# Equilibrium position

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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=46841
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article