Stationary Points—I: One-Dimensional p-Fuzzy Dynamical Systems

Abstract

p-Fuzzy dynamical systems are variational systems whose dynamic is obtained by means of a Mamdani type fuzzy rule-based system. In this paper, we will show the 1-dimensional p-fuzzy dynamical systems and will present theorems that establish conditions of existence and uniqueness of stationary points. Besides the obtained analytical results, we will present examples that illustrate and confirm the obtained mathematical results.

1. Introduction

Variational equations or deterministic dynamical systems (difference and differential equations) constitute a powerful tool for modeling when the state variables depend on variations throughout time. The efficiency of a deterministic model depends on knowledge of the relationships between variables and their variations. In addition, in many situations such relations are only partially known; therefore, the modeling with deterministic variational systems, or even with stochastic ones, may not be adequate. In addition, fuzzy systems derived from deterministic models, which have subjectivity regarding some parameters, are not appropriate when we have only incomplete information of the phenomenon being analyzed. Thus, the use of a rule-based system can be adopted as an alternative for modeling partially known phenomena or those carried with imprecision.

Fuzzy rule-based systems have been used with success in some areas as control, decision taking, recognition systems, and so forth. This success is due to its simplicity and interrelation with humans way of reasoning. Fuzzy rule-based systems are conceptually simple [1]. Such systems are basically threefold: an input (fuzzifier), an inference mechanism composed of a base of fuzzy rules together with an inference method, and, finally, an output (defuzzifier) stage (see Figure 1).

Figure 1 

Architecture of a fuzzy rule-based system.

There are two main types of fuzzy rule-based systems, the Mamdani fuzzy systems and the Takagi-Sugeno fuzzy systems [2]. The main characteristic of the Mamdani type systems is that both the antecedent and consequent are expressed by linguistic terms, while in the Takagi-Sugeno type systems only the antecedent is expressed by linguistic terms and the consequent is expressed by functions.

The Takagi-Sugeno fuzzy systems are more restrictive than the Mamdani fuzzy systems because they require an a priori function in the output. However, due to the existence of theoretical methods for Takagi-Sugeno fuzzy systems stability analysis [3–8], the latter has become more used. On the other hand, Mamdani type systems are used as a “black box” and are still criticized because they lack a study on its stability [9, 10].

Fuzzy variational equations have been used in different methods. Some attempts to contemplate subjectivity original from aleatory processes such as Hukuhara’s derivative, differential inclusions, and Zadeh’s extension [11] have been already proposed. In these methods, the adopted process for studying the variational systems is always derived from deterministic classical systems.

In this paper, we will present the p-fuzzy systems whose dynamics is not based in formal concepts of variations originated from derivative or explicit differences or differential inclusions. In the p-fuzzy dynamical systems the dynamic (iterative process) is obtained by means of a Mamdani’s fuzzy rule-based system. The main advantage of this method with respect to the other ones is the simplicity of the involved mathematics, just because the interactive method is supplied by the Mamdani controller.

Formally, a p-fuzzy system in ℝ^𝑛 is a discrete dynamic system: 𝑥_𝑘+1=𝐹𝑥_𝑘,𝑥_𝑜givenand𝑥_𝑘∈ℝ^𝑛, (1) where the 𝐹 function is given by 𝐹(𝑥_𝑘)=𝑥_𝑘+Δ(𝑥_𝑘) and Δ(𝑥_𝑘)∈ℝ^𝑛 is obtained by means of a fuzzy rule-based system; that is, Δ(𝑥_𝑖) is the defuzzification value of the rule-based system. The architecture of a p-fuzzy system can be visualized in Figure 2.

Figure 2 

Architecture of a p-fuzzy system.

The name p-fuzzy dynamical systems or purely fuzzy was chosen to differentiate it from other fuzzy systems given by variational equations.

In this paper, we will focus on the one-dimensional p-fuzzy systems, which are always associated with a Mamdani fuzzy system, where the defuzzification method is the centroid. We have chosen this method because it is widely used and more general to deal with weight mean of linguistic variables [12, pages 242, 243].

Analogous to the inhibited variational models in which one has stationary solutions, our objective is to present results that establish the necessary and sufficient conditions for the existence of a stationary point.

2. Preliminaries

In this section, we introduce the main concepts for the development of the work presented in this paper.

2.1. Definitions

Definition 1 (support). Let 𝐴 be a fuzzy subset of 𝑋; the support of 𝐴, denoted supp(𝐴), is the crisp subset of 𝑋 whose elements all have nonzero membership grades in 𝐴; that is, supp(𝐴)={𝑥∈𝑋;𝐴(𝑥)>0}. (2)

Definition 2 (𝛼-cut). An 𝛼-level set of a fuzzy subset 𝐴 of 𝑋 is a crisp set denoted by [𝐴]^𝛼 and defined by (i)[𝐴]⁰=supp(𝐴), (𝛼=0), (ii)[𝐴]^𝛼={𝑥∈𝑋;𝐴(𝑥)≥𝛼}, if 𝛼∈(0,1].

Definition 3 (fuzzy number). A fuzzy number 𝐴 is a fuzzy set 𝐴⊂ℝ satisfying the following conditions: (i)[𝐴]^𝛼≠Ø, ∀𝛼∈[0,1],(ii)[𝐴]^𝛼 is a closed interval, ∀𝛼∈[0,1],(iii)the supp(𝐴) is bounded.

Definition 4. Consider a one-dimensional p-fuzzy system given by  (1).
Consider that 𝑥^∗ is a stationary point of (1) if 𝐹𝑥^∗=𝑥^∗+Δ𝑥^∗=𝑥^∗⟺Δ𝑥^∗=0. (3)

Definition 5. Let {𝐴_𝑖}_{1≤𝑖≤𝑘} be any finite family of normal fuzzy subsets associated with the fuzzy variable 𝑥. Assume that {𝐴_𝑖}_{1≤𝑖≤𝑘} is a family of successive fuzzy subsets (Figure 3) if, (i)supp(𝐴_𝑖)∩supp(𝐴_𝑖+1)≠Ø, for each 1≤𝑖<𝑘; (ii)⋂_{𝑗=𝑖,𝑖+2}supp(𝐴_𝑗) has at maximum only one element for each 1≤𝑖<𝑘−1; that is, supp(𝐴_𝑖)∩supp(𝐴_𝑖+2)≠𝜙, if and only if, max{𝑥∈supp(𝐴_𝑖)}=min{𝑥∈supp(𝐴_𝑖+2)}; (iii)⋃_𝑖=1,𝑘supp(𝐴_𝑖)=𝑈, where 𝑈 is the domain of the fuzzy variable 𝑥; (iv)given 𝑧₁∈supp(𝐴_𝑖) and 𝑧₂∈supp(𝐴_𝑖+1), if 𝐴_𝑖(𝑧₁)=1 and 𝐴_𝑖+1(𝑧₂)=1, then necessarily 𝑧₁<𝑧₂ for each 1≤𝑖<𝑘.

