# Equilibrium position

*of a system of ordinary differential equations*

A point such that is a solution of

(constant in time). The solution itself is also called an equilibrium position. A point is an equilibrium position of

if and only if

Let be an arbitrary solution of . The change of variables transforms this solution into the equilibrium position of the system

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 .

The equilibrium position 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. N.Kh. Rozov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Equilibrium_position&oldid=18108