

Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844
Vol. VI (2011), No. 4 (December), pp. 749-760

On Fuzzy Sequences, Fixed Points and Periodicity in Iterated
Fuzzy Maps

H.N. Teodorescu

Horia Nicolai Teodorescu
Gheorghe Asachi Technical University,
Iasi, Romania
Institute for Computer Science,
Romanian Academy, Iasi Branch
E-mail: hteodor@etc.tuiasi.ro

Abstract: I exemplify various elementary cases of fuzzy sequences and results
related to the iteration of fuzzy mappings and to fuzzy logic systems (FLS).
Several types of fuzzy logic system iterations are exemplified in relationship
with oscillations in FLS and with the problem of stability in fuzzy logic control.
I establish several conditions for fixed points and periodicity of the iterations
based on fuzzy systems.
Keywords: modeling, control, fuzzy map iteration, fuzzy sequence, behavior.

1 Introduction

In this paper, I provide simple examples of iteration of fuzzy sets and I discuss the asymptotic
behavior of their successive iterations. I discuss several cases of iterated fuzzy logic and arithmetic
systems with and without defuzzification and determine conditions for fuzzy sets that are fixed
points. The topic was dealt with in the literature in the early approaches by Kloeden [1] and
in [2]- [9], but attracted little interest during the last two decades. However, sequences of
fuzzy numbers and various generalizations constituted recently the subject of a large number
of papers [10]- [21], while many applications indirectly referred to fuzzy sequences [22]- [24].
However, few recent papers, like [24], addressed fuzzy sequences and the problem of fixed points
in fuzzy mappings with relation to fuzzy systems and control.

In this paper, a fuzzy map is any application from a set of fuzzy sets to itself. A fuzzy system
is a function with fuzzy set-valued variables, also named inputs, and fuzzy set values, named
output. An iterated fuzzy system can be seen as a fuzzy system, in the sense used in control, but
without defuzzification, that has a connection from the fuzzy output to the fuzzy input through
a unitary delay loop. Analyzing fuzzy iterations has therefore applications in control, in decision
making, and in modeling, among others.

The connection between iterations and sequences is straightforward: while iterations are
defined by f[n+1] = f(f[n]), many sequences are obtained as an+1 = f(an), an+1 = f(an−1, an),
an+1 = f(g(an−1), an), or by similar formulas.

A fixed point of a specified fuzzy function (system) iteration is a fuzzy set, Ã, (membership
function, µ

Ã
) that is preserved under the iteration; in other words, when presented to the input of

the fuzzy system, it is found unchanged at its output.A periodical fuzzy system will recover after
a fixed number of iterations - the period - the initial input membership function as its output.
Assume that a metric space has been defined on a space of fuzzy sets. Subsequently, when
convergence of sequences is considered in the respective metric space, I mean that Ãn → Ã∞ iff
d(Ãn, Ã∞) −→n→∞ 0 . Whenever absolute convergence will be meant, the following definition
applies: Ãn → Ã∞ iff |µÃn(x) − µÃ∞| −→

n→∞ 0 ∀x. In this paper, the primary interest is in
absolute convergence.

Copyright c⃝ 2006-2011 by CCC Publications


750 H.N. Teodorescu

A basin of attraction of a specified input fuzzy set, which is a fixed point for a specified fuzzy
system, is the set of membership functions that determine the output of the fuzzy system to tend
during the iteration to the specified fuzzy set. The applications of the above concepts in the real
space are well known to control engineers, biologists, and virtually in all disciplines.

I denote by [0, 1]R the set of functions from R to [0,1]. Membership functions used at the
input and at the output of the fuzzy logic systems belong to [0, 1]R. Being given an application
φ : [0, 1]R → [0, 1]R, the successive iteration of φ with initial condition Ã0 is Ã0 → Ã1 =
φ(Ã0), Ã2 = φ(Ã1) = φ

[2](Ã0), Ã3 = φ(Ã2) = φ
[3](Ã0), ... .

A fuzzy logic system (FLS) is basically an application φ : [0, 1]R → [0, 1]R, which is defined
in a specific manner, using rules. When a discrete-time feedback loop is added to a fuzzy logic
system, the equivalent operation is an iteration of the function φ starting with an initial input
membership function Ã0. The operation of the loop is expressed by the successive iterations
φ[n](Ã0), n=1,2,3,... . From the application point of view, an iteration of a FLS can be seen as
a FLS with a discrete time, constant loop, like in Fig.1.

