Coupled map lattice

Remarks

The 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. More general coupled map lattices allow other local maps and other coupling rules.

Example: if $\varepsilon = 0$, then $F(x)_i = f(x_i)$, so each site evolves by iterating $f$ on its own. If 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 $i$. So the spatially uniform states form an invariant set on which the lattice reduces to the single map $f$.

Non-example: a cellular automaton has the same discrete space and time. But its states take values in a finite set, not in $\R$, so it is not a coupled map lattice.

A common choice is the logistic map $f(x) = r x (1 - x)$ or the quadratic map $f(x) = 1 - a x^2$, with $\varepsilon$ and the parameter of $f$ as the two control parameters.