EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 4, 2019, 1689-1700
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Sensitivity of fuzzy nonautonomous dynamical systems

Yaoyao Lan

Department of Mathematics, Chongqing University of Arts and Sciences, Yongchuan, China

Abstract. This paper is devoted to a study of relations between two forms of sensitivity of nonau-
tonomous dynamical system and its induced fuzzy systems. More specially, we study strong sen-
sitivity and mean sensitivity in an original nonautonomous system and its connections with the
same ones in its induced systems, including set-valued system and fuzzified system.

2010 Mathematics Subject Classifications: 03E72, 37B55

Key Words and Phrases: Nonautonomous, dynamical systems, fuzzy, strong sensitivity, mean
sensitivity

1. Introduction

Let fn : X → X be a sequence of continuous maps acting on a compact metric space
(X, d). A nonautonomous discrete dynamical systems is a pair (X, {fn}∞n=1) defined by:

xn+1 = fn(xn), n ≥ 1, (1)

Note that the autonomous dynamical system is a special case of system (1) when
fn = f for all n ≥ 1.

For other notions and notations mentioned in this section, we refer to Section 2.
The dynamics of autonomous dynamical system have been extensively studied and

many elegant results have been obtained [1, 2, and the references therein]. Nonautonomous
systems, also called sequences of dynamical systems, present situations that the dynamics
vary with time. These systems can be very complicated and naturally appear as a suitable
model to describe real processes. The rich dynamics of non-autonomous discrete systems
attract the interest of several researchers, obtaining results on chaotic properties [3]-[7].

Sensitivity is essential for the concept of chaos. A study of stronger forms of sensitivity
has been initiated by Moothathu [8]. Along this line, several elegant results have been
obtained [9, 10]. A series of research focus on mean sensitivity [11, 12]. Until very recently,
sensitivity of nonautonomous dynamical system has been discussed [13]. Motivated by the
idea in [9], we discuss different kinds of sensitivities in nonautonomous dynamical systems
in this paper.

DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3573

Email addresses: yylanmath@163.com (Y. Lan)

http://www.ejpam.com 1689 c© 2019 EJPAM All rights reserved.



Y. Lan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1689-1700 1690

On the other hand, it is well known that every given discrete dynamical system uniquely
induces its fuzzified counterpart, i.e., a discrete system on the space of fuzzy sets. It
is natural to investigate the relation between dynamical properties of the original and
fuzzified systems. Actually, there are quite a few elegant results have been obtained [14]-
[21].

In this paper, we initiate a preliminary study of relations between several forms of
sensitivity of the original and its fuzzified nonautonomous dynamical systems. Below,
basic notions are introduced in Section 2. Main results are presented in Section 3, where
the relations between two forms of sensitivity of the original and fuzzified systems have
been discussed, respectively.

2. Basic concepts and notations

2.1. Metric space of fuzzy sets

Let (X, d) denote a compact metric space and let K(X) be the class of all non-empty
and compact subsets of X. Define the ε-neighborhood of a nonempty subset A in X to
be the set

Ud(A, ε) = {x | d(x,A) < ε},

where d(x,A) = infa∈A ‖x− a‖.
The Hausdorff separation ρ(A,B) of A,B ∈ K(X) is defined by

ρ(A,B) = inf{ε > 0| A ⊆ U(B, ε)},

The Hausdorff metric dH on K(X) is defined by letting

dH(A,B) = max{ρ(A,B), ρ(B,A)}.

For a compact metric space X, the topology generated by dH coincides with the finite
topology. It is known that the set of all finite subsets of X, denote by L(X), is dense in
K(X).

Define F(X) as the class of all upper semicontinuous fuzzy sets u : X → [0, 1] such
that [u]α ∈ K(X), where α-cuts and the support of u are defined by

[u]α = {x ∈ X|u(x) ≥ α}, α ∈ [0, 1],

and
supp(u) = {x ∈ X|u(x) > 0},

respectively.
Moreover, for each x ∈ X, we denote x̂ the characteristic function of x, it is clear that

for for all x ∈ X, x̂ ∈ F(X) and [x̂]α = {x} for α ∈ (0, 1]. Denote ∅X the empty fuzzy set
(∅X(x) = 0 for all x ∈ X).