Figure 3 

Family of successive fuzzy subsets.

Definition 6. Consider a family of successive fuzzy subsets {𝐴_𝑖}_{1≤𝑖≤𝑘} that describe the antecedent of a fuzzy system associated with the p-fuzzy system (1). We say that 𝐴^∗ is an equilibrium viable set of (1) if 𝐴^∗ contains stationary points of (1).

If for some 1≤𝑖<𝑘 there are 𝑧₁,𝑧₂∈[𝐴_𝑖∪𝐴_𝑖+1]⁰ such that Δ(𝑧₁)⋅Δ(𝑧₂)<0, then 𝐴^∗ is given by 𝐴^∗=[𝐴_𝑖∩𝐴_𝑖+1]⁰. If for all 𝑧₁,𝑧₂∈[𝐴_𝑖∪𝐴_𝑖+1]⁰, 1≤𝑖<𝑘, Δ(𝑧₁)⋅Δ(𝑧₂)>0, then 𝐴^∗=[𝐴_𝑘]⁰.

A p-fuzzy system depends on the fuzzy system associated with it, that is, it depends on the rule-base, on the inference method and on the defuzzification method used. In Definition 6, a sufficient condition for Δ(𝑧₁)⋅Δ(𝑧₂)<0 is that the p-fuzzy system be associated with a fuzzy system whose rule-base in 𝐴^∗=[𝐴_𝑖∪𝐴_𝑖+1]⁰ is of the type𝑅₁: if 𝑥 is 𝐴_𝑖 then Δ is 𝐵, 𝑅₂: if 𝑥 is 𝐴_𝑖+1 then Δ is 𝐶, where supp(𝐵)⊂ℝ⁻ and supp(𝐶)⊂ℝ⁺ or vice versa. When supp(𝐵)⊂ℝ⁻ and supp(𝐶)⊂ℝ⁺ we have that the equilibrium viable set 𝐴^∗ is of the type (𝐴_𝑖,𝐴_𝑖+1)→(𝐶,𝐵). If, on the other hand, we have that supp(𝐵)⊂ℝ⁺ and supp(𝐶)⊂ℝ⁻ we say that 𝐴^∗ is of the type (𝐴_𝑖,𝐴_𝑖+1)→(𝐵,𝐶).

To understand the dynamics of the p-fuzzy system we need to understand how the rule-based system works more specifically, given 𝑥∈𝐴^∗, how is Δ(𝑥) obtained? In the following section we will describe this process.

2.2. Output Defuzzification of the Fuzzy System

Let 𝐴^∗=[𝐴_𝑖∩𝐴_𝑖+1]⁰=[𝑐₁,𝑐₂] be an equilibrium viable set of the p-fuzzy system. To facilitate the notation, we will indicate by 𝑟 the membership function of 𝐴_𝑖, by 𝑠 the membership function of 𝐴_𝑖+1, 𝑧₁=min_{𝑥∈supp(𝐴𝑖)}{𝑟(𝑥)=1},𝑧₂=max_{𝑥∈supp(𝐴𝑖+1)}{𝑠(𝑥)=1}, (4) and by 𝑓 and 𝑔 the membership functions of 𝐶 and 𝐷 (Figure 4), respectively. Assume that the p-fuzzy system, in the equilibrium viable set 𝐴^∗, is of the type (𝐴_𝑖,𝐴_𝑖+1)→(𝐶,𝐵).

Figure 4 

Mamdani’s inference process for 𝐴∗ of the type (𝐴𝑖,𝐴i+1)→(𝐶,𝐵).

For each 𝑥∈𝐴^∗, Δ(𝑥) is the 𝐑 region centroid abscise, with 𝐑 limited by the membership function of the fuzzy output, Δ𝑥, (see Figure 4). Thus, Δ(𝑥)=^𝑓−1_𝑏^(𝑚)𝑡𝑓(𝑡)𝑑𝑡+⁰_{𝑓−1(𝑚)}𝑚𝑡𝑑𝑡+^𝑔−1₀^(𝑛)+𝑛𝑡𝑑𝑡_{𝑎𝑔−1(𝑛)}(𝑡)×𝑡𝑔𝑑𝑡_𝑛0𝑔⁻¹(𝑡)𝑑𝑡−^𝑚₀𝑓⁻¹(𝑡)𝑑𝑡⁻¹, (5) where (𝑛,𝑚)=(𝑟(𝑥),𝑠(𝑥)). Equation (5) can be rewritten as Δ(𝑥)=ℎ₁(𝑛)+ℎ₂(𝑚)𝐴(),𝑚,𝑛 (6) where ℎ₁(𝑛)=^𝑔−1₀^(𝑛)𝑛𝑡𝑑𝑡+^𝑎_{𝑔−1(𝑛)}(𝑡)ℎ𝑡𝑔𝑑𝑡,₂(𝑚)=^𝑓−1_𝑏^(𝑚)𝑡𝑓(𝑡)𝑑𝑡+⁰_{𝑓−1(𝑚)}𝐴()=𝑚𝑡𝑑𝑡,𝑚,𝑛^𝑛₀𝑔⁻¹(𝑡)𝑑𝑡−^𝑚₀𝑓⁻¹(𝑡)𝑑𝑡. (7)(8)(9)

2.3. Preliminary Results

In this section, we will present some important technical results for the main demonstrations of the theorems that establish sufficient conditions for uniqueness and existence of the stationary point of the p-fuzzy systems. Here, the presented results are technical enough and are important only for demonstrating the theorems in the next section. Therefore, the reader will be able, without loss of continuity, to skip them if desired.

