Equilibrium of a difference equation

Summary

An equilibrium of an autonomous difference equation $x_{n+1} = f(x_n)$ is a point $x^*$ with $f(x^*) = x^*$, that is, a fixed point of $f$. Starting there, the solution stays there forever, so equilibria are the constant solutions. They are the discrete-time analogue of the equilibrium points of differential equations. The first question about a nonlinear difference equation is usually where its equilibria are and whether nearby solutions approach them.