CHAOTIC RESPONSES IN A SELF-RECURRENT FUZZY INFERENCE WITH NONLINEAR RULES

It is shown that a self-recurrent fuzzy inference can cause chaotic re sponses at least three membership functions, if the inference rules ar e set to represent nonlinear relations such as pie-kneading transforma tion. This system has single input and single output both with crisp v alues, in which membership functions is taken to be triangular. Extens ions to infinite memberships are proposed, so as to reproduce the cont inuum case of one-dimensional logistic map f(x) = Ax(1-x). And bifurca tion diagrams are calculated for number N of memberships of 3, 5, 9 an d 17. It is found from bifurcation diagrams that different periodic st ates, coexist at the same bifurcation parameter for N greater-than-or- equal-to 9. This indicates multistability necessarily accompanied with hysteresis effects. Therefore, it is concluded that the final states are not uniquely determined by fuzzy inferences with sufficiently larg e number of memberships.