The following results are referred to the equilibrium viable set of the type (𝐴_𝑖,𝐴_𝑖+1)→(𝐶,𝐵) (Figure 4). From now on, it will be assumed that the functions 𝑟, 𝑠, 𝑓, and 𝑔 (which represent the membership functions 𝜇_𝐴𝑖, 𝜇_𝐴𝑖+1, 𝜇_𝐵 and 𝜇_𝐶, resp.) are continuous.

Lemma 7. The function ℎ₁ (7) is increasing, and its range is given by, 𝐼𝑚(ℎ₁)=[0,∫^𝑎₀𝑡𝑔(𝑡)𝑑𝑡].

Proof. Since 𝑔 is continuous in [0,𝑎] (𝑔⁻¹ is limited in [0,𝑎]), then the function ℎ₁ is differentiable, and ℎ^′₁(𝑛)=^𝑔−1₀^(𝑛)𝑛𝑡𝑑𝑡^′+^𝑎_{𝑔−1(𝑛)}𝑡𝑔(𝑡)𝑑𝑡^′. (10) Using the derivative properties, ℎ^′₁(𝑛)=^𝑔−1₀^(𝑛)𝑡𝑑𝑡+𝑛^𝑔−1₀^(𝑛)𝑡𝑑𝑡^′−^𝑔−1_𝑎^(𝑛)𝑡𝑔(𝑡)𝑑𝑡^′, (11) and using the chain rule and the fundamental theorem of calculus, we obtain ℎ^′₁(𝑛)=^𝑔−1₀^(𝑛)𝑡𝑑𝑡+𝑛𝑔⁻¹(𝑛)𝑔^−1′(𝑛)−𝑛𝑔⁻¹(𝑛)𝑔^−1′(𝑛). (12) Hence, ℎ^′₁(𝑛)=^𝑔−1₀^(𝑛)𝑔𝑡𝑑𝑡=⁻¹(𝑛)²2>0. (13) Therefore, ℎ₁ is increasing.
Since, ℎ₁(0)=^𝑔−1₀⁽⁰⁾0𝑡𝑑𝑡+^𝑎_𝑔−1(0)𝑡𝑔(𝑡)𝑑𝑡=^𝑎_𝑎(𝑡)ℎ𝑡𝑔𝑑𝑡=0,₁(1)=^𝑔−1₀⁽¹⁾𝑡𝑑𝑡+^𝑎_𝑔−1(1)𝑡𝑔(𝑡)𝑑𝑡=^𝑎₀𝑡𝑔(𝑡)𝑑𝑡, (14) then 𝐼𝑚(ℎ₁)=[0,∫^𝑎₀𝑡𝑔(𝑡)𝑑𝑡].

Lemma 8. The function ℎ₂ is decreasing, and its range is given by 𝐼𝑚(ℎ₂)=[∫⁰_𝑏𝑡𝑓(𝑡)𝑑𝑡,0].

Proof. Analogous to the demonstration of Lemma 7.

Lemma 9. Let 𝜙∶𝐼=[𝑑₁,𝑑₂]→ℝ be a function of the class 𝐶². If 𝜙^′′(𝑧)>0, ∀𝑧∈(𝑑₁,𝑑₂) and 𝜙(𝑑₁)<0, then 𝜙 has at maximum one root in 𝐼.

Proof of Lemma 9. Assume there are 𝑧₁,𝑧₂∈𝐼 (𝑧₁<𝑧₂) such that 𝜙(𝑧₁)=𝜙(𝑧₂)=0. Since 𝜙^′′(𝑧)>0, we have that 𝜙 is not constant. Hence, by Rolle’s Theorem, ∃𝑐∈(𝑧₁,𝑧₂) such that 𝜙^′(𝑐)=0. Hence 𝑐 is a minimum point, because 𝜙^′′(𝑐)>0. But, 𝜙(𝑑₁)<0⇒𝜙(𝑐)>0. Since 𝜙 is continuous ∃𝑧_𝑜∈(𝑧₁,𝑧₂) such that 𝜙(𝑐)>𝜙(𝑧_𝑜)>0, it is nonsense! Therefore 𝜙 has at maximum one root.

Lemma 10. If 𝑔(𝑡)>𝑓(−𝑡), ∀𝑡∈[0,−𝑏], then 𝑔⁻¹(𝑘)>−𝑓⁻¹(𝑘)∀𝑘∈[0,1].

Proof. The proof is simple (see Figure 5).

Figure 5 

System p-fuzzy output with 𝑔(𝑡)>𝑓(−𝑡).

Lemma 11. If 𝑔(𝑡)>𝑓(−𝑡), ∀𝑡∈[0,−𝑏] then for 𝑚,𝑛∈[0,1] with 𝑚≤𝑛 one has that Δ(𝑥)=ℎ₁(𝑛)+ℎ₂(𝑚)𝐴(𝑚,𝑛)>0. (15)

Proof. The proof is simple.

Lemma 12. If 𝑔(𝑡)<𝑓(−𝑡), ∀𝑡∈[0,𝑎], and 𝑚,𝑛∈[0,1]𝑚≥𝑛, then Δ(𝑥)<0.

Proof. It is analogous to the proof of Lemma 11.

3. Stationary Point

In this section, we will enunciate and prove a theorem that guarantees the existence of at least one stationary point for each equilibrium viable set of the p-fuzzy system. For this, we will use again Figure 4 to motivate the presented results in this section.

Theorem 13 (existence). Let 𝑆 be a p-fuzzy system and 𝐴^∗ an equilibrium viable set of 𝑆 of the type (𝐴_𝑖,𝐴_𝑖+1)→(𝐶,𝐵). Then, there is at least one stationary point of 𝑆 in 𝐴^∗. That is, ∃𝑥^∗∈𝐴^∗ such that Δ(𝑥^∗)=0.