A levelwise metric d∞ on F(X) is defined by

d∞(u, v) = sup
α∈[0,1]

dH([u]α, [v]α),



Y. Lan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1689-1700 1691

for all u, v ∈ F(X). It is well known that if (X, d) is complete, then (F(X), d∞) is also
complete but is not compact and is not separable.

2.2. Zadeh’s and set-valued extension

The set-valued extension of a discrete dynamical system (X, f) is a map f̄ : K(X) →
K(X) defined by f̄(A) = f(A) for any A ∈ K(X). It is shown that f̄ is continuous in
Hausdorff metric if and only if f is continuous [14].

The Zadeh’s extension of (X, f) is a map f̂ : F(X)→ F(X) defined by

[f̂(u)](x) = sup
y∈f−1(x)

{u(y)}

for any u ∈ F(X) and x ∈ X. It is known that for compact X, f̂ : F(X) → F(X) is
continuous if and only if f : X → X is continuous [15].

Lemma 1 ([16],[17]). Let X be a metric space. If f : X → X is continuous, then
[f̂(u)]α = f([u]α).

A fuzzy set u is piecewise constant if there exists a strictly decreasing sequence of
closed subsets {C1, C2, · · ·, Ck} of X and a strictly increasing sequence of real numbers
{α1, α2, · · ·, αk} ⊆ (0, 1] such that [u]α = Ci+1, where α ∈ (αi, αi+1].

Lemma 2 ([18]). For any v ∈ F(X) and ε > 0 there exists a piecewise constant u ∈ F(X)
such that d∞(u, v) < ε, i.e., the set of piecewise constant fuzzy sets is dense in F(X).

Denote by SF(X) the set of piecewise constant fuzzy sets.

2.3. Nonautonomous discrete dynamical systems

For a compact metric space X, let {fn}∞n=1 be a sequence of continuous maps, where
fn : X → X. An orbit {xn}∞n=1 of a point x1 ∈ X is defined as follows:

xn+1 = fn(xn), n = 1, 2, · · ·

The set-valued extension of (X, {fn}∞n=1) is denoted by (K(X), {f̄n}∞n=1).
Denote Fn : X → X and F̄n : K(X)→ K(X) by

Fn(x) = fn ◦ fn−1 · · · ◦ f2 ◦ f1(x),

and
F̄n(x) = f̄n ◦ f̄n−1 · · · ◦ f̄2 ◦ f̄1(x),

respectively.



Y. Lan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1689-1700 1692

3. Main Results

In this section, we investigate the relations between several forms of sensitivity of
nonautonomous dynamical system and its induced fuzzy systems.

Let (X, d) be a compact metric space and {fn}∞n=1 be a sequence of continuous maps
on X.

For (X, {f̂n}∞n=1), its Zadeh’s extension (or fuzzification) is a sequence of continuous
maps f̂n : F(X)→ F(X) defined by

[f̂n(u)](x) = supy∈f−1
n (x){u(y)},

for any u ∈ F(X) and x ∈ X.
An orbit {un}∞n=1 of a point u1 ∈ F(X) is defined as follows:

un+1 = f̂n(un), n = 1, 2, · · · .

Define F̂n : F(X)→ F(X) by

F̂n(u) = f̂n ◦ f̂n−1 · · · ◦ f̂2 ◦ f̂1(u),

for any u ∈ F(X).

Definition 1. We say that {fn}∞n=1 is
strong sensitive if there is a constant δ > 0 such that for every point x and every

neighborhood A of x, there is a y ∈ A and an integer n0 such that d(Fk(x), Fk(y)) > δ for
every n ≥ n0.

mean sensitive if there is a constant δ > 0 such that for every point x ∈ X and every
neighborhood A of x, there is a y ∈ A such that

lim sup
n→∞

1

n

n−1∑
i=0

d(Fi(x), Fi(y)) > δ.

We call (x, y) a mean sensitive pair.

