View source for Equilibrium position

<!--
e0360101.png
$#A+1 = 11 n = 0
$#C+1 = 11 : ~/encyclopedia/old_files/data/E036/E.0306010 Equilibrium position
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''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.