We study the principal dynamical aspects of the cyclic automata on fin
ite graphs. We give bounds in the transient time and periodicity depen
ding essentially on the graph structure. It is shown that there exist
non-polynomial periods e(Omega)(root(\V\)), where \V\ denotes the numb
er of sites in the graph. To obtain these results we introduce some ma
thematical tools as continuity, firing paths, jumps and efficiency, wh
ich are interesting by themselves because they give a strong mathemati
cal framework to study such discrete dynamical systems.