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  "body": "\n## Definition\n\nLet $N \\ge 3$, let $f \\colon \\R \\to \\R$ be a map, and let $\\varepsilon \\in [0, 1]$ be a coupling strength.\nIndex the sites by $i \\in \\Z / N\\Z$.\nThe *diffusively coupled map lattice* with local map $f$ is the discrete-time [[dynamical system|BBZP1R872N]] on $\\R^{N}$ given by $x_{n+1} = F(x_n)$, where\n$$F(x)_i = (1 - \\varepsilon) f(x_i) + \\frac{\\varepsilon}{2} \\big( f(x_{i-1}) + f(x_{i+1}) \\big), \\qquad i \\in \\Z / N\\Z.$$\n\n## Remarks\n\nThe sites form a ring, which is the usual periodic boundary condition; the same rule makes sense on $\\Z$ or on higher-dimensional lattices, with the average over all nearest neighbours.\nMore general coupled map lattices allow other local maps and other coupling rules.\n\nExample: if $\\varepsilon = 0$, then $F(x)_i = f(x_i)$, so each site evolves by [[iterating|AYVTR2ES5B]] $f$ on its own.\nIf all sites start equal, $x_i = s$ for all $i$, then $F(x)_i = (1 - \\varepsilon) f(s) + \\varepsilon f(s) = f(s)$ for all $i$.\nSo the spatially uniform states form an invariant set on which the lattice reduces to the single map $f$.\n\nNon-example: a cellular automaton has the same discrete space and time.\nBut its states take values in a finite set, not in $\\R$, so it is not a coupled map lattice.\n\nA common choice is the [[logistic map|QN9X9BNKTX]] $f(x) = r x (1 - x)$ or the quadratic map $f(x) = 1 - a x^2$, with $\\varepsilon$ and the parameter of $f$ as the two control parameters.\n",
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