# Dynamic Behaviors of a Class of High-Order Fuzzy Difference Equations

Authors: Lili Jia
Journal: Journal of Mathematics
 (2020)

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## Abstract

The purpose of this paper is to give the conditions for the existence and uniqueness of positive solutions and the asymptotic stability of equilibrium points for the following high-order fuzzy difference equation:xn+1=Axn−1xn−2/B+∑i=3kCixn−in=0,1,2,…, where xn is the sequence of positive fuzzy numbers and the parameters A,B,C3,C4,…,Ck and initial conditions x0,x−1,x−2,x−ii=3,4,…,k are positive fuzzy numbers. Besides, some numerical examples describing the fuzzy difference equation are given to illustrate the theoretical results.


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## Body

## 1. Introduction

It is well known that difference equations are one of the most widely used equations in various subject areas. In particular, many phenomena in the real world often rely on the establishment and solution of mathematical models describing them. In the last twenty five years of the twentieth century, the study on the difference equation developed rapidly and occupied important positions in many applied analyses, such as demographic analysis, finance, and biology (see, e.g., [1–5] and the references therein). However, as the complexity of the modeling systems is increasing, the uncertainty factors in the model increases dramatically. Obviously, the fact that the ordinary difference equation model is used to describe many practical problems is incomplete. In view of this, a fuzzy difference system is a powerful tool to solve the mathematical model with fuzzy parameters and initial values in the real world, and it is an interesting and far-reaching research for modeling uncertainty and for processing vague or subjective information in a mathematical model.In 1998, DeVault et al. [6] discussed the periodic character of solutions of the following nonlinear difference equation:(1)xn+1=−1xn+Axn−1,n=0,1,…,where A>0.Papaschinopoulos and Schinas [7] researched the following difference equations:(2)xn+1=A+ynxn−p,yn+1=A+xnyn−q,n=0,1,…,where A,x−p,x−p+1,…,x0,y−q,y−q+1,…,y0 are the positive numbers and pandq are the positive integers.In 2014, He et al. [8] studied the periodicity of the positive solutions for the following fuzzy max-difference equation:(3)xn+1=maxAnxn−m,xn−k,n=0,1,…,where k,m∈N∗,An is a periodic sequence of fuzzy numbers, and initial values x0,x−1,…,x−d are the positive fuzzy numbers with d=k,m.Zhang et al. [9] investigated the boundedness, persistence, and asymptotic behavior of a positive fuzzy solution of the following third-order fuzzy difference equation using a generalization of division for fuzzy numbers:(4)xn+1=A+xn−1xn−1xn−2,n=0,1,2,…,where A,x0,x−1,x−2 are the positive fuzzy numbers.These bright spots in the above papers deserve our learning. Moreover, in 2017, Khastan [10] discussed the following fuzzy difference equation:(5)xn+1−q=wxn,n=0,1,…,where xn is a sequence of the positive fuzzy numbers and x0,q,andw are the positive fuzzy numbers. In this paper, the generalization of division for the fuzzy number is used to investigate the existence, uniqueness, and global behavior of the solution.Wang and Wang [11] analyzed the following nonlinear difference equation:(6)xn+1=axn−lxn−kbxn−s+cxn−t,n=0,1,…,where the initial conditions x0,x1,…,x−r are the positive real numbers, l,k,s,t are the nonnegative integers, r=maxl,k,s,t, and a,b,c are the positive constants. Further, in 2017, the author in [12] investigated the asymptotic behavior of the equilibrium points for the following fuzzy difference equation:(7)xn+1=Axn−lxn−2D+Bxn−3+Cxn−4,n=0,1,2,…,where xn is a sequence of the positive fuzzy numbers and the parameters A,B,C,D and initial conditions x−4,x−3,x−2,x−1,x0 are the positive fuzzy numbers. Besides, some interesting results can be found in [13–18] and the references therein.Based on the above valuable theoretical results, this paper studies the following high-order fuzzy difference equation:(8)xn+1=Axn−1xn−2B+∑i=3kCixn−i,n=0,1,2,…,where xn is a sequence of the positive fuzzy numbers and the parameters A,B,C3,C4,…,Ck and initial conditions x0,x−1,x−2,x−ii=3,4,…,k are the positive fuzzy numbers. The purpose of this paper is to study the asymptotic behavior of the equilibrium point of the fuzzy difference equation. The main method is to convert the fuzzy difference equation into a rational difference equation according to the fuzzy number theory, and then the properties of the solutions of the fuzzy difference equations are obtained by studying the corresponding constant difference equations. In addition, the theoretical results are verified by numerical examples.Remark 1.
It is well known that difference equations look simple in form, but the properties of their solutions are very complex, especially the dependence on parameters and initial values. The main contribution and innovation of this paper are as follows: (1) based on the practical application, fuzzy parameters and initial values are introduced to the known models, and the new model can better describe the objective natural phenomenon. Obviously, the model in reference [12] is a special case of the model in this paper. (2) To study the new model, some new analytical methods and techniques that is different from those mentioned in the references are obtained. (3) In this paper, the research contents are more rich than the related references. Firstly, the existence and uniqueness of positive fuzzy solutions are proved. Secondly, the nonzero equilibrium points of the corresponding ordinary difference equations which are unstable are obtained by using the linearization method. Finally, it was found that the zero trivial solution of the fuzzy difference equation (8) is asymptotically stable when the parameters of the system are positive trivial fuzzy numbers. (4) The sufficient conditions obtained herein are new, general, and easily verifiable, which provide flexibility for the application and analysis of the high-order fuzzy difference equation.


