Synchronization Phenomena in Rule Dynamical Systems

A rule-dynamical system is constructed using a simple one-dimensional cellular automata (CA), and a method to construct a generalized rule-dynamical system is also introduced. The various synchronization phenomena in one-dimensional elementary CA we observe are systematically studied. In the case of ‘autonomous rule dynamics’, rule changes are coupled to the temporal variation of the density, and it is shown that global statistical aspects of the dynamical attractor can be determined uniquely, independent of the initial conditions. Next, the response of CA to forced rule variation is studied. This is called ‘forced rule dynamics’. Under periodic forcing, density variations exhibit a wide variety of synchronization phenomena which depend not only on the class of rules employed, but also on the temporal ordering of the rules. If all rules are class 3, density synchronization occurs in all cases. For all density-synchronized cases, the density variation can be divided into two components. One oscillatory-type component can be characterized according to the constituent forcing rules and their periodic structure, while the other fluctuation-type component exhibits a standard deviation which has 1/ √ N system size dependence. This suggests that the rule variation completely determines the density variation as N →∞. The biological significance of the density synchronization is briefly discussed in relation to information processing in central nervous systems.