
Citation: Qianhong Zhang, Ouyang Miao, Fubiao Lin, Zhongni Zhang. On discrete-time laser model with fuzzy environment[J]. AIMS Mathematics, 2021, 6(4): 3105-3120. doi: 10.3934/math.2021188
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Mathematical models of physics, chemistry, ecology, physiology, psychology, engineering and social sciences have been governed by differential equation and difference equation. With the development of computers, compared with continuous-time model, discrete-time models described by difference equations are better formulated and analyzed in the past decades. Recently laser model has vast application in medical sciences, industries, highly security areas in army [1,2,3,4,5,6,7]. Laser, whose basic principal lies on the Einstein theory of light proposed in 1916, is a device that produces intense beam of monochromatic and coherent light. Since then it is developed by Gordon Gould in 1957. In 1960, the first working ruby laser was invented by Theodore Maiman. Laser light is coherent, highly directional and monochromatic which makes it different from ordinary light. The working principle of laser is based on the spontaneous absorption, spontaneous emission, stimulated emission, and population inversion are essential for the laser formation. The readers can refer to [8,9,10]. For instance, Hakin [11] proposed a simple continuous-time laser model in 1983. Khan and Sharif [12] proposed a discrete-time laser model and studied extensively dynamical properties about fixed points, the existence of prime period and periodic points, and transcritical bifurcation of a one-dimensional discrete-time laser model in R+.
In fact, the identification of the parameters of the model is usually based on statistical method, starting from data experimentally obtained and on the choice of some method adapted to the identification. These models, even the classic deterministic approach, are subjected to inaccuracies (fuzzy uncertainty) that can be caused by the nature of the state variables, by parameters as coefficients of the model and by initial conditions. In fact, fuzzy difference equation is generation of difference equation whose parameters or the initial values are fuzzy numbers, and its solutions are sequences of fuzzy numbers. It has been used to model a dynamical systems under possibility uncertainty. Due to the applicability of fuzzy difference equation for the analysis of phenomena where imprecision is inherent, this class of difference equation is a very important topic from theoretical point of view and also its applications. Recently there has been an increasing interest in the study of fuzzy difference equations (see [13,14,15,16,17,18,19,20,21,22,23,24,25]).
In this paper, by virtue of the theory of fuzzy sets, we consider the following discrete-time laser model with fuzzy uncertainty parameters and initial conditions.
xn+1=Axn+BxnCxn+H,n=0,1,⋯, | (1.1) |
where xn is the number of laser photon at the nth time, A,B,C,H and the initial value x0 are positive fuzzy numbers.
The main aim of this work is to study the existence of positive solutions of discrete-time laser model (1.1). Furthermore, according to a generation of division (g-division) of fuzzy numbers, we derive some conditions so that every positive solution of discrete-time laser model (1.1) is bounded. Finally, under some conditions we prove that discrete-time laser model (1.1) has a fixed point 0 which is asymptotically stable, and a unique positive fixed point x.
Firstly, we give the following definitions and lemma needed in the sequel.
Definition 2.1.[26] u:R→[0,1] is said to be a fuzzy number if it satisfies conditions (i)–(iv) written below:
(i) u is normal, i. e., there exists an x∈R such that u(x)=1;
(ii) u is fuzzy convex, i. e., for all t∈[0,1] and x1,x2∈R such that
u(tx1+(1−t)x2)≥min{u(x1),u(x2)}; |
(iii) u is upper semicontinuous;
(iv) The support of u, suppu=¯⋃α∈(0,1][u]α=¯{x:u(x)>0} is compact.
For α∈(0,1], the α−cuts of fuzzy number u is [u]α={x∈R:u(x)≥α} and for α=0, the support of u is defined as suppu=[u]0=¯{x∈R|u(x)>0}.
A fuzzy number can also be described by a parametric form.
Definition 2.2. [26] A fuzzy number u in a parametric form is a pair (ul,ur) of functions ul,ur,0≤α≤1, which satisfies the following requirements:
(i) ul(α) is a bounded monotonic increasing left continuous function,
(ii) ur(α) is a bounded monotonic decreasing left continuous function,
(iii) ul(α)≤ur(α),0≤α≤1.
A crisp (real) number x is simply represented by (ul(α),ur(α))=(x,x),0≤α≤1. The fuzzy number space {(ul(α),ur(α))} becomes a convex cone E1 which could be embedded isomorphically and isometrically into a Banach space (see [26]).
Definition 2.3.[26] The distance between two arbitrary fuzzy numbers u and v is defined as follows:
D(u,v)=supα∈[0,1]max{|ul,α−vl,α|,|ur,α−vr,α|}. | (2.1) |
It is clear that (E1,D) is a complete metric space.
Definition 2.4.[26] Let u=(ul(α),ur(α)),v=(vl(α),vr(α))∈E1,0≤α≤1, and k∈R. Then
(i) u=v iff ul(α)=vl(α),ur(α)=vr(α),
(ii) u+v=(ul(α)+vl(α),ur(α)+vr(α)),
(iii) u−v=(ul(α)−vr(α),ur(α)−vl(α)),
(iv) ku={(kul(α),kur(α)),k≥0;(kur(α),kul(α)),k<0,
(v) uv=(min{ul(α)vl(α),ul(α)vr(α),ur(α)vl(α),ur(α)vr(α)},max{ul(α)vl(α),ul(α)vr(α),ur(α)vl(α),ur(α)vr(α)}).