Definition 2. We say that {f̂n}∞n=0 is strong sensitive if there is a constant δ > 0 such
that for every fuzzy set u ∈ F(X) and every neighborhood U about u, there is a v ∈ U and
an integer n0 such that d∞(F̂k(u), F̂k(v)) ≥ δ for every n ≥ n0.

mean sensitive if there is a constant δ > 0 such that for every fuzzy set u ∈ F(X)
and every neighborhood U of u, there is a v ∈ U such that

lim sup
n→∞

1

n

n−1∑
i=0

d∞(F̂i(u), F̂i(v)) > δ.

Proposition 1. Let u ∈ F(X) and F̂n : F(X) → F(X). Then [F̂n(u)]α = Fn([u]α) for
α ∈ [0, 1].



Y. Lan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1689-1700 1693

Proof. Take u = ω1. Since [f̂(ω)]α = f([ω]α) and f̂n(ωn) = ωn+1 for n = 1, 2, · · ·, then

[F̂n(u)]α = [F̂n(ω1)]α

= [f̂n ◦ f̂n−1 ◦ · · · ◦ f̂1(ω1)]α)

= [f̂n ◦ f̂n−1 ◦ · · · ◦ f̂2(ω2)]α)

= [f̂n ◦ f̂n−1 ◦ · · · ◦ f̂3(ω3)]α)

= · · ·
= [f̂n(ωn)]α

= fn([ωn]α)

= fn([f̂n−1(ωn−1)]α)

= fn ◦ fn−1([ωn−1]α)

= fn ◦ fn−1 ◦ · · · ◦ f1([ω1]α)

= Fn([ω1]α)

= Fn([u]α).

Theorem 1. If {f̂n}∞n=1 is strongly sensitive, then {fn}∞n=1 is strongly sensitive.

Proof. Let x ∈ X. Take u = x̂ ∈ F(X). Since {f̂n}∞n=1 is strongly sensitive, there exist
δ > 0 and an integer n0 such that

d∞(F̂n(u), F̂n(ν)) = d∞(F̂n(x̂), F̂n(ν))

= sup
α∈[0,1]

dH([F̂n(x̂)]α, [F̂n(ν)]α)

= sup
α∈[0,1]

dH(Fn([x̂]α), Fn([ν]α))

= sup
α∈[0,1]

dH(F̄n({x}), F̄n([ν]α))

= sup
α∈[0,1]

{ sup
y∈[ν]α

d(Fn(x), Fn(y))}

= sup
y∈[ν]0

d(Fn(x), Fn(y)) > δ.

for all n ≥ n0. Thus it follows from the continuity of {fn}∞n=1 and the compactness of [ν]0
that there exists y∗ ∈ [ν]0 such that

d∞(F̂n(x̂), F̂n(ν)) = d(Fn(x), Fn(y∗)) > δ.

On the other hand, since ν ∈ Ud∞(x̂, ε), we have [ν]0 ⊂ UdH ({x}, ε) and then y∗ ∈ Ud(x, ε).
Consequently, {fn}∞n=1 is strongly sensitive in X.

Claim 1 If {f̄n}∞n=1 is strongly sensitive in L(X), then it is strongly sensitive in K(X).
Proof. Let B ∈ L(X). Since L(X) is dense in K(X), for any ε > 0, there exists

A ∈ L(X) such that A ∈ UdH (B, ε). Due to the strong sensitivity of {f̄n}∞n=1 in L(X),



Y. Lan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1689-1700 1694

there exist a constant δ > 0 and an integer n0 such that dH(F̄k(A), F̄k(B)) ≥ δ for all
n ≥ n0. This completes the proof.

Apply the similar technique to (F(X), {f̂n}∞n=1), the following result is obtained:
Claim 2 If {f̂n}∞n=1 is strongly sensitive in SF(X), then it is strongly sensitive in

F(X).

Proposition 2. The following conditions are equivalent:
(1) {f̂n}∞n=1 is strongly sensitive.
(2) {f̄n}∞n=1 is strongly sensitive.

Proof. (1)⇒ (2) Since L(X) is dense in K(X), by Claim 1, it is sufficient to show that
{f̄n}∞n=1 |L(X) is strongly sensitive.

Let {f̂n}∞n=1 be strongly sensitive with sensitive constant δ and A = {x1, x2, · · ·, xk} ∈
L(X). Take ui = x̂i for 1 ≤ i ≤ k, then ui ∈ F(X). Since {f̂n}∞n=1 is strongly sensitive, for
each ui, there exist vi ∈ Ud∞(ui, ε) and an integer ni such that d∞(F̂r(ui), F̂r(vi)) > 2δ
for all r > ni, where i = 1, 2, · · ·, k. Set N = max{ni : 1 ≤ i ≤ k}.

