Cellular automata (CA) and Boolean networks (BN) are time- and state-discrete dynamical systems with many degrees of freedom. They are useful as discretizations of continuous systems where all elements display a switch-like behaviour with fast transitions between a low and a high saturation value.
After the discretization, however, the usual stability criterion is no longer applicable because it requires consideration of arbitrarily small open neighbourhoods in state space. Previous studies have resorted to two methods for testing stability in discrete systems. (1) Damage spreading initiated by flipping the state of a single element; (2) Inducing stochasticity (noise) by asynchronous random update. As we show here, these methods do not define stability of discrete systems in consistency with the original continuous ones, not even in the limit of a large number of elements. Thus these stability tests do not predict if the observed behaviour may be expected from the continuous one as well, or if it is just an artifact from the discretization.
Here we introduce a stability criterion for discrete dynamics that coincides with the usual notion of stability. We show that several elementary CA rules give stable dynamics even though previously classified as chaotic. For random (Kauffman) BN, the order-disorder transition is not observed. According to the consistent criterion, instability in Kauffman networks is a singular behaviour occurring only for the choice of parameters previously called the critical line.