Proof. Given 𝑥∈𝐴^∗, from Definition 4, then 𝑥 is a stationary point if and only if Δ(𝑥)=0⟺ℎ₁(𝑛)+ℎ₂(𝑚)=0. (16) If 𝜇_𝐴𝑖(𝑐₁)=0, then Δ𝑐₁=ℎ₁𝜇_𝐴𝑖𝑐₁+ℎ₂𝜇_𝐴𝑖+1𝑐₁=ℎ₁(0)+ℎ₂(0)=0, (17) and, therefore 𝑐₁ is a stationary point. If 𝜇_𝐴𝑖+1(𝑐₂)=0 one has that Δ(𝑐₂)=0; hence 𝑐₂ is a stationary point. Now, assume that 𝜇_𝐴𝑖(𝑐₁)>0 and 𝜇_𝐴𝑖+1(𝑐₂)>0. Since 𝜇_𝐴𝑖+1(𝑐₁)=0, then from Lemmas 7 and 8 one has that ℎ₁(𝜇_𝐴𝑖(𝑐₁))>0 and ℎ₂(𝜇_𝐴𝑖+1(𝑐₁))=0. Therefore Δ𝑐₁=ℎ₁𝜇_𝐴𝑖𝑐₁+ℎ₂𝜇_𝐴𝑖+1𝑐₁=ℎ₁𝜇_𝐴𝑖𝑐₁>0. (18) Thus, from Lemma 7, Δ𝑐₂=ℎ₁𝜇_𝐴𝑖𝑐₂+ℎ₂𝜇_𝐴𝑖+1𝑐₂=ℎ₂𝜇_𝐴𝑖+1𝑐₂<0. (19) Since Δ is continuous, by Bolzano’s Intermediate Value Theorem, ∃𝑥^∗∈[𝑐₁,𝑐₂] such that Δ(𝑥^∗)=0; therefore, 𝑥^∗ is a stationary point.

Remark 14. If in Theorem 13 we consider 𝐴^∗ as an equilibrium viable set of the type (𝐴_𝑖,𝐴_𝑖+1)→(𝐵,𝐶), the result is analogous; that is, there exists a stationary point 𝑥^∗∈𝐴^∗.

3.1. Local Stationary Points-Symmetrical Output

If 𝐴^∗ is an equilibrium viable set, where the membership functions of the consequents, 𝐵 and 𝐶, are symmetrical functions, then the stationary point in 𝐴^∗ is unique. Except when 𝜇_𝐴𝑖(𝑐₁)=0 or 𝜇_𝐴𝑖+1(𝑐₂)=0 possibly occurs. Then we have the following proposition.

Proposition 15. Let 𝑆 be a p-fuzzy system and 𝐴^∗ an equilibrium viable set of 𝑆 of the type (𝐴_𝑖,𝐴_𝑖+1)→(𝐶,𝐵). If the membership functions of 𝐵 and 𝐶, respectively, 𝜇_𝐵 and 𝜇_𝐶, are monotonous and symmetric, that is, 𝜇_𝐶(𝑡)=𝜇_𝐵(−𝑡), then there exists an equilibrium point in 𝐴^∗: 𝑥^∗=max_{𝑥∈𝐴∗}min𝜇_𝐴𝑖(𝑥),𝜇_𝐴𝑖+1(𝑥). (20)

Proof. Since 𝜇_𝐵(𝑡)=𝜇_𝐶(−𝑡) then 𝜇_𝐵(−𝑎)=𝜇_𝐶(𝑎)=0=𝜇_𝐵(𝑏)⇒𝑏=−𝑎, because 𝜇_𝐵 Ú is monotonous. Yet, we have that 𝜇_𝐵(𝑡)=𝜇_𝐶(−𝑡)⇒𝜇_𝐵⁻¹(𝜇_𝐶(𝑡))=−𝑡=−𝜇_𝐵⁻¹(𝜇_𝐵(𝑡))⇒𝜇_𝐵⁻¹(𝑦)=−𝜇_𝐶⁻¹(𝑦).
Then, Δ(𝑧_𝑜)=0, if and only if, ℎ₁(𝑛)=−ℎ₂(𝑚). Since 𝑏=−𝑎, from (8), then we obtain ℎ₂(𝑚)=^{𝜇𝐵−1(𝑚)}_−𝑎𝑡𝜇_𝐵(𝑡)𝑑𝑡+⁰_{𝜇𝐵−1(𝑚)}𝑚𝑡𝑑𝑡. (21) If we perform a change in the variable 𝑢=−𝑡, we have ℎ₂(𝑚)=^{−𝜇𝐵−1(𝑚)}_a𝑢𝜇_𝐵(−𝑢)𝑑𝑢+⁰_{−𝜇𝐵−1(𝑚)}𝑚𝑢𝑑𝑢⟹ℎ₂(𝑚)=^𝜇𝐶−1_𝑎^(𝑚)𝑢𝜇_𝐶(𝑢)𝑑𝑢+⁰_{𝜇𝐶−1(𝑚)}𝑚𝑢𝑑𝑢=−ℎ₁(𝑚). (22) That is, ℎ₂=−ℎ₁. Hence, ℎ₁(𝑛)=−ℎ₂(𝑚)⇔ℎ₁(𝑛)=ℎ₁(𝑚)⇔𝑚=𝑛 (because ℎ₁ is increasing: Lemma 7); that proves the proposition.

Remark 16. If 𝜇_𝐴𝑖(𝑐₁)≠0, then 𝑥^∗=max_𝑥∈𝐴^∗[min(𝜇_𝐴𝑖(𝑥),𝜇_𝐴𝑖+1(𝑥))] is the only stationary point in 𝐴^∗. Besides that, if the system 𝑆 is of the type (𝐴_𝑖,𝐴_𝑖+1)→(𝐵,𝐶), then the result of Proposition 15 is the same.

4. Uniqueness of the Stationary Point

In this section we will enunciate and prove theorems that establish condition for uniqueness of the stationary point of a one-dimensional p-fuzzy system. Initially we will consider a simpler case, when 𝐴^∗⊂[𝑧₁,𝑧₂] (Figure 4), where 𝑧₁=min_{𝐴𝑥∈supp𝑖}{𝑟(𝑥)=1},𝑧₂=max_{𝐴𝑥∈supp𝑖+1}{𝑠(𝑥)=1}. (23)

Theorem 17. Let 𝑆 be a p-fuzzy system and 𝐴^∗ an equilibrium viable set of 𝑆 of the type (𝐴_𝑖,𝐴_𝑖+1)→(𝐶,𝐵). If the functions 𝜇_𝐴𝑖 and 𝜇_𝐴𝑖+1 are piecewise monotonous and 𝐴^∗⊂[𝑧₁,𝑧₂] then there exists only one stationary point in 𝐴^∗.

