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  "body": "\n## Definition\n\nLet $K$ be a field and let $k \\ge 1$.\nLet $f \\colon \\N \\times K^k \\to K$ be a function, and consider the [[recurrence relation|07TZV87Y0T]]\n$$u_{n+k} = f(n, u_n, u_{n+1}, \\dots, u_{n+k-1}) \\qquad (n \\in \\N).$$\nIt is *linear* if there are functions $a_0, \\dots, a_{k-1}, g \\colon \\N \\to K$ with\n$$f(n, y_0, \\dots, y_{k-1}) = a_0(n) y_0 + \\cdots + a_{k-1}(n) y_{k-1} + g(n)$$\nfor all $n \\in \\N$ and all $y_0, \\dots, y_{k-1} \\in K$.\nIt is a *nonlinear difference equation* if it is not linear.\n\n## Remarks\n\nExample: the [[logistic map|QN9X9BNKTX]] recurrence $u_{n+1} = r u_n (1 - u_n)$ over $\\R$ with $r \\ne 0$ is nonlinear.\nIndeed, for fixed $n$ an affine function $y \\mapsto a y + c$ has constant second difference zero, while $f(y) = r y - r y^2$ gives $f(2) - 2 f(1) + f(0) = -2r \\ne 0$.\nNon-example: the Fibonacci recurrence $u_{n+2} = u_{n+1} + u_n$ is linear, with $a_0 = a_1 = 1$ and $g = 0$.\nThe same definitions apply with $K$ replaced by $\\R^m$ and $f$ by a map to $\\R^m$, which gives nonlinear systems.\nAn autonomous first-order equation $u_{n+1} = f(u_n)$ is the iteration of the map $f$, so its study belongs to discrete dynamical systems.\n",
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