Figure 1: Iteration of a FLS represented as a FLS with a delay loop. The loop "gain" is unitary.
The operation is in discrete time.

The set [0, 1]R, when endowed with one of the usual definitions of distance between functions,
becomes a metric space and has the corresponding induced topologic structure. Therefore, we
are able to talk about limit points in that topologic space and about the asymptotic behavior
of the successive iteration of an application represented by a FLS. When a metric is needed, I
use d(Ã, B̃) =

∫ ∞
−∞(µÃ(x) − µB̃(x))

2dx, where Ã, B̃ stand for the output of the FLSs, assuming
integrability.

In the next section, which is preparatory, I exemplify various ways for building sequences
of fuzzy sets. In section 3, I discuss sequences generated by FLSs. Conclusions are derived in
Section 4. Throughout this paper, ♢ denotes the end of a proof, example, or a line of reasoning.

2 Elementary examples of fuzzy sequences

2.1 Games based on fuzzy sequences produced by logic operations over fuzzy
sets

I start illustrating the topic of fuzzy sequence by a simple example based on logic operations
with fuzzy sets. Consider the following game: Three school colleagues have a high spirit of
contradiction. When they start a discussion, they typically start with two disparate opinions;
then, each of them is ready to contradict what the last speaker told and partly agree and partly
disagree with the ante-precedent speaker; to do so, they answer with what at the same time
contradicts the predecessor, yet agrees with the ante-predecessor. They take turns in expressing
their opinions in that way. Can they ever come to a conclusion? If yes, after how many steps
of mutual contradictions? (I would say that the game is not far from what seldom happens in
real life, so this example is not devoid of interest. The problem is if a group of three skillful
contradictors still can agree under fuzzy logic.)


On Fuzzy Sequences, Fixed Points and Periodicity in Iterated Fuzzy Maps 751

This "game" is not possible in binary logic, but makes sense in the frame of fuzzy logic.
Consider two fuzzy sets, Ã0 and Ã1 representing the first two opinions. The third colleague’s
and the subsequent responses define a sequence of fuzzy sets by Ãn+1 = ¬Ãn ∩ Ãn−1, n ≥ 1,
where ¬Ãn denotes the complement of the fuzzy set Ãn. The above recurrence, together with
the initial conditions Ã0 and Ã1, defines the fuzzy set sequence. It is easy to see that the
sequence is Ã0, Ã1, Ã2 = ¬Ã1 ∩ Ã0, Ã3 = ¬Ã2 ∩ Ã1,... . The membership functions of Ã3 is
µ3(x) = min(µ1(x), 1 − µ2(x)), or µ3(x) = min(µ1(x), 1 − min(µ0(x), 1 − µ1(x))).
(i) Assume µ0(x) < 1 − µ1(x). Then µ3(x) = min(µ1(x), 1 − µ0(x)). Because µ0(x) < 1 − µ1(x),
µ1(x) < 1 − µ0(x), thus µ3(x) = min(µ1(x), 1 − µ0(x)) = µ1(x).
(ii) Assume µ0(x) ≥ 1 − µ1(x). Then µ3(x) = min(µ1(x), 1 − (1 − µ1(x))) = µ1(x). Therefore,
Ã3 = Ã1. Next, Ã4 = ¬Ã3 ∩ Ã2 = ¬Ã1 ∩ Ã2 and µ4(x) = min(µ2(x), 1 − µ1(x)), or µ4(x) =
min(min(1 − µ1(x); µ0(x)), 1 − µ1(x)).
(i) Assume µ0(x) < 1 − µ1(x). Then µ4(x) = min(µ0(x), 1 − µ1(x)) = µ0(x).
(ii) Assume µ0(x) ≥ 1 − µ1(x). Then µ4(x) = min(1 − µ1(x), 1 − µ1(x)) = 1 − µ1(x). Thus,
Ã4 = ¬Ã1. Next, Ã5 = ¬Ã4 ∩ Ã3 = Ã1 ∩ Ã1 = Ã1.

