Chaotic Responses in a Self–Recurrent Fuzzy Inference with Nonlinear Rules

Kazuo SAKAI  Tomio MACHIDA  Masao MUKAIDONO  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E77-A   No.11   pp.1736-1741
Publication Date: 1994/11/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Nonlinear Theory and Its Applications)
Category: Fuzzy System--Theory and Applications--
Keyword: 
fuzzy inference,  chaos,  logistic map,  bifurcation diagram,  hysteresis,  

Full Text: PDF>>
Buy this Article




Summary: 
It is shown that a self–recurrent fuzzy inference can cause chaotic responses at least three membership functions, if the inference rules are set to represent nonlinear relations such as pie–kneading transformation. This system has single input and single output both with crisp values, in which membership functions is taken to be triangular. Extensions to infinite memberships are proposed, so as to reproduce the continuum case of one–dimensional logistic map f(x)=Ax(1–x). And bifurcation diagrams are calculated for number N of memberships of 3, 5, 9 and 17. It is found from bifurcation diagrams that different periodic states coexist at the same bifurcation parameter for N9. 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 large number of memberships.