Equilibrium point

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 $x' = f(x)$ if $f(x^*) = 0$.

Remarks

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

Explanations

L1 human

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