The recurrence of dynamic fuzzy systems

Abstract This paper analyses a recurrent behavior of dynamic fuzzy systems defined by fuzzy relations on a Euclidean space. By introducing a recurrence for crisp sets, we prove probability-theoretical properties for the fuzzy systems. In the contractive case in Kurano et al. [Fuzzy Sets and Systems 51 (1992) 83–88], the existence of the maximum recurrent set is proved. As another case, we introduce a monotonicity for fuzzy relations, which is extended from the linear structure in Yoshida et al. [Fuzzy Sets and Systems 66 (1994) 83–95]. In the monotone case we prove the existence of the arcwise connected maximal recurrent sets.