Difference between revisions of "Equilibrium position"
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| + | $#C+1 = 11 : ~/encyclopedia/old_files/data/E036/E.0306010 Equilibrium position | ||
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''of a system of ordinary differential equations | ''of a system of ordinary differential equations | ||
| − | + | $$ \tag{* } | |
| + | \dot{x} = \ | ||
| + | f ( t, x),\ \ | ||
| + | x \in \mathbf R ^ {n} | ||
| + | $$ | ||
'' | '' | ||
| − | A point | + | 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 | + | (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 | ||
| − | + | $$ | |
| + | f ( t, \xi ) = \ | ||
| + | 0 \ \textrm{ for } \ | ||
| + | \textrm{ all } t. | ||
| + | $$ | ||
| − | Let | + | 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 | + | 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 | + | 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.
Equilibrium position. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equilibrium_position&oldid=18108