---
tag: 0G4X36AE0X
kind: definition
name: Equilibrium point
aliases: [critical-point]
msc: [34C05, 34A34]
defines: 0N8NGKDZDZ
uses: [07BNRFQ0PJ, 04SW7WECS8]
sources:
  - url: https://en.wikipedia.org/wiki/Equilibrium_point_(mathematics)
    locator: Formal definition
---

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be a function.
A point $x^* \in U$ is an equilibrium point of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ if $f(x^*) = 0$.

## Remarks

The point $x^*$ is an equilibrium point exactly when the constant function $x(t) = x^*$ is a [[solution|04SW7WECS8]].
For the [[logistic equation|0CC4VC7KR3]] $x' = r x (1 - x/K)$ with $r, K > 0$, the equilibrium points are $0$ and $K$.
The equilibrium points are the fixed points of the [[flow|02903SSA48]]: $\varphi_t(x^*) = x^*$ for each $t$.
The behaviour of solutions near an equilibrium point is the subject of [[Lyapunov stability|092S6VBECQ]].

## Explanations

### L1 human

An equilibrium point is a state where the velocity is zero.
A system that starts there stays there for all time.