Definition 2.5. [27] Suppose that u,v∈E1 have α-cuts [u]α=[ul,α,ur,α],[v]α=[vl,α,vr,α], with 0∉[v]α,∀α∈[0,1]. The generation of division (g-division) ÷g is the operation that calculates the fuzzy number s=u÷gv having level cuts [s]α=[sl,α,sr,α](here [u]α−1=[1/ur,α,1/ul,α]) defined by
[s]α=[u]α÷g[v]α⟺{(i)[u]α=[v]α[s]α,or(ii)[v]α=[u]α[s]α−1 | (2.2) |
provided that s is a proper fuzzy number sl,α is nondecreasing, sr,α is nonincreasing, sl,1≤sr,1.
Remark 2.1. According to [27], in this paper the fuzzy number is positive, if u÷gv=s∈E1 exists, then the following two cases are possible
Case I. if ul,αvr,α≤ur,αvl,α,∀α∈[0,1], then sl,α=ul,αvl,α,sr,α=ur,αvr,α,
Case II. if ul,αvr,α≥ur,αvl,α,∀α∈[0,1], then sl,α=ur,αvr,α,sr,α=ul,αvl,α.
Definition 2.6. [26] A triangular fuzzy number (TFN) denoted by A is defined as (a,b,c) where the membership function
A(x)={0,x≤a;x−ab−a,a≤x≤b;1,x=b;c−xc−b,b≤x≤c;0,x≥c. |
The α−cuts of A=(a,b,c) are described by [A]α={x∈R:A(x)≥α}=[a+α(b−a),c−α(c−b)]=[Al,α,Ar,α], α∈[0,1], it is clear that [A]α are closed interval. A fuzzy number A is positive if suppA⊂(0,∞).
The following proposition is fundamental since it characterizes a fuzzy set through the α-levels.
Proposition 2.1.[26] If {Aα:α∈[0,1]} is a compact, convex and not empty subset family of Rn such that
(i) ¯⋃Aα⊂A0.
(ii) Aα2⊂Aα1 if α1≤α2.
(iii) Aα=⋂k≥1Aαk if αk↑α>0.
Then there is u∈En(En denotes n dimensional fuzzy number space) such that [u]α=Aα for all α∈(0,1] and [u]0=¯⋃0<α≤1Aα⊂A0.
The fuzzy analog of the boundedness and persistence (see [15,16]) is as follows:
Definition 2.7. A sequence of positive fuzzy numbers (xn) is persistence (resp. bounded) if there exists a positive real number M (resp. N) such that
supp xn⊂[M,∞)(resp. supp xn⊂(0,N]),n=1,2,⋯, |
A sequence of positive fuzzy numbers (xn) is bounded and persistence if there exist positive real numbers M,N>0 such that
supp xn⊂[M,N],n=1,2,⋯. |
A sequence of positive fuzzy numbers (xn),n=1,2,⋯, is an unbounded if the norm ‖xn‖,n=1,2,⋯, is an unbounded sequence.
Definition 2.8. xn is a positive solution of (1.1) if (xn) is a sequence of positive fuzzy numbers which satisfies (1.1). A positive fuzzy number x is called a positive equilibrium of (1.1) if
x=Ax+BxCx+H. |
Let (xn) be a sequence of positive fuzzy numbers and x is a positive fuzzy number, xn→x as n→∞ if limn→∞D(xn,x)=0.
Lemma 2.1. [26] Let f:R+×R+×R+×R+→R+ be continuous, A,B,C,D are fuzzy numbers. Then
[f(A,B,C,D)]α=f([A]α,[B]α,[C]α,[D]α), α∈(0,1]. | (2.3) |
Firstly we give the existence of positive solutions of discrete-time laser model with fuzzy environment.
Theorem 3.1. Let parameters A,B,C,H and initial value x0 of discrete-time laser model be fuzzy numbers. Then, for any positive fuzzy number x0, there exists a unique positive solution xn of discrete-time laser model with initial conditions x0.
Proof. The proof is similar to those of Proposition 2.1 [25]. So we omit the proof of Theorem 3.1.
Noting Remark 2.1, taking α-cuts, one of the following two cases holds
Case I
[xn+1]α=[Ln+1,α,Rn+1,α]=[Al,αLn,α+Bl,αLn,αCl,αLn,α+Hl,α,Ar,αRn,α+Br,αRn,αCr,αRn,α+Hr,α] | (3.1) |
Case II
[xn+1]α=[Ln+1,α,Rn+1,α]=[Al,αLn,α+Br,αRn,αCr,αRn,α+Hr,α,Ar,αRn,α+Bl,αLn,αCl,αLn,α+Hl,α] | (3.2) |
To study the dynamical behavior of the positive solutions of discrete-time laser model (1.1), according to Definition 2.5, we consider two cases.
First, if Case I holds true, we need the following lemma.