If the response strategy is changed, that is Ãn+1 = Ãn∩¬Ãn−1, n ≥ 1, the game is less simple.
Indeed, the membership functions of Ã2 and Ã3 are µ2 = min(µ1, 1−µ0), µ3 = min(min(µ1, 1−
µ0), 1 − µ1). Whenever µ1(x) < 1 − µ0(x), µ3(x) = min(µ1(x), 1 − µ1(x)), Ã3 = Ã1 ∩ ¬Ã1, else
Ã3 = ¬Ã0 ∩ ¬Ã1. The next step is Ã4, µ4 = min(µ3, 1 − µ2).

2.2 Fuzzy sequences produced by arithmetic operations over values of the
membership functions

Arithmetic operations with fuzzy numbers have been used to build chaotic processes by
Kloeden [1]. Here I exemplify how, based on sequences of fuzzy sets and operating them through
logic operations and using arithmetic manipulations of the elements of the sequences, one can
obtain new sequences with the desired properties. Consider the fuzzy set Ã0 with

µ0(x) =

{
x2 x ∈ [0, 1]
0 elsewhere.

and the sequence of singletons with the membership functions

µn(x) =

{
1 x = 1

n

0 elsewhere.

Define C̃n by C̃0 = Ã0 and C̃n = C̃n−1 ∪ B̃n for n > 0. The graph of the fuzzy set C̃n looks
like in Figure 2, where, for convenience, I have drawn a line to show where a singleton occurs.
Recall that a small full circle on the graph conventionally denotes the actual value at a point of
discontinuity of a function, while a small empty circle denotes the lack of value on the graph of
a continuous function. The number of singularities (discontinuities) of the membership function
of the sets C̃n is equal to n.♢

Define a sequence of fuzzy sets, {ũn, n = 1, 2, ...}, based on the fuzzy numbers ν0, ν1,

µν̃0(x) =

{
0 if x < 0 or x > 3

1 − x/3 else ,

µν̃1(x) =

{
0 if x ̸= 1
1 else ,


752 H.N. Teodorescu

Figure 2: .

as ũn = ν̃0 ∩ ¬(3 · ν̃1)n.

I propose to graphically and analytically find the first two sets in the series, then to write
their expressions, determining if there is a limit of the sequence. According to the operations
with fuzzy numbers in the framework of the fuzzy arithmetic,

µ(3−ν1)n(x) =

{
1 if x = 3n

0 elsewhere.

This expression is obtained by applying the definition of the product of fuzzy numbers:

m̃ · ñ = q̃, µq(w) = max
all y,z,s,t;y·z=w

min(µm(y), µn(z)).

For singletons ν̃, the expression min(µm(y), µn(z)) ̸= 0 only if z corresponds to the value of
the singleton. From the defining relations, ũ1 = ν̃0 ∩ ¬(3 · ν̃1)1 and

µ3ν1(x) =

{
1 if x = 3

0 elsewhere,

µ−3ν1(x) =

{
1 if x ̸= 3
0 if x = 3,

µu1(x) =




0 if x < 3 or x > 3

1 − x/3 if x ∈ [0, 3]|{3}
0 if x = 3.

Therefore, µu1 = µν0. Similarly,

ũ2 = ν̃0 ∩ ¬(3 · ν̃1)2,

µu1(x) =

{
0 if x < 0 or x > 3

1 − x/3 else

Thus, µu1 = µν0 . Moreover, for all n > 2, µun = µν0 (see Figure 3). Therefore, ν̃0 is the limit
of the constant sequence {ũn}.

Further sequences are obtained by varying the manipulation of the membership functions in
the precedent sequence. For example, changing the definition condition ũn = ν̃0 ∩¬(3 · ν̃1)n with
the condition ũn = ν̃0 ∩ ¬

(
2
3
· ν̃1
)n

, a new sequence is obtained.


On Fuzzy Sequences, Fixed Points and Periodicity in Iterated Fuzzy Maps 753

Figure 3: Explanatory graphs .