Proof. Given 𝑥∈𝐴^∗ one has Δ(𝑥)=ℎ₁(𝑛)+ℎ₂(𝑚)=ℎ₁𝜇_𝐴𝑖(𝑥)+ℎ₂𝜇_𝐴𝑖+1(𝑥). (24) Using Lemmas 7 and 8 and the chain rule we find that the derivative of Δ is Δ^′(𝑥)=𝜇_𝐶⁻¹𝜇_𝐴𝑖(𝑥)²2𝜇^′_𝐴𝑖(𝑥)−𝜇_𝐵⁻¹𝜇_𝐴𝑖+1(𝑥)²2𝜇^′_𝐴𝑖+1(𝑥). (25) Since in 𝐴^∗=[𝑐₁,𝑐₂]𝜇_𝐴𝑖 is not increasing and 𝜇_𝐴𝑖+1 is not decreasing, then 𝜇^′_𝐴𝑖(𝑥)≤0 and 𝜇^′_𝐴𝑖+1(𝑥)≥0, and, besides that, if 𝜇^′_𝐴𝑖(𝑥)=0, we have that 𝜇^′_𝐴𝑖+1(𝑥)≠0 and if 𝜇^′_𝐴𝑖+1(𝑥)=0 we obtain 𝜇^′_𝐴𝑖(𝑥)≠0. Then, from (25), Δ^′(𝑥)<0. This shows that Δ is decreasing. From Theorem 13, there exists a stationary point in 𝐴^∗ then this point is unique.
Now, consider the more general case, when 𝐴^∗⊄[𝑧₁,𝑧₂], and divide it into two theorems. Initially, consider the case that the membership functions 𝜇_𝐶 and 𝜇_𝐵 are such that 𝜇_𝐶(𝑡)>𝜇_𝐵(−𝑡); next, consider the case where 𝜇_𝐶(𝑡)<𝜇_𝐵(−𝑡).

4.1. Case 1: 𝜇_𝐶(𝑡)>𝜇_𝐵(−𝑡)

Theorem 18 (uniqueness). Let 𝑆 be a p-fuzzy system and 𝐴^∗ an equilibrium viable set of 𝑆 of the type (𝐴_𝑖,𝐴_𝑖+1)→(𝐶,𝐵). If the functions 𝜇_𝐴𝑖, 𝜇_𝐴𝑖+1, 𝜇_𝐵 and 𝜇_𝐶 are continuously differentiable, 𝜇_𝐴𝑖 and 𝜇_𝐴𝑖+1 are piecewise monotonous 𝜇_𝐵 and 𝜇_𝐶 are strictly monotonous, such that (i)𝜇_𝐶(𝑡)>𝜇_𝐵(−𝑡), ∀𝑡∈(0,−𝑏),(ii)𝜇^′_𝐶(𝑞)/𝜇^′_𝐵(𝑝)<(𝑝³/𝑞³), ∀𝑝∈supp(𝐵), 𝑞∈supp(𝐶) and 𝜇_𝐵(𝑝)>𝜇_𝐶(𝑞),(iii)[𝜇^′_𝐴𝑖+1(𝑥)/𝜇^′_𝐴𝑖(𝑥)]≤0, ∀𝑥∈(𝑧_𝑜,𝑐₂), 𝜇_𝐴𝑖(𝑥)≠𝜇_𝐴𝑖+1(𝑥). Then, 𝑆 has only one stationary point, 𝑥^∗ in 𝐴^∗, and 𝑥^∗∈(𝑧_𝑜,𝑐₂].

