Recurrent Points of Fuzzy Dynamical Systems

Abstract

The main goal of the present paper is to reconsider classical results from the theory of dynamical systems, making them appropriate for treating systems about which we have imprecise, vague, uncertain or incomplete information. Here we study qualitative properties of dynamical systems related to recurrent points. According to the basic Poincaré recurrence theorem, the set of recurrent points has full measure in any subset of the classical dynamical system space. In general, this property is invalid for fuzzy dynamical systems. In this paper, we consider fuzzy dynamical systems of two types. It is demonstrated that for fuzzy dynamical systems of the first type, the Poincaré property of recurrent points remains true if the fuzzy measure in the system space satisfies some, rather weak, additional conditions. It is also proved that infinitely recurrent points have the same property. At the same time, for fuzzy dynamical systems of the second type, the Poincaré property of recurrent points is invalid, and it is possible to find only sufficiently big subsets of recurrent points. In contrast to fuzzy dynamical systems of the first type, infinitely recurrent points of fuzzy dynamical systems of the second type do not have even this property.