Now we show that dH(F̄n(A), F̄n(B)) > δ for all B ∈ UdH (A, ε) and n > N .
Let n > N . Then for any ui, there exists vi ∈ Ud∞(ui, ε) such that d∞(F̂n(ui), F̂n(vi)) >

2δ. Set C = {wi}ki=1. Without loss of generality, let

wi =

{
vi, if d∞(F̂n(u1), F̂n(ui)) ≤ δ,
ui, if d∞(F̂n(u1), F̂n(ui)) > δ.

More specifically,
if wi = ui, then d∞(F̂n(u1), F̂n(wi)) = d∞(F̂n(u1), F̂n(ui)) > δ;
if wi = vi, then

2δ < d∞(F̂n(ui), F̂n(vi)) = d∞(F̂n(ui), F̂n(wi))

< d∞(F̂n(ui), F̂n(u1)) + d∞(F̂n(u1), F̂n(wi))

≤ δ + d∞(F̂n(u1), F̂n(wi)).

Thus d∞(F̂n(u1), F̂n(wi)) > δ and then

d∞(F̂n(u1), F̂n(wi)) = d∞(F̂n(x̂1), F̂n(wi))

= sup
α∈[0,1]

dH([F̂n(x̂1)]α, [F̂n(wi)]α)

= sup
α∈[0,1]

dH(Fn([x̂1]α), Fn([wi]α))

= sup
α∈[0,1]

dH({Fn(x1)}, Fn([wi]α)) > δ.

Therefore, there exists yi ∈ [wi]α such that d(Fn(x1), Fn(yi)) > δ for each i. Take B =
{yi}ki=1. Then dH(F̄n(A), F̄n(B)) > δ holds for all B ∈ UdH (A, ε) and n > N .

(2) ⇒ (1) Assume {f̄n}∞n=1 is strongly sensitive with sensitive constant δ. To show
that {f̂n}∞n=1 is strongly sensitive in F(X), it is sufficient to prove that {f̂n}∞n=1 |SF(X)



Y. Lan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1689-1700 1695

is strongly sensitive, as SF(X) is dense in F(X). Let u ∈ SF(X), then there exist a
sequence of nested closed subsets {A1, A2, · · ·, Ak} of X and a sequence of real numbers
{α1, α2, · · ·, αk} such that

[u]α = Ai+1, where α ∈ (αi, αi+1], 1 ≤ i ≤ k.

Since {f̄n}∞n=1 is strongly sensitive, for Ak and any B ∈ K(X) with B ∈ UdH (Ak,
ε
2), there

exists an integer n0 such that for all n > n0,

dH(F̄n(Ak), F̄n(B)) > δ. (2)

Set X1 = X and C1 = u−1(αk)
⋂
X1. In general, define {Xi}ki=1 and {Ci}ki=1 by the

following

Xi = Xi−1 \ UdH (Ci−1,
ε

4
), Ci = u−1(αk−i+1)

⋂
Xi.

Let Di =
⋃i
j=1UdH (Ci,

ε
4), then we obtain an incresing sequence D1 ⊂ D2 ⊂ · · · ⊂ Dk

of closed sets in K(X). Consequently, we have a piecewise constant fuzzy set ω ∈ SF(X)
satisfying

[ω]α = Di+1, where α ∈ (αi, αi+1].

It follows from the construction and Lemma 2.2 that

d∞(u, ω) <
ε

4
. (3)

Thus we have for each i = 1, 2, · · ·, k,

dH(B,Di) ≤ dH(B,Ak) + dH(Ak, Di) <
ε

2
+
ε

4
=

3ε

4
. (4)

Take ν ∈ SF(X) such that

[ν]α =

{
B, if α ∈ (αk−1, αk]

B
⋃
Di, if α ∈ (αi, αi+1], i = 1, 2, · · ·, k − 2

Then from (4), we have

d∞(ν, ω) <
3ε

4
. (5)

Hence it follows from (3) and (5) that

d∞(u, ν) < d∞(u, ω) + d∞(ω, ν) <
ε

4
+

3ε

4
= ε.