2.3 Sequences defined by conditions on the membership functions

Sequences of fuzzy sets can be conveniently defined based on sequences of real numbers
standing for α-cuts. Another fuzzy sequence with limit is exemplified subsequently. Consider a
fuzzy set Ã0 satisfying the condition µ0(x) > 0 if x ∈ (0,1) and µ(x) = 0 elsewhere. I denoted
µ
Ã0

by µ0. Apply the following iterative procedure to generate a sequence of fuzzy sets:

µn(x) =

{
µn−1(x) if µn−1(x) <

(
2
3

)n
, n ≥ 1,

0 elsewhere,

where µn represents the membership function of the fuzzy set obtained at the n-th iteration. It
is easy to check that µn −→n→∞ ∅ (see Figure 4.)

Figure 4: A sequence of fuzzy sets tending to the null fuzzy set, µ(x) = 0 ∀ x ∈ R.

Periodical fuzzy sequences are defined as a direct extension of periodicity for real valued
sequences. I define a sequence of fuzzy sets as follows:

Ã0 : µÃ0(x) =

{
0 if x < 0 or x > 3

1 − x/3 else,

Ã1 : µÃ1(x) =

{
0 if x < 0 or x > 3

x/3 if x ≥ 0 and x ≤ 3,

Ãn+1 = Ãn−1 ∩ ¬Ãn.

The first two sets in the sequence are shown in Fig.5. I start by drawing the graphs of
the membership functions of the first two sets in the sequence (see Figure 5.) From Ãn+1 =
Ãn−1 ∩ ¬Ãn, derive Ã2 = Ã0 ∩ ¬Ã1. Notice that on the interval [0,3], ¬Ã1 = Ã0, ¬Ã0 = Ã1.
Moreover, for x < 0 and for x > 3, µ

Ã0∩¬Ã0(x) = 0, µÃ1∩¬Ã1(x) = 0 and µÃ0∩¬Ã1(x) = 0.
Therefore, Ã2 = Ã0 ∩ Ã0 = Ã0.♢

Next, write the expression of the membership functions of Ã2 and Ã3 like

Ã2 : µÃ2(x) =

{
0 if x < 0 or x > 3

1 − x/3 if x ≥ 0 and x ≤ 3;


754 H.N. Teodorescu

Figure 5: Explanatory graphs.

also, Ã3 = Ã1 ∩ ¬Ã2, Ã3 = Ã1, where

Ã3 :

{
0 if x < 0 or x > 3

x/3 if x ≥ 0 and x ≤ 3;

There is no limit for this sequence. Indeed, Ãn = Ãn−2 ∀ n. Concluding, this sequence of
fuzzy sets has the form Ã0, Ã1, Ã0, Ã1,..., that is, it oscillates with period 2.

2.4 Fuzzification-defuzzification-based sequences

Consider the set of fuzzy sets Φ = {Ã|µA : R → [0, 1]}. Define an operation ◦ : R×Φ×Φ → Φ,
where R is the set of real numbers. An example of such operation is the fuzzification followed
by an inference (applying rules and truncation). The operation ◦ is defined by (x ∈ R, (Ã, B̃) ∈
Φ × Φ) → Φ, the result being the fuzzy set with the membership function µ(u; (Ã, B̃, x)) =
min(µ

Ã
(x), µ

B̃
(u)). Then, define a "seed" x0 ∈ R and the sequence Ã0, Ã1 = x0 ◦ ¬Ã0, Ã2 =

x0 ◦ ¬Ã1,... .
Consider, as above, the sets R and Φ, and the operation ◦ : R × Φ → Φ. Moreover, consider

a map ◦̂ : Φ → R. An example of such a map is the defuzzification, but a more interesting
case would be the defuzzification followed by a map from R to R. Define a "seed" x0 ∈ R,
then the sequence Ã0, x1 = ◦̂(◦(x0, Ã0, Ã0)), Ã1 = (◦(x1, Ã0, Ã0)), x2 = ◦̂(◦(x1, Ã0, Ã0)),...
The generation of this sequence relates to the main subject of study in this topic. Indeed, the
procedure above corresponds to an "identity (Mamdani-type) fuzzy logic system", that is, a
system based on a single rule "If input is Ã, then output is Ã, followed by a defuzzifier, the
defuzzified output value being fed back at the input for fuzzification. Both the defuzzification
and the feedback loop are included in the same ◦ operation. This amounts to a simplified
fuzzy system in a discrete-time feedback loop configuration. As a matter of example, consider
a Mamdani-type fuzzy system with a single rule, Ã → Ã, where the graph of the membership
function of Ã is shown in Figure 6 a) or b).

