Relationship between stability and capacity for synchronization of linear automata defined on different fields

Linear automata defined on the field GF(p) and on the field R of real numbe rs are considered. It is proved that, for an automaton that is defined on C F(p), the existence of an asymptotically stable state of equilibrium is equ ivalent to asserting the automaton can be synchronized. The concept of epsi lon-synchronizability is introduced for a linear automaton defined on the f ield R. It is proved that the epsilon-synchronizability of such an automato n follows from stability of the automaton. A condition is presented under w hich an automaton defined on R is simultaneously stable and epsilon-synchro nizable.