## 2. Preliminaries and Notations

For the convenience of readers, the definitions and preliminary results related to the theoretical proof of the paper are given, see [19–23].Definition 1.
For a setB, B¯ is denoted as the closure of B and a function A:R⟶0,1 is a fuzzy number if the following conclusions are true:(i)
A is normal, i.e., there exists x∈R such that Ax=1;(ii)
A is a fuzzy convex set, i.e.,(9)Atx1+1−tx2≥minAx1,Ax2,∀t∈0,1,x1,x2∈R;(iii)
A is upper semicontinuous on R;(iv)
The support ofA, i.e., supp A=∪α∈0,1Aα¯=x∈R:Ax>0¯, is compact,where the α−cuts of A are closed intervals, defined as Aα=x∈R:Ax≥α, and if supp A⊂0,∞, then the fuzzy number A is obviously positive.Definition 2.
LetAandB be the fuzzy numbers which satisfy Aα=Al,α,Ar,α and Bα=Blα,Brα,α∈0,1, then the following metric is denoted:(10)DA,B=supmaxAl,α−Bl,α,Ar,α−Br,α,where sup is taken for all α∈0,1. Then, Rf,D is a complete metric space. For the convenience of application in the future, 0^∈Rf is defined as(11)0^x=1,x=0,0,x≠0.
Thus,0^α=0,0,0<α≤1.Definition 3.
Persistence (resp., boundedness) of fuzzy numbers is defined if there exists a positive real numberMresp.N, such that the following conclusions are true: suppxn⊂M,∞resp.suppxn⊂0,N,n=1,2,…, where xn is a sequence of the positive fuzzy numbers.
Further,xn is bounded and persistent if there exist positive real numbers M,N, such that suppxn⊂M,N,n=1,2,….Lemma 1.
LetIx,Iy be some intervals of real numbers and let f:Ixk+1×Iyl+1⟶Ix, g:Ixk+1×Iyl+1⟶Iy be continuously differentiable functions. Then, for every set of initial conditions xi,yj∈Ix×Iy, i=−k,−k+1,…,0,j=−l,−l+1,…,0, the following system of difference equations:(12)xn+1=fxn,xn−1,…,xn−k,yn,yn−1,…,yn−l,yn+1=gxn,xn−1,…,xn−k,yn,yn−1,…,yn−l,n=0,1,2,…,has a unique solution xi,yji=−k,j=−l+∞,+∞.Definition 4.
A pointx¯,y¯∈Ix×Iy is called an equilibrium point of system (12) if x¯=fx¯,x¯,…,x¯,y¯,y¯,…,y¯,y¯=gx¯,x¯,…,x¯,y¯,y¯,…,y¯, that is, xn,yn=x¯,y¯ for n≥0 is the solution of difference system (12), or equivalently, x¯,y¯ is a fixed point of the vector map f,g.Definition 5.
Suppose thatx¯,y¯ is an equilibrium point of system (12), then the following is obtained:(i)
x¯,y¯ is called locally stable if for every ε>0, there exists δ>0 such that for any initial conditions xi,yi∈Ix×Iyi=−k,…,0,j=−l,…,0 with ∑i=−k0xi−x¯<δ,∑j=−l0yj−y¯<δ, there is s for any n>0;(ii)
x¯,y¯ is called an attractor if limn⟶∞xn=x¯,limn⟶∞yn=y¯ for any initial conditions xi,yi∈Ix×Iyi=−k,…,0,j=−l,…,0;(iii)
x¯,y¯ is called asymptotically stable if it is stable and an attractor;(iv)
x¯,y¯ is called unstable if it is not locally stable.Definition 6.
Letx¯,y¯ be an equilibrium point of the vector map F=f,xn,…,xn−k,g,yn,…,yn−l, where f and g are continuously differential functions at x¯,y¯. The linearized system of (12) about the equilibrium point x¯,y¯ is Xn+1=FXn=Fj⋅Xn, where FJ is the Jacobian matrix of system (12) about x¯,y¯ and Xn=xn,…,xn−k,yn,…,yn−lT.Definition 7.
Letp,q,s,t be four nonnegative integers such that p+q=n and s+t=m. Split x=x1,x2,…,xn into x=xp,xq and y=y1,y2,…,ym into y=ys,yt, where xσ denotes a vector with σ-components of x. The function fx1,x2,…,xn,y1,y2,…,ym possesses a mixed monotone property in subsets Ixn×Iym of Rn×Rm if fxp,xq,ys,yt is monotone nondecreasing in each component of xq,yt for x,y∈Ixn×Iym. In particular, if q=0,t=0, then it is said to be monotone nondecreasing in Ixn×Iym.Lemma 2.
Assume thatXn+1=FXn,n=0,1,…, is a system of difference equations and X¯ is the equilibrium point of this system, i.e., FX¯=X¯, then the following is obtained:(i)
If all eigenvalues of the Jacobian matrixJF about X¯ lie inside the open unit disk λ<1, then X¯ is locally asymptotically stable;(ii)
If one of the eigenvalues of the JacobianJF matrix about X¯ has a norm greater than one, then X¯ is unstable.Lemma 3.
Assume thatXn+1=FXn,n=0,1,… is a system of difference equations and X¯ is the equilibrium point of this system, and the characteristic polynomial of this system about the equilibrium point X¯ is Pλ=a0λn+a1λn−1+⋯+an−1λ+an=0, with the real coefficients and a0>0. Then, all roots of the polynomial Pλ lie inside the open unit disk λ<1 if and only if(13)Δk>0,fork=1,2,…,n,where Δk is the principal minor of order k of the n×n matrix:(14)Δn=a1a3a5⋯0a0a2a4⋯00a1a3⋯0⋮⋮⋮⋱⋮000⋯an.