Figure 6: .

For case (a), the sequence Ã0, x1 = ◦̂(◦(x0, Ã0, Ã0)), Ã1 = (◦(x1, Ã0, Ã0)), x2 =
◦̂(◦(x1, Ã0, Ã0)),... becomes constant (Ã) after two steps, for whatever x0 such that µÃ(x0) > 0.


On Fuzzy Sequences, Fixed Points and Periodicity in Iterated Fuzzy Maps 755

For case (b), there is a limit when x = ◦̂(◦(x, Ã, Ã)) has a solution, that is, the application
◦̂(◦(·, Ã, Ã)) has a fixed point.

3 Iterated fuzzy logic systems

Essentially, a fuzzy logic system without defuzzification operation at the output is a function
from a set of fuzzy sets (the "input" fuzzy sets) to another set of fuzzy sets (the "output" fuzzy
sets). A FLS is described by:

i) A finite set of "input" fuzzy sets; I denote these sets by Ãi, i = 1, ..., p.
ii) A finite set of "output" fuzzy sets; I denote these sets by B̃j, j = 1, ..., q.
iii) An index function ı : {1, ..., p} → {1, ..., q}.
iv) An actual "input" fuzzy set (equivalently, membership function), denoted by X̃; X̃ is the

value of the variable of the function represented by the fuzzy logic system.
v) The equations defining the "output" (the fuzzy set value of the function) corresponding

to the input are:

wj=ı(i) = max
x

min⌊µ
X̃
(x), µ

Ãi
(x)⌋,

µ
Ỹ
(x) = max

j
⌊min(w1, µB̃1(x)), ..., (w2, µB̃2(x)), ..., (wq, µB̃q(x))⌋.

In applications, fuzzy logic systems are described by rules, in the form IF "input" X̃ is Ãi
THEN "output" Ỹ is B̃j.
where Ãi corresponds to an input fuzzy set and B̃j to an output fuzzy set, according to the
rule. In principle, nothing prevents that two rules, with the same antecedent have different
consequents. Also, rules in the form:

IF "input" X̃ is Ã THEN "output" Y is B̃ AND Ỹ is B̃′;
are allowed. However, in the discussion in this section, I do not include such cases. With the

above definition, introduce iterated fuzzy systems as follows:
i) Assume the sets of the input and output fuzzy sets are defined on the same universe of

discourse. This condition allows us to perform the iteration. While it is not required, we may
require that the input fuzzy sets are identical to the output ones; that is, the function represented
by the fuzzy logic system maps a set of fuzzy sets into itself.

ii) At time moment "0", present the fuzzy set X̃[0] at the input of the fuzzy system.
iii) Assign to the actual input fuzzy set X̃[n], which is presented at the input of the fuzzy

system at the time moment n, the output at time moment n-1, Ỹ [n − 1], for n > 0.
This procedure produces the repeated iteration φn of the mapping represented by the fuzzy

system φ : MU → MU , where MU is the set of fuzzy sets on the universe of discourse U.
In what follows, I present some examples related to typical applications of FLSs. I first

address the general case of Sugeno iterated system; next, I address an example involving Sugeno
systems; finally, I address an example involving an iterated Mamdani system.

Consider a directed graph G=(V,E), where V is the set of vertices and E is the set of
directed edges. The graph can be interpreted as the pictorial description (graph) of the index
function used in the definition of the input-output mapping represented by the FLS. When the
input membership functions differ from the output membership functions, the graph is bipartite.
Intuition may induce us to believe that the iteration of a FLS system can have a fixed point only
if the graph has a node with a self-loop. This is not true. Even FLSs with bipartite graphs may
have a fixed point.

Iterated fuzzy logic systems with singleton-type output. General case.


756 H.N. Teodorescu

Figure 7: (a) Bipartite graph and loop for a fuzzy logic system with the index functions 1 →
b, 2 → c, 3 → a. (b) respectively 1 → 2, 2 → 3, 3 → 1. (c) Example of membership functions for
the case (b). Arrows indicate the input to output correspondence. Mamdani-type system. (d)
Sugeno-type system.