On the other hand, by (2) and Lemma 2.1, the following

d∞(F̂n(u), F̂n(ν)) = sup
α∈[0,1]

dH([F̂n(u)]α, [F̂n(ν)]α)

= sup
α∈[0,1]

dH(Fn([u]α), Fn([ν]α))



Y. Lan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1689-1700 1696

= sup
α∈[0,1]

dH(F̄n([u]α), F̄n([ν]α))

≥ dH(F̄n([u]αk), F̄n([ν]αk))

= dH(F̄n(Ak), F̄n(B)) > δ

holds, the strong sensitivity of {f̂n}∞n=1 follows.

Theorem 2. If (F(X), {f̂n}∞n=1) is mean sensitive, then (X, {fn}∞n=1) is also mean sensi-
tive.

Proof. Let (F(X), {f̂n}∞n=1) be mean sensitive with sensitive constant δ, then for every
u ∈ F(X) and every ε > 0 there exists v1 ∈ Ud∞(u, ε) such that

lim sup
n→∞

1

n

n−1∑
i=0

d∞(F̂iu, F̂iv1) > δ.

Taking u = x̂ ∈ F(X) we have that

lim sup
n→∞

1

n

n−1∑
i=0

d∞(F̂ix̂, F̂iv1) = lim sup
n→∞

1

n

n−1∑
i=0

sup
α∈[0,1]

dH([F̂i(x̂)]α, [F̂i(v1)]α)

= lim sup
n→∞

1

n

n−1∑
i=0

sup
α∈[0,1]

dH(Fi([x]α), Fi([v1]α))

= lim sup
n→∞

1

n

n−1∑
i=0

sup
α∈[0,1]

dH(F̄i({x}), F̄i([v1]α))

= lim sup
n→∞

1

n

n−1∑
i=0

sup
y∈[v1]0

d(Fi(x), Fi(y)) > δ.

Thus it follows from the continuity of {fn}∞n=1 and the compactness of [v1]0 that there
exist y1 ∈ [v1]0 and an integer n1 such that

n1−1∑
i=0

d(Fi(x), Fi(y1)) > n1δ.

If (x, y1) forms a mean sensitive pair, then the proof is done. If not, then there exists an
integer k1 with k1 > n1 such that

∑n−1
i=0 d(Fi(x), Fi(y1)) ≤ n1δ for all n ≥ k1. Thus we can

find a neighborhood U1 of y1 with U1 ⊂ Ud(x, ε) such that
∑n1−1

i=0 d(Fi(x), Fi(z)) > n1δ
for all z ∈ U1. Furthermore, there exists ε1 > 0 such that

Ud(y1, ε1) ⊂ U1.

Using the mean sensitivity of {f̂n}∞n=1 again, we have v2 ∈ Ud∞(ŷ1, ε1) such that (ŷ1, v2)
is a mean sensitive pair, that is,

lim sup
n→∞

1

n

n−1∑
i=0

d∞(F̂iŷ1, F̂iv2) = lim sup
n→∞

1

n

n−1∑
i=0

sup
α∈[0,1]

dH([F̂i(ŷ1)]α, [F̂i(v2)]α)



Y. Lan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1689-1700 1697

= lim sup
n→∞

1

n

n−1∑
i=0

sup
α∈[0,1]

dH(F̄i(y1), F̄i([v2]α))

= lim sup
n→∞

1

n

n−1∑
i=0

sup
y∈[v2]0

d(Fi(y1), Fi(y)) > δ.

Therefore, there exist y2 ∈ [v2]0 and an integer n2 > k1 > n1 such that

n2−1∑
i=0

d(Fi(y1), Fi(y2)) > n2δ,

and then

n2−1∑
i=0

d(Fi(x), Fi(y2)) >

n2−1∑
i=0

d(Fi(y1), Fi(y2))−
n2−1∑
i=0

d(Fi(x), Fi(y1)) ≥ (n2 − n1)δ.