Proof. For the sake of simplicity notation, we make 𝑟=𝜇_𝐴𝑖, 𝑠=𝜇_𝐴𝑖+1, 𝑓=𝜇_𝐵 and 𝑔=𝜇_𝐶.
Initially, we see that given 𝑥∈(𝑧_𝑜,𝑐₂] (Figure 4), 𝑥 determines only one (𝑛,𝑚)∈[0,1]² such that 𝑛=𝑟(𝑥) and 𝑚=𝑠(𝑥). By monotonicity of 𝑟, we have that for each 𝑛∈[0,𝑟(𝑧_𝑜)) there exists only one 𝑚∈[0,1] such that 𝑛=𝑟(𝑥) and 𝑚=𝑠(𝑥). That is, each (𝑛,𝑚), in this situation, determines only one 𝑥∈(𝑧_𝑜,𝑐₂].
By Theorem 13, there exists a stationary point 𝑥^∗∈[𝑐₁,𝑐₂]=[𝑐₁,𝑧_𝑜]∪(𝑧_𝑜,𝑐₂]. Given 𝑥∈[𝑐₁,𝑧_𝑜]⇒𝑚=𝑠(𝑥)≤𝑛=𝑟(𝑥). Then, by Lemma 11𝑥^∗∉[𝑐₁,𝑧_𝑜]⇒𝑥^∗∈(𝑧_𝑜,𝑐₂]. That is equivalent to the existence of only one (𝑛^∗,𝑚^∗), with 𝑛^∗∈[0,𝑟(𝑧_𝑜)) such that 𝐻(𝑛^∗,𝑚^∗)=0.
Since for each 𝑛∈[0,𝑟(𝑧_𝑜)) there exists only one 𝑚∈[0,1] such that 𝑛=𝑟(𝑥) and 𝑚=𝑠(𝑥), then we may define a function 𝛿₂∶[0,𝑟(𝑧_𝑜))→[0,1] such that 𝑚=𝛿₂(𝑛) (Figure 7). We observe that 𝛿₂ is continuous, because 𝑟 and 𝑠 are continuous. Using the chain rule we get the derivative of 𝛿₂ to be 𝛿^′₂(𝑛)=𝑠^′(𝑥)𝑟^′(𝑥)⁽ⁱⁱⁱ⁾⇒𝛿^′′₂(𝑛)≤0,∀𝑛∈𝐷_𝛿2. (26) By Lemmas 7 and 8, ℎ₁ and −ℎ₂ are increasing, and, by condition (i), it follows that (Figure 6) 𝑡𝑔(^𝑎₀𝑡)𝑑𝑡>−⁰_𝑏𝑡𝑓(𝑡)𝑑𝑡⟺ℎ₁(1)>−ℎ₂(1). (27) Then, given 𝑛∈[0,ℎ₁⁻¹(−ℎ₂(1))], there exists only one 𝑚∈[0,1] such that ℎ₁(𝑛)=−ℎ₂(𝑚)⟺ℎ₁(𝑛)+ℎ₂(𝑚)=0. (28) Therefore, we may define an injective function 𝜉, 𝑚=𝜉(𝑛) (Figure 7) so that the inverse range of 0 by 𝐻 is given by 𝐻⁻¹(0)={(𝑛,𝑚);𝑚=𝜉(𝑛)}, (29) where 𝐻∶[0,ℎ₁⁻¹(−ℎ₂(1))]×[0,1]→ℝ is given by 𝐻(𝑛,𝑚)=ℎ₁(𝑛)+ℎ₂(𝑚).
Since 𝜕𝐻/𝜕𝑛=ℎ^′₁(𝑛)=(𝑔⁻¹(𝑛))²/2>0 (Lemma 7) and 𝜕𝐻/𝜕𝑚=ℎ^′₂(𝑚)=−(𝑓⁻¹(𝑚))²/2<0 (Lemma 8) then, by the Implicity Function Theorem, 𝜉 is 𝑘 times differentiable, and, besides that, 𝜉^′(𝑛)=−𝑑ℎ₁/𝑛𝑑ℎ₂=𝑔/𝑚⁻¹(𝑛)𝑓⁻¹(𝑚)²>0,∀𝑛∈0,ℎ₁⁻¹−ℎ₂(1),𝑚∈(0,1),𝑚=𝜉(𝑛). (30) Thus, 𝜉 is a strictly increasing function, and since 𝐻(0,0)=0 and 𝐻(ℎ₁⁻¹(−ℎ₂(1)),1)=0, then 𝐷_𝜉=[0,ℎ₁⁻¹(ℎ₂(1))] and 𝐼𝑚_𝜉=[0,1].
Given 𝑚,𝑛∈(0,1), there is only one 𝑝∈(𝑏,0) such that 𝑝=𝑓⁻¹(𝑚), and there exists only one 𝑞∈(0,𝑎) such that 𝑞=𝑔⁻¹(𝑛), since by assumption 𝑓 and 𝑔 are strictly monotonous.
From Lemma 11 we have that 𝑚≤𝑛⇒𝐻(𝑚,𝑛)>0. Hence, 𝐻(𝑚,𝑛)=0⇒𝑚>𝑛. Therefore, we are interested in the pairs (𝑚,𝑛) such that 𝑚>𝑛. Thus, we have 𝑚>𝑛⟺𝑓𝑝>𝑔𝑞. (31) Since 𝑓 and 𝑔 are monotonous, by Lagrange’s Medium Value Theorem, we obtain 𝑝=𝑓⁻¹(𝑚)⟺𝑓^−1′(𝑚)=1𝑓^′𝑝𝑞=𝑔⁻¹(𝑛)⟺𝑔^−1′(𝑛)=1𝑔^′𝑞. (32)(33) Then, 𝑚>𝑛⟺⁽⁾³¹𝑓𝑝>𝑔𝑞⇒⁽ⁱⁱ⁾𝑔^′𝑞𝑓^′𝑝<𝑝³𝑞^{3()𝑒()3233}⟺𝑓^−1′(𝑚)𝑔^−1′(𝑛)<𝑓⁻¹(𝑚)³𝑔⁻¹(𝑛)³. (34) Therefore, 𝑚>𝑛⟹𝑓^−1′(𝑚)𝑔⁻¹(𝑛)³−𝑔^−1′(𝑛)𝑓⁻¹(𝑚)³>0. (35)
By differentiation of (30), we obtain 𝜉^′′(𝑛)=−2𝑔⁻¹(𝑛)𝑓⁻¹(𝑚)⁵×𝑓^−1′(𝑚)𝑔⁻¹(𝑛)³−𝑔^−1′(𝑛)𝑓⁻¹(𝑚)³ (36) and since −2𝑔⁻¹(𝑛)/[𝑓⁻¹(𝑚)]⁵>0, ∀𝑚,𝑛∈(0,1), from (35), we have 𝜉^′′(𝑛)>0,∀𝑛∈𝐷^𝑜_𝜉. (37) Now we take 𝐼=𝐷_𝜉∩𝐷_𝛿2=𝐷_𝜉∩[0,𝑟(𝑧_𝑜)), and we define the function 𝜙∶𝐼→[0,1] such that 𝜙(𝑛)=𝜉(𝑛)−𝛿₂(𝑛). (38) Then, from (26) and (37) we have 𝜙^′′(𝑛)>0, ∀𝑛∈𝐼^𝑜. Since 𝜉(0)=0 and by condition (iii) of Theorem 18, it follows that 𝛿₂(0)>0; then we have 𝜙(0)<0. Consequently, from Lemma 9, we have that there is only one 𝑛^∗∈𝐼 such that 𝜙𝑛^∗=0⟺⁽⁾³⁸𝜉𝑛^∗=𝛿₂𝑛^∗. (39) Since 𝜉=𝐻⁻¹(0), then we obtain 0=𝐻𝑛^∗,𝜉𝑛=^∗()39𝐻𝑛^∗,𝛿₂𝑛^∗. (40)
So, there exists only one 𝑥^∗∈(𝑧_𝑜,𝑐₂], 𝑛^∗=𝑟(𝑥^∗) and 𝑚^∗=𝛿₂(𝑛^∗)=𝑠(𝑥^∗) such that Δ𝑥^∗=𝐻𝑛^∗,𝑚^∗𝐴(𝑛^∗,𝑚^∗)=0. (41) This finally proves the theorem.

Figure 6 

Functions ℎ1 and ℎ2.

Figure 7 

Functions 𝜉, 𝛿1 and 𝛿2.

4.2. Case 2: 𝜇_𝐶(𝑡)<𝜇_𝐵(−𝑡)

Theorem 19 (uniqueness). Let 𝑆 be a p-fuzzy system and 𝐴^∗ an equilibrium viable set of 𝑆 of the type (𝐴_𝑖,𝐴_𝑖+1)→(𝐶,𝐵). If the functions 𝜇_𝐴𝑖, 𝜇_𝐴𝑖+1, 𝜇_𝐵, and 𝜇_𝐶 are continuously differentiable, 𝜇_𝐴𝑖 and 𝜇_𝐴𝑖+1 are piecewise monotonous, 𝜇_𝐵 and 𝜇_𝐶 are strictly monotonous, such that (i)𝜇_𝐶(𝑡)<𝜇_𝐵(−𝑡), ∀𝑡∈(0,𝑎),(ii)𝜇^′_𝐶(𝑞)/𝜇^′_𝐵(𝑝)>𝑝³/𝑞³, ∀𝑝∈supp(𝐵), 𝑞∈supp(𝐶) and 𝜇_𝐵(𝑝)<𝜇_𝐶(𝑞), (iii)[𝜇^′_𝐴𝑖(𝑥)/𝜇^′_𝐴𝑖+1(𝑥)]≤0, ∀𝑥∈(𝑐₁,𝑧_𝑜), 𝜇_𝐴𝑖(𝑥)≠𝜇_𝐴𝑖+1(𝑥). Then, 𝑆 has only one stationary point, 𝑥^∗ in 𝐴^∗, and 𝑥^∗∈[𝑐₁,𝑧_𝑜).

