Qualitative analysis of second-order fuzzy difference equation with quadratic term

In this paper, we explore the qualitative features of a second-order fuzzy difference equation with quadratic term xn+1=A+Bxnxn-12,n=0,1,….\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x_{n+1}=A+\frac{Bx_{n}}{x_{n-1}^2},\ \ n=0,1,\ldots . \end{aligned}$$\end{document}Here the parameters A,B∈ℜF+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A, B\in \Re _F^+$$\end{document} and the initial values x0,x-1∈ℜF+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0, x_{-1}\in \Re _F^+$$\end{document}. Utilizing a generalization of division (g-division) of fuzzy numbers, we obtain some sufficient condition on the qualitative features including boundedness, persistence, and convergence of positive fuzzy solution of the model, Moreover two simulation examples are presented to verify our theoretical analysis.


Introduction
Difference equation is one of the most important dynamical model. It is the analogue of corresponding differential equation and delay differential equation having an extensive applications in computer science, control engineering, chemistry, biology, economics, etc. (see [1][2][3][4][5][6][7][8][9][10][11]). Recently, many authors are interested in studying qualitative features of rational difference equation. For example, Bešo et al. [12] proposed a second-order rational difference equation with quadratic term where γ > 0, δ > 0, and the initial values x 0 > 0, x −1 > 0. In 2017, Khyat et al. [13] explored a similar model with quadratic term in both the numerator and denominator Furthermore, they obtained the globally asymptotically stability of (2) and the direction of the Neimark-Sacker bifurcation.
In the last few decades, there are many publications on the stability, oscillatory, periodicity, and boundedness of nonlinear rational difference equations. Moreover a lot of similar qualitative features also appear in nonlinear rational difference equation systems (see [14][15][16][17]).
Although these models are very simple in their forms, we can not understand fully the qualitative features of their solutions. In fact, these models inevitably implicit inherent uncertainty or vague. It is well known that fuzzy set is a powerful tool to cope with these uncertainties or subjective information in mathematical model. Therefore, it is a natural method to explore dynamical model with uncertainty or impression by establishing fuzzy difference equation (FDE) or fuzzy differential equation.
FDE is a special kind of difference equation whose coefficients and the initial condition are fuzzy numbers, and its' solution is a sequence of fuzzy number. Due to the advantage of FDE in dealing with inherent imprecision, the study on qualitative features of these models has become an important research topic both from theoretical viewpoint and in applications. Therefore, in the last decades, there has been an increasing results on the study of FDE (see [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]). It is found that fuzzy set theory has potential in the application of fuzzy differential equations and fuzzy time series (see [34][35][36][37][38][39][40]).
Inspired by previous works, in this paper, utilizing a generalization of division (g-division) of fuzzy numbers, we explore the qualitative features of positive fuzzy solution to the following FDE with quadratic term where the initial condition x −1 , x 0 , and the parameters A, B are positive fuzzy numbers. The organization of this paper is arranged as follows. Section 2 gives some basic concepts of fuzzy numbers used throughout the paper. In Sect. 3, the qualitative features of the positive fuzzy solution to FDE (3) are obtained by virtue of g-division of fuzzy numbers. Section 4 presents two illustrative examples to verify our theoretic results. A general conclusion and discussion are drawn in Sect. 5.

Preliminary and definitions
In this section, we first review some basic concepts and definitions to be used in the sequel. These particular descriptions are found in many publications [20][21][22].
A function defined as U : R → [0, 1] is called a fuzzy number if it is normal, fuzzy convex, upper semi-continuous, and compactly support on R.
For α ∈ (0, 1], we denote the α−cuts of U by [U ] α = {x ∈ R : U (x) ≥ α}, and for α = 0, the support of U is written as suppU It is easy to see that the [U ] α is a closed interval. Provided that suppU ⊂ (0, ∞), then U is said to be a positive fuzzy number. Particularly, if U is a positive real number, then it is said to be a trivial fuzzy number, i.e., Addition and multiplication of fuzzy numbers are defined as follows.
The family of fuzzy numbers with addition and multiplication defined by Eqs. (4) and (5) is written as F . Particularly, the family of positive (resp. negative) fuzzy numbers is denoted by + F ( resp. − F ).

Definition 2.1
Let U , V ∈ F , the metric is defined as follows.
Obviously, ( F , D) is a complete metric space.
if W is a proper fuzzy number.
That is, (x n , y n ) = (x, y) for n ≥ 0 is the solution of (9), or equivalently, (x, y) is a fixed point of the vector map ( f , g). Definition 2.6 [8] Let (x, y) be an equilibrium point of (9). (i) The equilibrium point (x, y) is called locally stable if for every ε > 0, there exists δ > 0 such that for all The equilibrium (x, y) of (9) is called locally asymptotically stable if it is locally stable, and if there exists γ > 0, such that for all ( (iv) The equilibrium (x, y) of (9) is called globally asymptotically stable if it is locally stable and a global attractor.
(v) The equilibrium (x, y) of (9) is called unstable if it is not stable.