Consider the set of membership functions MR = {µ : R → [0, 1]} and the set of weighted
singletons on R, denoted by SR = {(β, w)|β ∈ R, w ∈ [0, 1]}. In (β, w), β are the values where
the singletons occur, that is, x ̸= β ⇒ µβ(x) = 0, and w ∈ [0,1] are the corresponding weights
(truth-values), that is x = β ⇒ µβ(x) = w.

Subsequently, formalize the definition of fuzzy logic systems with singleton-type output, which
is similar to the classic 0-order Sugeno system, but "deprived" of the defuzzification of the output.
Such a system always produces at the output a set of singletons. Thus, the system is a fuzzy-
input, fuzzy-output system. When iterated, such a system may have as fixed point a set of
singletons that, when presented at the input of the system, produces the same set of singletons
at the output.

First, define:
i) A finite set of "input" fuzzy sets; denote these sets by Ãi, i = 1, ..., p.
ii) A finite set of "output" singleton-type fuzzy sets; denote these sets by (Bj, wj = 1), j =

1, ..., q.
iii) An index function ı : {1, ..., p} → {1, ..., q}.
vi) An actual "input" fuzzy set (equivalently, membership function), denoted by X̃; X̃ is the

value of the variable of the function represented by the fuzzy logic system.
iv) The equation defining the truncation of the "output" singletons corresponding to the

input X̃ is wj=ı(i)(X̃) = maxx min⌊µX̃(x), µÃi(x)⌋. Here, wj=ı(i)(X̃) is the truncation value of
the singleton with the current number j, produced by the rule which for an antecedent Ãi has
consequent βj .

Notice that the above relation determines the weights of the output singletons, based on the
intersection of the fuzzy sets X̃ and Ãi. FLSs as above will be named Sugeno-type.

Proposition. The iteration of a FLS as above defined may have a fixed point.
The proof is based on an example. Consider the FLS defined as in Fig. 8. Assume the

membership functions and the rules are:

Ã1 → β2 = 2.5, Ã2 → β1 = 1.5, µÃ1(x) =




x − 1 x ∈ [1, 2]
3 − x x ∈ [2, 3],
0 else

µ
Ã2

(x) =




x − 2 x ∈ [2, 3]
4 − x x ∈ [3, 4].
0 else

Then, X̃, where µ
X̃
(x) =




0.5 x = 1.5

0.5 x = 2.5 is a fixed point of the FLS.♢
0 else

It is not difficult to see that:


On Fuzzy Sequences, Fixed Points and Periodicity in Iterated Fuzzy Maps 757

Figure 8: .

Consider a Sugeno-type FLS as above, with continuous input membership functions and with
two rules Ãi → βj Ãi+1 → βk where µÃi(βj) ̸= 0, µÃi+1(βk) ̸= 0 and ∀ h such that h ̸= i&h ̸= i+
1 µ

Ãh
(βj) = 0 and µÃh(βk) = 0. Then, the FLS has the fixed point (βj, µÃi(βj))∪(βk, µÃi+1(βk)).

The condition of continuity is needed only to prevent undefined values.
Subsequently, I consider Mamdani-type FLS, that is, systems having "complete" (not single-

tons) fuzzy sets at the input and the output. Assume that the input and the output membership
functions are the same and are isosceles triangular, overlapping two by two (typical case in ap-
plications, see figures 7,8). Also assume that every rule has a single consequent, that is rules
have the form Ãi → Ãj, while rules like Ãi → Ãj ∨ Ãk are not allowed. Also assume the output
of the FLS is defuzzified by the center of gravity (c.o.g.) method. Then,

Proposition. If there is a cycle in the graph of the above described FLS, then there is an
initial state of the system that produces an oscillatory iteration.

Indeed, there is an input real value where only one input membership function is non-null
and thus selects a single output membership function. Defuzzification of the output by c.o.g.
produces a real value; by the assumed conditions, for that value only one input membership
function is non-null. Continuing, the cycle in the graph is thus completed.♢

As a last case, I deal with Sugeno-type FLS. The equation defining the output fuzzy (overall)
set (the fuzzy set value of the function represented by the fuzzy system so defined) is:

µY (y, X̃) = max
j

[
min(w1(X̃), µβ1(y), ..., (wj(X̃), µβj(y), ..., (wq(X̃), µβq(y)

]
Notice that the last relation represents the union of the weighted output singletons.
The iteration is obtained by imposing µX[n](x) = µY [n − 1](y). For allowing the iteration,

the fuzzy system must satisfy the condition that the input and output membership functions
must be the defined and take values on the same space. It is also apparent that, whatever the
initial input membership function is, after the first iteration, the input membership function can
be but a union of singletons. Consequently, with no loss of generality, I assume that the initial
input in the iteration is a set of singletons with the same positions as the output singletons,
(βj, wj), j=1,...,q. These singletons determine a set of p×q values of the input membership
functions, µj(βj), some of them possibly null.

