We redefine the transition function of elementary cellular automata (ECA) in
terms of discrete operators. The operator representation provides a clear hint
about the way systems behave both at the local and the global scale. We show
that mirror and complementary symmetric rules are connected to each other via
simple operator transformations. It is possible to decouple the representation
into two pairs of operators which are used to construct a periodic table of ECA
that maps all unique rules in such a way that rules having similar behavior are
clustered together. Finally, the operator representation is used to implement a
generalized logistic extension to ECA. Here a single tuning parameter scales
the pace with which operators iterate the rules. We show that, as this
parameter is tuned, many rules of ECA undergo multiple phase transitions
between periodic, locally chaotic, chaotic and complex (Class 4) behavior