^{1, 2}

^{2}

^{1}

^{3}

^{1}

^{2}

^{3}

Letting

Throughout this paper, let

Motivated by this question, the study of set-valued discrete systems has recently become active [

As a partial response to this question, in the case of Devaney chaos, Román-Flores and Chalco-Cano [

In addition, by analyzing connections between the fuzzified dynamical system related to the original one, the authors have pointed out that this kind of investigation should be useful in many real problems, such as in ecological modelling and demographic sciences. Some recent works along these lines appear, for example, we refer to [

In this paper, unless otherwise stated a chaotic map is always Devaney chaotic. We investigate relations between

Below, Section

In this section, we complete notations and recall some known definitions. Let

The Hausdorff separation

The Hausdorff metric on

Define

Moreover, let

A

We say that

A point

The family

Let

If

For any continuous map

In this section, some conditions are discussed, under which

Let

Let

Since

If

On one hand, by Proposition

On the other hand, the map

The conditions on

If

If

Suppose

On one hand, since

It is known that a totally transitive system having dense period points is weakly mixing. The following proposition shows that a totally transitive map with dense small period sets is also weakly mixing.

If

Let

Concerning the transitivity of fuzzy dynamical systems, the authors in [

We say that a map

Let

If

Suppose

Theorem

Let

By Proposition

In this present investigation, we discuss relations between dynamical properties of the original and fuzzified dynamical systems. More specifically, we study transitivity, periodic density, and weakly mixing and so forth. And we show that the dynamical properties of the original system and its fuzzy extension mutually inherits some global characteristics. More precisely, the following implications are obtained

Actually the open question raised in [

On the other hand, it is well known that any given discrete dynamical system uniquely induces its fuzzified counterpart, that is, a discrete fuzzy dynamical system. There have been various attempt to “fuzzify” the discrete dynamical systems. One of these methods appeared in [

This work was partially supported by the National Natural Science Foundation of China (no. 11226268, no. 11071061, no. 61202349), the National Basic Research Program of China (no. 2011CB311808), NNSF of Hunan Province (no. 10JJ2001), and NSF of CUAS (no. Z2010ST14).