## 3. Main Results and Their Proofs

The following lemmas are applied to study the existence and uniqueness of a positive solution of the fuzzy difference equation (8).Lemma 4 (see [24]).
Letf be a continuous function from R+×R+×R+ into R+ and A,B,C be the fuzzy numbers, then(15)fA,B,Cα=fAα,Bα,Cα,α∈0,1.Lemma 5 (see [19, 25]).
Let us denoteu∈Rf such that uα=ul,α,ur,α,α∈0,1, then ul,α and ur,α can be seen as functions on α∈0,1 which satisfy the following conclusions:(i)
ul,α is nondecreasing and left continuous;(ii)
ur,α is nonincreasing and left continuous;(iii)
ul,α≤ur,α.Conversely, for any functionsaα and bα defined on 0,1 which satisfy the above (i)–(iii), there exists a unique u∈Rf such that uα=aα,bα for any α∈0,1.Theorem 1.
Consider equation (8), where A,B,Cii=3,4,…,k are the positive fuzzy numbers, and then for any positive fuzzy numbers x0,x−1,x−2,x−ii=3,4,…,k, there exists a unique positive solution xn of (8) with the initial conditions x0,x−1,x−2,x−ii=3,4,…,k.Proof.
Assume that there exists a sequencexn that is the positive solution of (8) with the positive parameters and initial conditions A,B,Ci,x0,x−1,x−2,x−ii=3,4,…,k. Consider the α−cuts,α∈0,1,(16)Aα=Al,α,Ar,α,Bα=Bl,α,Br,α,Ciα=Cil,α,Cir,α,i=3,4,…,k,xnα=Ln,α,Rn,α,n=−k,−k+1,…,0.
From (8) and (16) and Lemma 4, the following is obtained:(17)xn+1α=Ln+1,α,Rn+1,α=Axn−1xn−2B+∑i=3kCixn−iα=Axn−1xn−2αB+∑i=3kCixn−iα=Al,α,Ar,αLn−1,α,Rn−1,αLn−2,α,Rn−2,αBl,α,Br,α+∑i=3kCil,α,∑i=3kCir,α∑i=3kLn−i,α,∑i=3kRn−i,α=Al,αLn−1,αLn−2,αBr,α+∑i=3kCir,αRn−i,α,Ar,αRn−1,αRn−2,αBl,α+∑i=3kCil,αLn−i,α.
Hence, forα∈0,1,n=−k,−k+1,…,0, according to the above result, it follows that(18)Ln+1,α=Al,αLn−1,αLn−2,αBr,α+∑i=3kCir,αRn−i,α,Rn+1,α=Ar,αRn−1,αRn−2,αBl,α+∑i=3kCil,αLn−i,α.
And then, for any initial conditionsLj,α,Rj,α,j=−k,−k+1,…,0,α∈0,1, it is evident from Lemma 1 that there exists a unique solution Ln,α,Rn,α of the ordinary difference equations (18).
Conversely, it is proved that the positive solutionxn of equation (8) is determined by Ln,α,Rn,α with initial conditions Lj,α,Rj,α,j=−k,−k+1,…,0,α∈0,1 and satisfies the following condition:(19)xnα=Ln,α,Rn,α,α∈0,1,n=−k,−k+1,….
For anyα1,α2∈0,1,α1<α2, according to Lemma 5, the following is obtained:(20)0<Al,α1≤Al,α2≤Ar,α2≤Ar,α1,0<Cil,α1≤Cil,α2≤Cir,α2≤Cir,α1,i=3,4,…,k,0<Bl,α1≤Bl,α2≤Br,α2≤Br,α1,0<Lj,α1≤Lj,α2≤Rj,α2≤Rj,α1,j=−k,−k+1,…,0,where A,B,Ci,xji=3,4,…,k,j=−k,−k+1,…,0 are the positive fuzzy numbers.