Existence of positive fuzzy solution
In this section, we first discuss the existence of the positive fuzzy solution of FDE (3). Proof The proof of Theorem is similar to those of Proposition 2.1 [19]. Assume that (x n ) is a sequence of fuzzy numbers and satisfies FDE (3) with initial values Applying Lemma 2.1, it follows from (3) and (10) that Utilizing g-division of fuzzy numbers, one of the following two cases occurs.
Secondly, we will prove that suppx Hence from (17) and (18), one can get, for n = 1, From which, it follows that And since L n,α , R n,α are left continuous, from (16) and (21), we get that [L n,α , R n,α ] makes certain a positive sequence x n satisfying (14). Now we show that, for the initial conditions x i (i = 0, −1), x n is the solution of (3). Since, ∀α ∈ (0, 1], Therefore, x n is the solution of FDE (3) with initial conditions x i , i = 0, −1.
Suppose that, for the initial values x i ∈ + F , i = 0, −1, x n is another solution of (3). Then, deducing as above, it has, for n ∈ N + Then, (14) and (22) If Case (ii) occurs, The proof is similar to the proof above. This completes the proof of Theorem 3.1.

Dynamics of FDE (3)
In order to obtain results on qualitative features of the positive solutions, Case (i) and Case (ii) are considered respectively.
If Case (i) occurs, the following lemma is required.

Lemma 3.1 Consider the following difference equations
where p 2 > c > 0, y i ∈ (0, +∞), i = 0, −1. Then Proof It is clear that, for n ≥ 1, y n > p, z n > q from (23). Moreover, for n ≥ 4, By induction, it can get that, for n − k ≥ 3 Noting n − k ≥ 3 is equal to k ≤ n − 3. The proposition is true.
Then every positive fuzzy solution x n of FDE (3) is bounded and persists.

Lemma 3.2 Consider the following difference equation
Assume p 2 > 4c 3 . Then the equilibrium y of (31) is locally asymptotically stable.
Thus 1 = λ 1 . Namely lim n→∞ y n exists. Since y of (23) is the unique positive equilibrium, then lim n→∞ y n = y. Combining two lemmas above, we have the following theorem Proof Suppose that there is a fuzzy number x satisfying in which L α , R α ≥ 0. Then, from (33), one gets Hence we have from (34) that Let x n be a positive fuzzy solution of FDE (3) satisfying (11). Since A 2 l,α > 4 3 B l,α , A 2 r ,α > 4 3 B r ,α , α ∈ (0, 1]. Utilizing Lemma 3.3 to system (13), then Then, from (35) The following lemmas are required.
Lemma 3.4 Consider the following difference equations system.

Lemma 3.5 Consider Eq. (37), if condition
Proof From (37), we obtain a positive equilibrium (y, z) = The linearized equation of (37) at the equilibrium (y, z) is here n = (y n , y n−1 , z n , z n−1 ) T , Let λ i , i = 1, 2, 3, 4 be the eigenvalues of matrix B, L = diag(l 1 , l 2 , l 3 , l 4 ) be a diagonal matrix, Clearly, L is an invertible matrix. Calculating L B L −1 , one has Furthermore, noting (45), we have It is clear that B has the same eigenvalues as L B L −1 , and Therefore the equilibrium (y, z) of (37) is locally asymptotically stable.
We claim that Suppose contrarily that H 1 > h 1 , then from the first inequality of (47), it can conclude h 2 > H 2 , which is a contradiction. So H 1 = h 1 . Similarly we can get H 2 = h 2 . Noting (37) and (48), then lim n→∞ y n = y, lim n→∞ z n = z. The proof of Lemma 3.6 is completed. Combining Lemma 3.5 with Lemma 3.6. We have the following theorem.

Proof
The proof is similar to those of Theorem 3.4. Assume that there is a positive fuzzy number x satisfying (33). From (33), condition (49) and (50), we can get Hence we can from (51) have that Suppose that x n is a positive fuzzy solution of FDE (3) such that (10) holds. Noting (50) and (51), utilizing Lemma 3.5 and Lemma 3.6 to (36), we have The proof of the theorem is completed.
Remark 3. 1 In fuzzy discrete dynamical systems. To find qualitative behavior of solutions for discrete fuzzy difference equation, it is very vital to utilize which operations such as addition, scalar multiplication, division of fuzzy numbers. In [19-22, 25, 26, 28-31], Using Zadeh extension principle, the authors obtained dynamical behaviors of some fuzzy difference equations. However, utilizing g-division of fuzzy numbers, Zhang et al. [23,24] studied the dynamical behaviors of some nonlinear fuzzy difference equations. Compared with the former, The advantage is that the support sets of positive fuzzy solution of latter is smaller than those of former. In fact, it is obvious that the degree of fuzzy uncertainty is reduced by virtue of g-division of fuzzy numbers. Based on this fact above, Therefore, we consider the qualitative behaviors of the fuzzy difference equation with quadratic term by virtue of g-division of fuzzy numbers.

Numerical examples
In this section, two illustrative examples are presented to verify the effectiveness of theoretic results.

Example 4.1
Consider the following fuzzy difference equation where A, B ∈ + F and the initial conditions x i ∈ + F , i = 0, −1 are as follows From (55), one has From (56), one has     From (61), one has From (62), one has  It is clear that the initial value x i ∈ + F (i = 0, −1), and (47) is satisfied, so applying Theorem 3.6, there is a unique positive equilibrium x = (6.3879, 7.2749, 7.7348). Data Availability Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Conflict of interest
The authors declare that they have no conflict of interest.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.