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  "body": "\n## Definition\n\nLet $X$ be a set and let $k \\ge 1$.\nLet $F \\colon X^k \\to X$ be a function, and consider the autonomous [[recurrence|07TZV87Y0T]]\n$$x_{n+1} = F(x_n, x_{n-1}, \\dots, x_{n-k+1}) \\qquad (n \\ge k - 1).$$\nA point $x^* \\in X$ is an *equilibrium* (or *equilibrium point*) of this difference equation if $F(x^*, x^*, \\dots, x^*) = x^*$.\nFor $k = 1$, this says that $x^*$ is a [[fixed point|W46BP9W06H]] of $F$.\n\n## Remarks\n\nThe point $x^*$ is an equilibrium if and only if the constant sequence $x_n = x^*$ is a solution.\nExample: 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 $1 - 1/r$.\nFor $r = 2$, check: $2 \\cdot \\tfrac12 \\cdot \\tfrac12 = \\tfrac12$.\nNon-example: $x_{n+1} = x_n + 1$ on $\\R$ has no equilibrium, since $x + 1 \\ne x$ for every $x$.\nFor a non-autonomous equation $x_{n+1} = F(n, x_n)$, an equilibrium is a point $x^*$ with $F(n, x^*) = x^*$ for all $n$.\nOther names are fixed point, steady state and stationary solution.\n",
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