Next, from the mathematical induction, it is proved that the following conclusion is true, that is,(21)0<Ln,α1≤Ln,α2≤Rn,α2≤Rn,α1,n=1,2,….
According to (20), (21) is true for n=−k,−k+1,…,0. Suppose that (21) is true for any n≤m,m∈1,2,…, then from (18)–(21), it follows that for n=m+1,(22)Lm+1,α1=Al,α1Lm−1,α1Lm−2,α1Br,α1+∑i=3kCir,α1Rm−i,α1≤Al,α2Lm−1,α2Lm−2,α2Br,α2+∑i=3kCir,α2Rm−i,α2=L,m+1,α2L≤m+1,α2Ar,α2Rm−1,α2Rm−2,α2Bl,α2+∑i=3kCil,α2Lm−i,α2=Rm+1,α2≤Ar,α1Rm−1,α1Rm−2,α1Bl,α1+∑i=3kCil,α1Lm−i,α1=Rm+1,α1.
Hence, (21) is true.
Furthermore, from (18), it follows that(23)L1,α=Al,αL−1,αL−2,αBr,α+∑i=3kCir,αR−i,α,R1,α=Ar,αR−1,αR−2,αBl,α+∑i=3kCil,αL−i,α,α∈0,1.
Then, in view ofA,B,Ci,xji=3,4,…,k,j=−k,−k+1,…,0 which are the positive fuzzy numbers, it is known thatAl,α,Ar,α,Bl,α,Br,α,Cil,α,Cir,α,L−k,α,R−k,α,L−k+1,α,R−k+1,α,…,L0,α,R0,α,i=3,4,…,k are left continuous from Lemma 5. So, L1,αandR1,α are also left continuous from (23). Moreover, Ln,α,Rn,α,n=1,2,… are left continuous using mathematical induction.
Now, it is proven thatxn is supported, i.e., supp xn=∪α∈0,1Ln,α,Rn,α¯ is compact. It is easy to know that it is enough to prove that ∪α∈0,1Ln,α,Rn,α is bounded.
Letn=1, from A,B,Ci,xji=3,4,…,k,j=−k,−k+1,…,0 which are the positive fuzzy numbers, then there exist constants Mi,Ni>0,i=3,4,…,k, such that for all α∈0,1,(24)Al,α,Ar,α⊂M1,N1,Bl,α,Br,α⊂M2,N2,Cil,α,Cir,α⊂M1,N1,i=3,4,…,k,Lj,α,Rj,α⊂M1,N1,j=−k,−k+1,…,0.
Therefore, relations (23) and (24) imply that(25)L1,α,R1,α⊂M1M−1M−2N2+∑i=3k∑j=−3−kNiNj,N1N−1N−2M2+∑i=3k∑j=−3−kMiMj,α∈0,1,from which the following is obtained:(26)∪α∈0,1L1,α,R1,α⊂M1M−1M−2N2+∑i=3k∑j=−3−kNiNj,N1N−1N−2M2+∑i=3k∑j=−3−kMiMj,α∈0,1.
From (26), ∪α∈0,1L1,α,R1,α¯ is compact and ∪α∈0,1L1,α,R1,α¯⊂0,∞. Moreover, under the mathematical induction method, it is also proven that ∪α∈0,1L1,α,R1,α¯ is compact and(27)∪α∈0,1L1,α,R1,α¯⊂0,∞,n=1,2,….
Since (21) and (24) and Ln,α,Rn,α are left continuous, from Lemma 5, a sequence of positive fuzzy numbers xn is determined Ln,α,Rn,α such that (8) holds.
Furthermore, it is proven thatxn is the solution of (6) with initial conditions x0,x−1,x−2,x−ii=3,4,…,k. Since for all α∈0,1,(28)xn+1α=Ln+1,α,Rn+1,α=Al,αLn−1,αLn−2,αBr,α+∑i=3kCir,αRn−i,α,Ar,αRn−1,αRn−2,αBl,α+∑i=3kCir,αLn−i,α=Axn−1xn−2B+∑i=3kCixn−iα,it is claimed that xn is the solution of (6) with initial conditions x0,x−1,x−2,x−i,i=3,4,…,k.
