View source for Equilibrium position

''of a system of ordinary differential equations

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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

(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

if and only if

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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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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" />.

The equilibrium position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036010/e03601011.png" /> of a non-autonomous system

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