Therefore, at iteration n > 1, µi(βj[n − 1]). The input truncation values become
max(µi(βj[n − 1]), wj[n − 1]). Thus, the new values of (truth-degree of) the singletons are:

wh[n] = max
j s.t.h=ı(j)

max
i

(µi(βj[n − 1]), wj[n − 1]);

The fixed point condition becomes:

wh = max
j s.t.h=ı(j)

max
i

(µi(βj), wj);


758 H.N. Teodorescu

because the equality must hold for whatever time moment is considered at the two sides. The
p-period condition is obtained by requesting that

wh[n] = max
j s.t.h=ı(j)

max
i

(µi(βj[n − p − 1], wj[n − p − 1]).

Example. In this example I deal with a fuzzy logic system as above, with two fuzzy sets
defining the input and two singletons defining the output. Consider the two input functions

µA1(x) =

{
4x(1 − x) if x ∈ [0, 1]
0 elsewhere ,

µA2(x) =




3x if x ∈ [0, 1/3]
1 − (3/2)(x − 1/3) if x ∈ [1/3, 1].
0 elsewhere

The index function is described by 1→2, 2→1. The output singletons are at β1 = 0.27 and
β2 = 0.6 (see Figure 9).

Figure 9: Explanatory graphs for the iteration.

The initial input is µX(0) =




1 if x = 0.2

1 if x = 0.8.

0 elsewhere

The first value of the output is (0.27, w1[0])(0.6, w2[0]) where:
w1[0] = max [min(µA2(0.2), 1), min(µA2(0.8), 1)] = max(0.6, 1 − .7) = 0.6,

w2[0] = max [min(µA1(0.2), 1), min(µA1(0.8), 1)] = max(0.64, 0.64) = 0.64.

The output becomes the new value of the variable of the function represented by the fuzzy
system. The second value of the output is (β1 = 0.27, w1[1] = 0.6) ∪ (β2 = 0.6, w2[1] = 0.64)
where:

w1[1] = max [min(µA2(0.27), w1[0]), min(µA2(0.6), w2[0])] =

max[min(0.81, 0.6), min(0.6, 0.64)] = max(0.6, 0.6) = 0.6,

w2[1] = max [min(µA1(0.27), w1[0]), min(µA1(0.6), w2[0])] =

max(0.64, 0.64) = 0.64.

Therefore, the output for the second iteration is identical to the output of the first, (β1 =
0.27, w1[1] = 0.6) ∪ (β2 = 0.6, w2[1] = 0.64); thus, the iteration result is constant. The result is
a fixed point of the system.


On Fuzzy Sequences, Fixed Points and Periodicity in Iterated Fuzzy Maps 759

4 Conclusions

In this paper, I focused on preparatory topics, namely fuzzy sequences, the iteration of FLS,
which are functions defined on sets of (membership) functions, and on their fixed points and their
periodicity. Whatever narrow is this topic, it is central in the theory of dynamic fuzzy systems,
because both fuzzy logic systems without defuzzifier, and fuzzy algebraic systems apply sets of
functions to sets of functions, while fuzzy logic systems with defuzzifier apply sets of functions
to real numbers.

The iteration may be convergent (either in a specified metric, or uniformly in the space of
the output, that is in the space of the membership functions values) to a limit fuzzy set, or may
be periodical, as it has been illustrated using simple examples. More complex behaviors can
appear when the output of the FLS is defuzzified. I primarily dealt with fuzzy systems without
defuzzifier at the output. The systems with defuzzified output will be dealt with in subsequent
papers.

Acknowledgment. I thank several colleagues and the referees for their comments and
suggestions. The paper was not supported by, but is related to research performed in the frame
of the Romanian Academy.

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