Now, its uniqueness by contradiction is proven in the following. Assume that there exists another positive solutionxn∗ of (8) with initial conditions x0,x−1,x−2,x−ii=3,4,…,k, and according to a similar discussion above, it holds that(29)xn∗α=Ln,α,Rn,α,α∈0,1,n=0,1,…,and then, from (19) and (29), one has that(30)xnα=xn∗α,α∈0,1,n=−k,−k+1,…,and from the above result, it is obtained that the sequence xn is a unique solution of (6) with initial conditions x0,x−1,x−2,x−ii=3,4,…,k, so the proof is completed.
In the following result, the asymptotic behavior of the equilibrium point of (6) is further investigated. From the above proof process, it is known that if xn is the unique positive solution of (6) with the initial values x0,x−1,x−2,x−ii=3,4,…,k, then the following is obtained:(31)xnα=Ln,α,Rn,α,α∈0,1,n=0,1,….
Hence, it is known thatLn,α,Rn,α satisfies the family of system (18). The following corresponding ordinary parametric systems to study the asymptotic behavior of equation (8) are constructed:(32)yn+1=a1yn−1yn−2b1+∑i=3kc1izn−i,zn+1=a2zn−1zn−2b2+∑i=3kc2iyn−i,n=0,1,…,where the parameters a1,a2,b1,b2,c1i,c2ii=3,4,…,k and initial conditions y0,y−1,y−2,y−i,z0,z−1,z−2,z−ii=3,4,…,k are the positive real constants. Clearly, system (32) has a unique solution yn,zn for any initial conditions from Lemma 1.
Now, it is easily obtained that system (32) has the following three equilibrium points:(33)X¯1=y¯1,z¯1=0,0,X¯2=y¯2,z¯2=0,b2a2,X¯3=y¯3,z¯3=b1a1,0,if a1a2>∑i=3kc1i∑i=3kc2i, and system (32) has the fourth positive equilibrium point X¯4:(34)X¯4=y¯4,z¯4=a2b1+b2∑i=3kc1ia1a2−∑i=3kc1i∑i=3kc2i,a1b2+b1∑i=3kc2ia1a2−∑i=3kc1i∑i=3kc2i.
Next, the asymptotic behavior of these four equilibrium points in detail is analyzed.Theorem 2.
The equilibrium pointX¯1=0,0 is locally asymptotically stable.Proof.
LetF:R+k⟶R+,G:R+k⟶R+ be multivariate functions defined by(35)Fyn−1,yn−2,zn−3,…,zn−k=a1yn−1yn−2b1+∑i=3kc1izn−i,Gzn−1,zn−2,yn−3,…,yn−k=a2zn−1zn−2b2+∑i=3kc2iyn−i,n=0,1,…,thus, it holds that(36)Fyn−1=a1yn−2b1+∑i=3kc1izn−i,Fyn−2=a1yn−1b1+∑i=3kc1izn−i,Fzn−i=−a1c1iyn−1yn−2b1+∑i=3kc1izn−i2;Gzn−1=a2zn−2b2+∑i=3kc2iyn−i,Gzn−2=a2zn−1b2+∑i=3kc2iyn−i,Gyn−i=−a2c2izn−1zn−2b2+∑i=3kc2iyn−i2.
Moreover, the linearized system of (32) about the equilibrium point X¯1 is provided by(37)φn+1=D1φn,where(38)φn=ynyn−1⋮yn−kznzn−1⋮zn−k,D1=00⋯000⋯0010⋯000⋯00⋯⋯⋯⋯⋯⋯⋯⋯⋯00⋯100⋯0000⋯000⋯0000⋯001⋯00⋯⋯⋯⋯⋯⋯⋯⋯⋯00⋯000⋯10,and the corresponding characteristic polynomial of D10,0 is as follows:(39)fλ=λ2k+1=0.
Obviously, all eigenvalues ofD10,0 about 0,0 lie in an open unit disk λ<1; hence, from Lemma 2, the equilibrium point X¯1 is locally asymptotically stable, and then the proof ends.Theorem 3.
