This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by others.
(Thanks to Pavel Pokorny for Contributions to this answer)
A Cellular automaton (CA) is a dynamical system with discrete time (like
a map, see [2.6]), discrete state space and discrete geometrical space (like
an ODE), see [2.7]). Thus they can be represented by a state s(i,j) for
spatial state i, at time j, where s is taken from some finite set. The update
rule is that the new state is some function of the old states, s(i,j+1) =
f(s). The following table shows the distinctions between PDE's, ODE's, coupled
map lattices (CML) and CA in taking time, state space or geometrical space
either continuous (C) of discrete (D):
time state space geometrical space PDE C C C ODE C C D CML D C D CA D D D
 
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