A fully-discrete general method to approximate any real map in the unit interval by a cellular automaton (CA) to arbitrary precision is presented. This result leads to establish a one-to-one correspondence between the qualitative behavior found in bifurcation diagrams of real nonlinear maps and the Wolfram classes of CAs. The local, nonlocal and global dynamical behaviors of CAs are systematically addressed and universal maps are derived for the three levels of description showing their direct interrelationships and elucidating some essential aspects of their dynamics. None of the maps contain any freely adjustable parameter and they are valid for any number of symbols in the alphabet p and neighborhood range ρ. The method is applied to the logistic map, for which a logistic CA is derived. All dynamical behavior present in the former is shown to be exactly reproduced by the latter.
Nonlocal and global dynamics of cellular automata: A theoretical computer arithmetic for real maps