= Equilibrium point of a dynamical system
{title2=$x^*$}
= Equilibrium points of a dynamical system
{synonym}
= Equilibrium point
{synonym}
= Critical point of a dynamical system
{synonym}
= Equilibrium
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= Equilibria
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An <equilibrium point of a dynamical system> is a state $x_*$ satisfying $f(x_*)=0$ for an <autonomous differential equation> $\dot x=f(x)$. Starting exactly there gives the constant solution $x(t)=x_*$. A mechanical equilibrium requires all state derivatives to vanish: a mass momentarily passing through the zero-force <position> with nonzero <velocity> is not at an <equilibrium point> in its full <phase plane>.
Existence and stability are separate questions. For $\dot x=x-x^3$, the points $-1,0,1$ are all equilibria. The linear derivatives are negative at $\pm1$ and positive at zero, giving locally attracting outer states and a repelling middle state. This simple example connects algebraic roots with <multistability> and distinct <basins of attraction>. Small perturbations reveal stability; the exact unperturbed equilibrium solution alone does not.
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