The Length Of Limit Cycles For Fuzzy Bidirectional Associative Memories And Algorithms To Compute The Period Index Of Fuzzy Matrix

The maximum length of limit cycles for fuzzy bidirectional associative memories ( FBAM ) is studied in this paper. The decomposition theorem becomes a bridge connecting fuzzy matrix theory with Boolean matrix theory, so we start this problem from Boolean matrix. Based on the characters of circular vertex, we classify this question into three cases and educe different results respectively. Firstly, the connection matrices are strongly connected Boolean matrix, it is shown that the maximum length of limit cycles for FBAM is the period index of the product of the connection matrices. Thus, the maximum limit cycle is reachable. Secondly, two strongly connected components are connected by walks, it is proved that the maximum length of limit cycles for FBAM is the max. of the period index of the two strongly connected components if they are prime, otherwise, it is the least common multiple of the two period index. Finally, n strongly connected component are connected in succession, the method for computing the maximum length is described. By different patterns are presented to the networks, diminishing the number of the strongly connected components until only two components. At last, we show two algorithms to compute the period index of fuzzy matrix. Algorithm one is directly based on the power sequence, which overcomes the limitlessness of the power sequence and is determined by its powers from 1 to 2 n ? 1. Algorithm two makes use of Warshall algorithm and Balcer ? Veinott algorithm, which can compute reachable matrix and the period index of strongly connected graphs, respectively. Warshall algorithm is only applied to Boolean matrix ago. It is extended to fuzzy matrix in this paper and detailed proof is given.