Convergence of q-state vector-valued fuzzy cellular automata with weighted-averaging rules

Fuzzy cellular automata are dynamical systems that are continuous counterparts of the usual cellular automata (CA). Compared with the binary case, defining a fuzzy CA with three or more states is challenging because defining mixed states is difficult. Recently, this difficulty was resolved by representing multiple states as independent vectors in higher dimensions, and the concept of vector-valued fuzzy CA (VFCA) was introduced. In this study, we theoretically analyze and discuss the asymptotic behavior of three-neighbor VFCA. First, we define the weighted-averaging rules of VFCA and show how many rules exist up to the equivalence relations. According to these rules, each state vector in the next step is determined by the weighted average of the vectors in its neighboring cells. Next, we prove that three-state VFCA with weighted-averaging rules converge to a periodic configuration characterized by the symmetric group of order 3. In particular, the non-commutativity of the group action provides an interesting behavior that is not observed in fuzzy CA arising from binary states. Finally, we extend the results to VFCA with more than three states.