||In the literature the relations between chaotic and fuzzy systems have been deeply inspected, with the aim to provide more insights on the relation between these two branches.
However the focus of such researches is mainly on the approximation or control of chaotic behaviors by means of fuzzy rule-based systems.
In this paper an exact fuzzy representation of a chaotic logistic maps is provided by letting the initial condition (i.e., the initial population) be defined by a fuzzy set. Then, the properties of the system are inspected by resorting to the framework of Discrete-time Fuzzy Systems (DFS), proving some stability results and discussing the practical level-wise representation.
The proposed framework allows to capture how the fuzziness further increases the chaotic behavior of the system under analysis, since such a formalism highlights that close points in the membership function representing the state of the system evolve in significantly different ways.|