Proof. It is analogous to the proof of Theorem 18.

4.3. Some Comments about Uniqueness Theorems

When we do not have 𝜇_𝐶(𝑡)>𝜇_𝐵(−𝑡) or 𝜇_𝐶(𝑡)<𝜇_𝐵(−𝑡), it is impossible to establish general conditions for uniqueness of stationary points. For example, consider the p-fuzzy system in Figure 8. This system has an equilibrium viable set, 𝐴^∗=[𝐴₁∩𝐴₂]⁰.

Figure 8 

p-Fuzzy system: existence of more than one stationary point.

The sets that describe the input variable have membership functions 𝜇_𝐴1 and 𝜇_𝐴2: 𝜇_𝐴1(𝑥)=140𝑥,if0<𝑥≤40−1950𝑥+5𝜇,if40<𝑥≤900,otherwise,_𝐴2(𝑥)=1155𝑥−11,if5<𝑥≤60−1540𝑥+2,if60<𝑥≤1000,otherwise, (42) and the fuzzy sets that describe the output variable have membership functions: 𝜇_𝐶(𝑡)=(−1/2)𝑡+1 and 𝜇_𝐵(𝑡)=13𝑡+1,if−3<𝑡≤3𝑚₁𝜀−13𝑚₁−1𝑡+𝑚₁+𝜀,if3𝑚₁≤−1<𝑡3𝜀𝜀+𝑚₁−1𝑚₁−13𝑚₁−1+𝜀²1𝜀𝑡+1,if3𝜀𝜀+𝑚₁−1𝑚₁−13𝑚₁−1+𝜀²<𝑡≤00,otherwise. (43)

For example, if we take in (43) 𝑚₁=0.1 and 𝜀=0.1, the p-fuzzy system obtained has three stationary points in 𝐴^∗, which can be visualized in Figure 9, which depicts the graphic of function Δ.

Figure 9 

Function Δ with 𝑚1=𝜀=0.1.

Remark 20. Note that the function 𝜇_𝐵 is not differentiable in all points into supp(𝐵), which was a requirement made in the previous cases. However, it can be clearly constructed a function 𝜇_𝐵 derivable in all points of supp(𝐵). For example, if we substitute the second sentence of 𝜇_𝐵 by an adequate fourth degree polynomial, obviously 𝜇_𝐵 will be derivable into supp(𝐵).

Remark 21. If we take, for example, 𝜀=0.3 and 𝑚₁=0.3, we have that the obtained p-fuzzy system has only one stationary point (Figure 10). This shows that Theorems 18 and 19 establish only sufficient conditions for uniqueness of the stationary point.

Figure 10 

Function Δ with 𝜀=0.4, 𝑚1=0.1.

4.4. Uniqueness for Triangular and Trapezoidal Membership Functions

In this section, we will list some important consequences of Theorems 18 and 19. We will also show that for triangular and trapezoidal membership functions, the stationary point is only one in 𝐴^∗. However, before doing so, let us take a look at the following lemmas.

Lemma 22. If 𝜇_𝐵(𝑝)>𝜇_𝐶(𝑞) then 𝑞>−𝑝, where 𝑝=𝜇_𝐵⁻¹(𝑚) and 𝑞=𝜇_𝐶⁻¹(𝑛).

Proof. In fact, we have that 𝜇_𝐵(𝑝)>𝜇_𝐶(𝑞)⇒𝑚>𝑛 hence using Lemma 10 and the fact that −𝜇_𝐵⁻¹ isincreasing, once 𝜇_𝐵 is increasing, then we have that 𝑞=𝜇_𝐶⁻¹(𝑛)>−𝜇_𝐵⁻¹(𝑛)>−𝜇_𝐵⁻¹(𝑚)=−𝑝. (44)

Lemma 23. If 𝜇_𝐵(𝑝)<𝜇_𝐶(𝑞), then 𝑞<−𝑝, where 𝑝=𝜇_𝐵⁻¹(𝑚) and 𝑞=𝜇_𝐶⁻¹(𝑛).

Proof. Analogous to previous proof.

Corollary 24. Let 𝑆 be a p-fuzzy system and 𝐴^∗ an equilibrium viable set of 𝑆. If 𝜇_𝐴𝑖, 𝜇_𝐴𝑖+1, 𝜇_𝐵 and 𝜇_𝐶 are triangular fuzzy numbers, then 𝑆 has only one stationary point in 𝐴^∗.

Proof. We will prove the case where 𝑆 is (𝜇_𝐴𝑖,𝜇_𝐴𝑖+1)→(𝜇_𝐶,𝜇_𝐵). If 𝑆 is (𝜇_𝐴𝑖,𝜇_𝐴𝑖+1)→(𝜇_𝐵,𝜇_𝐶), then the proof is analogous.
If 𝑎=𝑏, then 𝜇_𝐴𝑖 and 𝜇_𝐴𝑖+1 are symmetrical and by Proposition 15 the stationary point is only one: 𝑥^∗=max_{𝑥∈𝐴∗}min𝜇_𝐴𝑖(𝑥),𝜇_𝐴𝑖+1(𝑥). (45)
Assume that 𝑎>𝑏; then 𝜇_𝐴𝑖, 𝜇_𝐴𝑖+1, 𝜇_𝐵, and 𝜇_𝐶 satisfy Theorem 18. In fact, (i) and (iii) are trivial. Since 𝜇_𝐵(𝑡)=−(1/𝑏)𝑡+1 and 𝜇_𝐶(𝑡)=−(1/𝑎)𝑡+1, then 𝜇_𝐵(𝑝)>𝜇_𝐶(𝑞)⇒𝑏/𝑎<𝑝/𝑞⇒𝜇^′_𝐶(𝑞)/𝜇^′_𝐵(𝑝)<𝑝/𝑞. From Lemma 22, we have that 𝑞>−𝑝⇒𝑝/𝑞<𝑝³/𝑞³ and, therefore, we obtain 𝜇^′_𝐶(𝑞)/𝜇^′_𝐵(𝑝)<𝑝³/𝑞³, which satisfies (ii).
Now, assume that 𝑎<𝑏; then 𝜇_𝐴𝑖, 𝜇_𝐴𝑖+1, 𝜇_𝐵, and 𝜇_𝐶 satisfy the Theorem 19. In fact, (i) and (iii) are trivial and 𝜇_𝐵(𝑝)<𝜇_𝐶(𝑞)⇒𝑏/𝑎>𝑝/𝑞⇒𝜇^′_𝐶(𝑞)/𝜇^′_𝐵(𝑝)>𝑝/𝑞. From Lemma 23 we get 𝑞<−𝑝⇒𝑝/𝑞>𝑝³/𝑞³, and, therefore, 𝜇^′_𝐶(𝑞)/𝜇^′_𝐵(𝑝)>𝑝³/𝑞³, which satisfies (ii). This concludes the proof.