The equilibrium pointX¯2 is unstable.Proof.
From (36), the linearized systems of (32) about the equilibrium point X¯2 is as follows:(40)φn+1=D2φn,where(41)φn=ynyn−1yn−2⋮yn−kznzn−1⋮zn−k,D2=0000⋯0000⋯001000⋯0000⋯000100⋯0000⋯00⋯⋯⋯⋯0000⋯1000⋯00000−c23a2⋯−c2ka2011⋯000000⋯0010⋯00⋯⋯⋯⋯0000⋯0000⋯10,and the characteristic polynomial of (40) is given by(42)fλ=λ2k−1λ3−λ−1=0.
It is obvious that there existsλ>1 so that λ2k−1λ3−λ−1=0. Thus, a root of the characteristic equation (42) lies outside the unit disk. According to Lemma 2, the equilibrium point X¯2 of (32) is unstable, and the proof is complete.Theorem 4.
The equilibrium pointX¯3 is unstable.Proof.
From (36), the linearized systems of (36) about the equilibrium point X¯3 is provided by(43)φn+1=D3φn,where(44)φn=ynyn−1yn−2⋮yn−kznzn−1⋮zn−k,D3=0110⋯000−c13a1⋯−c1k−1a1−c1ka11000⋯0000⋯000100⋯0000⋯00⋯⋯⋯⋯0000⋯1000⋯000000⋯0000⋯000000⋯0010⋯00⋯⋯⋯⋯0000⋯0000⋯10,and the characteristic polynomial of (43) is given by(45)fλ=λ2k−1λ3−λ−1=0,which is the same as with (42); therefore, the equilibrium point X¯3 of (32) is unstable,, and the proof is complete.Theorem 5.
Ifa1a2>∑i=3kc1i∑i=3kc2i, then the equilibrium point X¯4 is unstable.Proof.
From (36), the linearized systems of (32) about the equilibrium point X¯4 is provided by(46)φn+1=D4φn,where(47)φn=ynyn−1yn−2⋮yn−kznzn−1⋮zn−k,D4=0110⋯000−c13a1⋯−c1k−1a1−c1ka11000⋯0000⋯000100⋯0000⋯00⋯⋯⋯⋯0000⋯1000⋯00000−c23a2⋯−c2ka2011⋯000000⋯0010⋯00⋯⋯⋯⋯0000⋯0000⋯10,and the characteristic polynomial of (46) is given by(48)fλ=λ2k+2−2λ2k−2λ2k−1+λ2k−2+2λ2k−3+λ2k−4−1a1a2∑i=3i=2k−3∑j=3j=2k−3c1ic2jλ2k−i−j=0.
According to Lemma3, the following is obtained:(49)Δ1=a1=0;Δ2=0−21−2=2>0;Δ3=0−221−2100−2=4>0;Δ4=0−2201−21100−2200−21=−4<0.
It is obvious that not allΔi>0,i=1,2,…,2k+2. From Lemma 2, the equilibrium point X¯4 is unstable, and then the proof is completed.Lemma 6 (see [12]).
LetIx,Iy be some intervals of real numbers and assume that f:Ixk+1×Iyl+1⟶Ix and g:Ixk+1×Iyl+1⟶Iy be continuously differentiable functions satisfying a mixed monotone property. If there exists(50)m0≤minx−k,…,x0,y−l,…,y0≤maxx−k,…,x0,y−l,…,y0≤M0,n0≤minx−k,…,x0,y−l,…,y0≤maxx−k,…,x0,y−l,…,y0≤N0,such that(51)m0≤fm0p,M0q,n0s,N0t≤fM0p,m0q,N0s,n0t≤M0,n0≤gm0p1,M0q1,n0s1,N0t1≤gM0p1,m0q1,N0s1,n0t1≤N0,then there exist m,M∈m0,M02 and n,N∈n0,N02 satisfying(52)M=fMp,mq,Ns,nt,m=fmp,Mq,ns,Nt,N=gMp1,mq1,Ns1,nt1,n=gmp1,Mq1,ns1,Nt1.Moreover, ifm=Mandn=N, then equation (12) has a unique equilibrium point x¯,y¯∈m0,M0×n0,N0 and every solution of (12) converges to x¯,y¯.Theorem 6.
