). The solution itself is also called an equilibrium position. A point $  \xi \in \mathbf R  ^ {n} $
 +
is an equilibrium position of
  
 
if and only if
 
if and only if
  
<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/e/e036/e036010/e0360105.png" /></td> </tr></table>
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$$
 +
f ( t, \xi )  = \
 +
0 \  \textrm{ for } \
 +
\textrm{ all }  t.
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036010/e0360106.png" /> be an arbitrary solution of . The change of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036010/e0360107.png" /> transforms this solution into the equilibrium position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036010/e0360108.png" /> of the system
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Let $  x = \phi ( t) $
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be an arbitrary solution of . The change of variables $  x = \phi ( t) + y $
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transforms this solution into the equilibrium position $  y = 0 $
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of the 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/e/e036/e036010/e0360109.png" /></td> </tr></table>
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$$
 +
\dot{y}  = \
 +
F ( t, y),\ \
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F ( t, y)  \equiv \
 +
f ( t, \phi ( t) + y) -
 +
f ( t, \phi ( t)).
 +
$$
  
Therefore, in stability theory, for example, it is possible to assume, without loss of generality, that the problem always consists of investigating the stability of an equilibrium position at the origin in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036010/e03601010.png" />.
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Therefore, in stability theory, for example, it is possible to assume, without loss of generality, that the problem always consists of investigating the stability of an equilibrium position at the origin in $  \mathbf R  ^ {n} $.
  
The equilibrium position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036010/e03601011.png" /> of a non-autonomous system
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The equilibrium position $  x = 0 $
 +
of a non-autonomous system
  
 
is often called the trivial solution, zero solution, singular point, stationary point, rest point, equilibrium state, or fixed point.
 
is often called the trivial solution, zero solution, singular point, stationary point, rest point, equilibrium state, or fixed point.

Latest revision as of 19:37, 5 June 2020


of a system of ordinary differential equations

$$ \tag{* } \dot{x} = \ f ( t, x),\ \ x \in \mathbf R ^ {n} $$

A point $ \xi \in \mathbf R ^ {n} $ such that $ x = \xi $ is a solution of

(constant in time). The solution itself is also called an equilibrium position. A point $ \xi \in \mathbf R ^ {n} $ is an equilibrium position of

if and only if

$$ f ( t, \xi ) = \ 0 \ \textrm{ for } \ \textrm{ all } t. $$

Let $ x = \phi ( t) $ be an arbitrary solution of . The change of variables $ x = \phi ( t) + y $ transforms this solution into the equilibrium position $ y = 0 $ of the system

$$ \dot{y} = \ F ( t, y),\ \ F ( t, y) \equiv \ f ( t, \phi ( t) + y) - f ( t, \phi ( t)). $$

Therefore, in stability theory, for example, it is possible to assume, without loss of generality, that the problem always consists of investigating the stability of an equilibrium position at the origin in $ \mathbf R ^ {n} $.

The equilibrium position $ x = 0 $ of a non-autonomous system

is often called the trivial solution, zero solution, singular point, stationary point, rest point, equilibrium state, or fixed point.

How to Cite This Entry:
Equilibrium position. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equilibrium_position&oldid=18108
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article