Corollary 25. Let 𝑆 be a p-fuzzy system and 𝐴^∗ an equilibrium viable set of 𝑆. If 𝜇_𝐴𝑖 and 𝜇_𝐴𝑖+1 are trapezoidal fuzzy numbers and 𝜇_𝐵 and 𝜇_𝐶 are triangular fuzzy numbers, then 𝑆 has only one stationary point in 𝐴^∗.

Proof. It is analogous to the proof of Corollary 24.

5. Examples

In this section we will present some computational experiments that confirm the mathematical theory presented in the previous sections. The experiments had been carried out with Matlab software. For the experiments we will consider inhibited one-dimensional p-fuzzy systems. These systems can be used to model situations where the state variable is increasing (resp., decreasing) with a carrying capacity (resp. lower bound). These situations, in population dynamics, are described by inhibited models such as Gompertz’s model, Verhulst’s model, von Bertallanffy’s models, and Asymptotic Exponential model.

The inhibited one-dimensional p-fuzzy systems are composed of the variables “Population” (Figure 11) and “Variation” (Figure 12). The rule-base of these systems is(1)if Population is low (𝐴₁) then Variation is low positive (𝐶); (2)if Population is medium low (𝐴₂) then Variation is medium positive (𝐷); (3)if Population is medium (𝐴₃) then Variation is high positive (𝐸); (4)if Population is medium high (𝐴₄) then Variation is medium positive (𝐷); (5)if Population is high (𝐴₅) then Variation is low positive (𝐶); (6)if Population is the highest (𝐴₆) then Variation is low negative (𝐵).

Figure 11 

Population (𝑥).

Figure 12 

Variation (Δ): 𝜇𝐶(𝑡)=𝜇𝐵(−𝑡).

5.1. Example 1

In this system the membership functions of 𝐵 and 𝐶 are 𝜇_𝐵(𝑡)=𝑡+1 and 𝜇_𝐶(𝑡)=1−𝑡. These functions are symmetric (Figure 12). Observing the rules we can identify an equilibrium viable set, 𝐴^∗=[200,280], where 𝐴^∗=𝐴₅∩𝐴₆, in which membership functions are given by size: 𝜇_𝐴5(𝑥)=1()50𝑥−150,if150<𝑥≤2001,if200<𝑥≤220−1()𝜇60𝑥−280,if220<𝑥≤2700,otherwise,_𝐴6(𝑥)=1()70𝑥−200,if200<𝑥≤2701,if270<𝑥≤3000,otherwise. (46)

A simple calculation shows that 𝜇_𝐴5∩𝜇_𝐴6=243.07, which is the stationary point of the system, as shown in Proposition 15. This is the same result as that obtained from numerical experiments in Figure 13, where it is possible to observe the solution of the p-fuzzy system with initial condition 𝑥_𝑜=50 converging to the stationary point 𝑥^∗=243.07 (dotted line curve).

Figure 13 

Equilibrium of the p-fuzzy systems.

5.2. Example 2 (Application)

Losses caused by fungi attacks reach up to the amazing and worrisome figure of 20% of the total harvested fruits. Among the recognized pathogens of these degenerative processes are the Colletotrichum sp. (bitter putrefaction); Alternaria sp and Fusarium sp (carpelar putrefaction); Alternaria alternata (black putrefaction) and Penicillium sp. (blue mould). In general, putrefaction occurs always after harvesting, when the apples are kept in boxes awaiting the refrigeration process.

Let us consider a situation where an apple box filled with approximately 3000 fruits exists and there is a rotten apple in the center of the box that will contaminate the other apples. The dispersion of the disease occurs through the contact of the rotten apple with a healthy one. If we want to model the dispersion of the disease, we can only use the intuition because the available information we have is that after 𝑛 days all the apples will be contaminated. We do not have in hand a “force of infection of the disease” parameter, and we know little about the possible contacts between the rotten apple and a healthy one. Thus, any mathematical model or simulation of the phenomenon only will produce an approximate result. On the other hand, if we simply use the intuition, we can formulate some relative rules to the dispersion process such as the following:

“If the rotten apple population is “low” then the variation (incidence) of rotten apples is “small”.”

Let us consider then the p-fuzzy system with the following linguistic variables: (i)𝐴: “rotten apple population” (Figure 14); L: low; Ml: medium low; M: medium; medium high; H: high, (ii)Δ “Variation” (incidence of the disease, Figure 15); S: small; M: medium; G: great, where the rule-base is: (1)if Population (𝐴) is low (L), then Variation is small (S); (2)if Population (𝐴) is medium low (L), then Variation is medium (M); (3)if Population (𝐴) is medium (M), then Variation is great (G); (4)if Population (𝐴) is medium high (M), then Variation is medium (M); (5)if Population (𝐴) is medium (G), then Variation is small (S).

Figure 14 

Population (𝐴).

Figure 15 

Variation (Δ).

This p-fuzzy system can be used to adjust the parameters of a deterministic differential equation model of the type 𝑑𝐴𝐴𝑑𝑡=𝑟𝐴1−𝐾, (47) for example, by the method of least squares. In Figure 16 it is possible to observe the solution of the p-fuzzy system (dotted line curve) and the adjust model (continuous line curve).

Figure 16 

Solutions: p-fuzzy system and adjust model.

Acknowledgment

The authors would like to thank CAPES for the financial support.