Ifa1=a2,b1=b2,c1i=c2i, then the equilibrium point X¯1=0,0 of system (32) is a global attractor for any conditions y−i,z−i∈0,b1/2a1×0,b1/2a1,i=−k,−k+1,…,0.Proof.
In view ofa1=a2,b1=b2,c1i=c2i, system (32) can be changed to(53)yn+1=a1yn−1yn−2b1+∑i=3kc1izn−i,zn+1=a1zn−1zn−2b1+∑i=3kc1iyn−i,n=0,1,….
Letf,g:0,b1/2a12k+2×0,b1/2a12k+2⟶0,∞×0,∞ be a function defined by(54)fyn,yn−1,…,yn−i,zn,zn−1,…,zn−i=a1yn−1yn−2b1+∑i=3kc1izn−i,gyn,yn−1,…,yn−i,zn,zn−1,…,zn−i=a1zn−1zn−2b1+∑i=3kc1iyn−i.
Set(55)f=a1uvb1+∑i=3kc1iwi,g=a1u∗v∗b1+∑i=3kc1iwi∗,fu=a1vb1+∑i=3kc1iwi>0,fv=a1ub1+∑i=3kc1iwi>0,fwi=−a1c1iuvb1+∑i=3kc1iwi2<0,gu∗=a1v∗b1+∑i=3kc1iwi∗>0,gv∗=a1u∗b1+∑i=3kc1iwi∗>0,gwi∗=−a1c1iu∗v∗b1+∑i=3kc1iwi2<0,which means that f and g have a mixed monotone property.
Let(56)M0=N0=maxy0,y−1,y−2,y−i,z0,z−1,z−2,z−i,i=3,4,…,k,a1M0−b1∑i=3kc1i≤m0=n0<0,then,(57)m0≤a1m02b1+N0∑i=3kc1i≤a1M02b1+n0∑i=3kc1i≤M0,n0≤a1n02b1+M0∑i=3kc1i≤a1N02b1+m0∑i=3kc1i≤N0.
Obviously, sincemi=niandMi=Ni,i=0,1,…, then from system (32) and Lemma 6, there exist m,M∈m0,M0,n=m,N=M such that(58)m=a1m2b1+N∑i=3kc1i,n=a1n2b1+M∑i=3kc1i,M=a1M2b1+n∑i=3kc1i,N=a1N2b1+m∑i=3kc1i.
Thus,a1m+M−b1m−M=0. Moreover, from 2a1M0<b1, then M=mandN=n.
Hence, it is obtained that the equilibrium point0,0 of system (32) is a global attractor from Lemma 6. The proof is completed.
Moreover, from Definition5, it is obtained that the equilibrium point X¯1 of system (32) is asymptotically stable.
It is obvious that the fuzzy difference (6) has the trivial solution 0^. Next, the stability of the trivial solution is discussed. To do this, first, the following definition is introduced.Definition 8.
The trivial solutionx=0^ of the fuzzy difference equation (8) is said to be(i)
Stable, if givenε>0, and there exists δε>0 with Dxi,0^<δ,i=−k,−k+1,…,0, which implies Dxn,0^<ε, for any n>0, such that for any xi∈Dδ,i=−k,−k+1,⋯,0, the solution xn∈Dε,n>0;(ii)
Attractive if there isδ>0 such that Dxi,0^<δ,i=−k,−k+1,…,0,, and one has(59)limn⟶∞Dxn,0^=0;(iii)
Asymptotically stable if (i) and (ii) hold simultaneously.Theorem 7.
If the parametersA,B,Ci are the positive trivial fuzzy numbers, i.e., positive real numbers, and the initial conditions are the positive fuzzy numbers with xiα⊂0,B/2A,i=−k,−k+1,…,0,α∈0,1, then the trivial solution x=0^ of the fuzzy difference equation (8) is asymptotically stable with respect to D as n⟶∞.Proof.
The result follows from Theorem2 and Theorem 6.


