Equilibrium of a difference equation

Remarks

The point $x^*$ is an equilibrium if and only if the constant sequence $x_n = x^*$ is a solution. Example: for the logistic recurrence $x_{n+1} = r x_n (1 - x_n)$ on $\R$ with $r \ne 0$, the equation $r x (1 - x) = x$ gives the equilibria $0$ and $1 - 1/r$. For $r = 2$, check: $2 \cdot \tfrac12 \cdot \tfrac12 = \tfrac12$. Non-example: $x_{n+1} = x_n + 1$ on $\R$ has no equilibrium, since $x + 1 \ne x$ for every $x$. For a non-autonomous equation $x_{n+1} = F(n, x_n)$, an equilibrium is a point $x^*$ with $F(n, x^*) = x^*$ for all $n$. Other names are fixed point, steady state and stationary solution.