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Consider the following equation: $$ \frac{d{\mathbf{x}}}{dt} = F({\mathbf x}) $$ Let's $F({\mathbf x}_0) = 0$. Is ${\mathbf x}_0$ is equilibrium point or stationary point? What is the difference?

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2 Answers

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(i) When ${\bf F}({\bf x}_0)={\bf 0}$ then ${\bf x}_0$ is an equilibrium point of the ODE (system) $\dot{\bf x}={\bf F}({\bf x})$.

(ii) When $f:\ {\mathbb R}^n\to{\mathbb R}$ is a scalar function of $n\geq1$ variables then a point ${\bf p}\in{\rm dom}(f)$ is a stationary point of $f$ if $\nabla f({\bf p})={\bf 0}$, i.e., if all partial derivatives ${\partial f\over\partial x_k}$ vanish at ${\bf p}$.

(iii) Sometimes an ODE system has the form $$\dot{\bf x}=\nabla f({\bf x})\ ,$$ where $f$ is as in (ii). In this case the equilibrium points of the ODE system coincide with the (hopefully isolated) stationary or critical points of $f$.

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There are a number of terms that convey the same meaning of an equilibrium point. The term "stationary point" with respect to a vector field $\boldsymbol F$ has exactly the same meaning as an equilibrium point of a dynamical system $\boldsymbol {\dot x}=\boldsymbol{F(x)}$: this is a point at which $\boldsymbol F$ vanishes. Sometimes the terms fixed point, rest point, or critical point are also used in the same context. While I agree with the other answers that it would be much better to keep the term equilibrium point for continuous time dynamical systems (and fixed point for maps) and stationary point for study functions $f\colon {\bf R}^n\to{\bf R}$, the truth is that the context has to be used to infer what is meant.

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