If (x, y2) forms a mean sensitive pair, then the proof is done. If not, then there ex-
ists an integer k2 with k2 > n2 such that

∑n−1
i=0 d(Fi(x), Fi(y2)) ≤ (n2 − n1)δ for all

n ≥ k2. Again, we can find a neighborhood U2 of y2 with U2 ⊂ Ud(y1, ε1) such that∑n2−1
i=0 d(Fi(x), Fi(z)) > (n2 − n1)δ for all z ∈ U2. Thus, there exists ε2 > 0 such that

Ud(y2, ε2) ⊂ U2.

Proceeding inductively, we eventually obtain either the mean sensitive pair (x, yk) or
a sequence {yn} in Ud(x, ε). It follows from the construction that the sequence {yn}
converges to a point y0. Thus

y0 ∈ Ud(yi, εi) ⊂ Ud(yi, εi) ⊂ Ui ⊂ Ud(x, ε).

Hence for each i, we have
ni−1∑
i=0

d(Fi(x), Fi(y0)) > riδ,

where

ri =

{∑i
k=1(−1)k−1nk, i = 2m− 1∑i
k=1(−1)knk, i = 2m.

Therefore,

lim sup
n→∞

1

n

n−1∑
i=0

d∞(Fi(x), Fi(y0)) > δ

and then {fn}∞n=1 is mean sensitive.

The following example shows that, in general, the converse of Theorem 3.4 is not true.



Y. Lan / Eur. J. Pure Appl. Math, 12 (4) (2019), 1689-1700 1698

Example 1. Let S1 be a circle. It is known that the Denjoy map Dλ : S∗ → S∗ is
an orientation preserving homeomorphism of the constructed circle S∗. There exists a
Cantor set Cλ ⊂ S∗ on which Dλ acts minimally. There exists a continuous surjection
hλ : S∗ → S1 that semi-conjugates Dλ with Rλ. In [22], the authors show that the system
(K(Cλ), Dλ) is not sensitive. Hence it is not mean sensitive, as the mean sensitivity is
stronger than sensitivity.

Let fn = Dλ, n = 1, 2, · · · . Define iλ : K(Cλ) → F(Cλ) by iλ(K) = λχK for any
K ∈ K(Cλ) and any λ ∈ (0, 1], where χK is the characteristic function of K. Hence,

iλ ◦Dλ = D̂λ ◦ iλ. Note that iλ is continuous. We show that the mean sensitivity of Dλ

cannot be inherited by D̂λ as follows.
Since (K(Cλ), Dλ) is not mean sensitive, for every δ > 0, there exist a nonempty set

A ∈ K(Cλ) and a neighborhood U of A such that for all B ∈ U ,

lim sup
n→∞

1

n

n−1∑
i=0

dH(D
n
λ(A), D

n
λ(B)) ≤ δ. (6)

Suppose u ∈ e(A) (recall that e(A) = {u ∈ F(Cλ) | [u]0 ⊆ A}), by continuity of iλ and
(3.5), we have

lim sup
n→∞

1

n

n−1∑
i=0

dH(D
n
λ([u]0), D

n
λ(B)) ≤ δ

⇒ lim sup
n→∞

1

n

n−1∑
i=0

dH(iλ ◦D
n
λ([u]0), iλ ◦D

n
λ(B)) ≤ δ

⇒ lim sup
n→∞

1

n

n−1∑
i=0

d∞(D̂λ
n
◦ iλ([u]0), D̂λ

n
◦ iλ(B))

= lim sup
n→∞

1

n

n−1∑
i=0

d∞(D̂λ
n
(u), D̂λ

n
(ν)) ≤ δ,

where ν = iλ(B) ∈ F(Cλ). It follows that (F(Cλ), D̂λ) is not mean sensitive.

4. Conclusions

In this paper, we introduce the notions of strong sensitivity and mean sensitivity
for nonautonomous systems and investigate these two forms of sensitivity in an original
nonautonomous system and its connections with the same ones in its fuzzified system.
More precisely, we prove that the strong sensitivity of original system and its induced
systems, including set-valued system and fuzzified system, are equivalent. The mean
sensitivity of induced fuzzy system implies the same one in original nonautonomous system,
however, the converse is not true.



REFERENCES 1699

Acknowledgements

This work was supported by the National Natural Science Foundation of China (NO.
11601051) and China Scholarship Council Contract (NO. 201608505146).

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