## 4. Numerical Simulation

Numerical examples are given in this section to confirm the results of the previous sections and to support the theoretical discussion. The example gives the asymptotic behavior of the solution of the fuzzy difference system (8). The following fuzzy difference equation is considered.Example 1.
Letk=5, and the following fuzzy difference equation is considered:(60)xn+1=Axn−1xn−2B+∑i=35Cixn−i,n=0,1,2,…,where A,B,Cii=3,4,5 are the positive real numbers. By Theorem 7, Aα=A,A=0.2,Bα=B,B=16,C3α=C,C=2,C4α=C,C=2,C5α=C,C=3,α∈0,1. In addition, from Theorem 7, the initial conditions x−5,x−4,x−3,x−2,x−1,x0 with xiα⊂0,B/2A,i=−5,−4,…,0,α∈0,1, are denoted such that(61)x0x=12x−52,5≤x≤7,−15x+125,7≤x≤12,x−1x=18x−38,3≤x≤11,−12x+132,11≤x≤13,x−2x=18x−18,1≤x≤9,−14x+134,9≤x≤13,x−3x=16x−13,2≤x≤8,−14x+3,8≤x≤12,x−4x=15x−45,4≤x≤9,−15x+145,9≤x≤14,x−5x=14x−32,6≤x≤10,−16x+83,10≤x≤16.
From (61), the corresponding triangular fuzzy numbers are obtained:(62)x0α=5+2α,12−5α,x−1α=3+8α,13−2α,x−2α=1+8α,13−4α,x−3α=2+6α,12−4α,x−4α=4+5α,14−5α,x−5α=6+4α,16−6α.
From (60), the parameters A,B,Cii=3,4,5 and initial values x−5,x−4,x−3,x−2,x−1,x0 satisfy the following system of nonlinear difference equation with parameter α:(63)Ln+1,α=Al,αLn−1,αLn−2,αBr,α+∑i=35Cil,αRn−i,α,Rn+1,α=Ar,αRn−1,αRn−2,αBl,α+∑i=35Cil,αLn−i,α,α∈0,1,n=0,1,….
It is easy to prove thatxiα⊂0,B/2A,i=−5,−4,…,0, for α∈0,1, namely, the conditions of Theorem 7 is satisfied. So from Theorem 7, the trivial solution x=0^ of the fuzzy difference (6) is asymptotically stable with respect to D as n⟶∞ (see Figures 1–4).Figure 1
The dynamics of system (63).Figure 2
The solution of system (63) when α=1.Figure 3
The solution of system (63) when α=0.5.Figure 4
The solution of system (63) when α=0.


## 5. Conclusion

In this paper, a variational iterative method for the nonlinear fuzzy difference equations is proposed. This method is a powerful tool for solving various fuzzy difference equations and can also be applied to other nonlinear differential equations or difference equations in mathematical physics. Numerical simulation results show that this method is effective and simple. Variational iterative method is an effective method to deal with nonlinear structures. MATLAB 2016 software package was used for calculation.The purpose of this paper was to study the dynamic behavior for a class of nonlinear high-order fuzzy difference equations. Firstly, the existence and uniqueness of positive fuzzy solutions are proved. Secondly, it is obtained that the nonzero equilibrium points of the corresponding ordinary difference equations (32) are unstable by using the linearization method. Finally, it is found that the trivial solution 0^ of fuzzy difference (6) is asymptotically stable when the parameters A,B,Ci are positive trivial fuzzy numbers. In particular, an illustrate example is given to show the effectiveness of the obtained results. In addition, the sufficient conditions obtained in this paper are very simple and provide some flexibility for the application and analysis of nonlinear fuzzy difference equations.



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