Lemma 3.1. Consider the following difference equation
yn+1=ayn+byncyn+h, n=0,1,⋯, | (3.3) |
where a∈(0,1),b,c,h∈(0,+∞),y0∈(0,+∞), then the following statements are true:
(i) Every positive solution of (3.3) satisfies
0<yn≤bc(1−a)+y0. | (3.4) |
(ii) The equation has a fixed point y∗=0 if b≤(1−a)h.
(iii) The equation has two fixed points y∗=0,y∗=bc(1−a)−hc if b>(1−a)h.
Proof. (i) Let yn be a positive solution of (3.3). It follows from (3.3) that, for n≥0,
0<yn+1=ayn+byncyn+h≤ayn+bc. |
From which we have
0<yn≤bc(1−a)+(y0−bc(1−a))an+1≤bc(1−a)+y0. |
This completes the proof of (i).
If y∗ is fixed point of (3.3), i.e., yn=y∗. So from (3.3), we have
y∗=ay∗+by∗cy∗+h. | (3.5) |
After some manipulation, from (3.5), we can get
y∗=0, y∗=bc(1−a)−hc. | (3.6) |
From (3.3), we can summarized the existence of fixed points as follows
(ii) If b<h(1−a), then y∗=bc(1−a)−hc is not a positive number. And hence if b≤h(1−a) then (3.3) has a fixed point y∗=0.
(iii) If b>h(1−a), then y∗=bc(1−a)−hc is a positive number. And hence if b>h(1−a) then (3.3) has two fixed points y∗=0,y∗=bc(1−a)−hc.
Proposition 3.1. The following statements hold true
(i) The fixed point y∗=0 of (3.3) is stable if b<(1−a)h.
(ii) The fixed point y∗=0 of (3.3) is unstable if b>(1−a)h.
(iii) The fixed point y∗=0 of (3.3) is non-hyperbolic if b=(1−a)h.
Proof. From (3.3), let
f(y):=ay+bycy+h | (3.7) |
From (3.7), it follows that
f′(y)=a+bh(cy+h)2 | (3.8) |
From (3.8), it can get
|f′(y)|y∗=0|=|a+bh| | (3.9) |
Therefore from (3.9), it can conclude that y∗=0 is stable if b<(1−a)h, unstable if b>(1−a)h, non-hyperbolic if b=(1−a)h.
Proposition 3.2. The fixed point y∗=bc(1−a)−hc of (3.3) is globally asymptotically stable if b>(1−a)h.
Proof. From (3.8), it can get
|f′(y)|y∗=bc(1−a)−hc|=|a+h(1−a)2b|. | (3.10) |
Therefore from (3.10), it can conclude that, if a+h(1−a)2b<1, i.e., b>(1−a)h then the fixed point y∗=bc(1−a)−hc is stable.
On the other hand, it follows from (3.4) that (yn) is bounded. And from (3.8), we have f′(y)>0. Namely (yn) is monotone increasing. So we have
limn→∞yn=y∗=bc(1−a)−hc. | (3.11) |
Therefore the fixed point y∗=bc(1−a)−hc is globally asymptotically stable.
Proposition 3.3. The fixed point y∗=0 of (3.3) is globally asymptotically stable if b<(1−a)h.
Proof. From (3.3), we can get that
yn+1≤(a+bh)yn | (3.12) |
From (3.12), it follows that
y1≤(a+bh)y0y2≤(a+bh)2y0⋮yn≤(a+bh)ny0 | (3.13) |
Since b<(1−a)h, then limn→∞yn=0. Therefore the fixed point y∗=0 of (3.3) is globally asymptotically stable.
Theorem 3.2. Consider discrete-time laser model (1.1), where A,B,C,H and initial value x0 are positive fuzzy numbers. There exists positive number NA, ∀α∈(0,1], Ar,α<NA<1. If
Bl,αLn,αBr,αRn,α≤Cl,αLn,α+Hl,αCr,αRn,α+Hr,α,∀α∈(0,1]. | (3.14) |
and
Bl,α<Hl,α(1−Al,α), Br,α<Hr,α(1−Ar,α), ∀α∈(0,1]. | (3.15) |
Then (1.1) has a fixed point x∗=0 which is globally asymptotically stable.
Proof. Since A,B,C,H are positive fuzzy numbers and (3.14) holds true, taking α-cuts of model (1.1) on both sides, we can have the following difference equation system with parameters
Ln+1,α=Al,αLn,α+Bl,αLn,αCl,αLn,α+Hl,α, Rn+1,α=Ar,αRn,α+Br,αRn,αCr,αRn,α+Hr,α. | (3.16) |
Since (3.15) holds true, using Proposition 3.2, we can get
limn→∞Ln,α=0, limn→∞Rn,α=0. | (3.17) |
On the other hand, let xn=x∗, where [xn]α=[Ln,α,Rn,α]=[Lα,Rα]=[x∗]α,α∈(0,1], be the fixed point of (1.1). From (3.16), one can get
Lα=Al,αLα+Bl,αLαCl,αLα+Hl,α, Rα=Ar,αRα+Br,αRαCr,αRα+Hr,α. | (3.18) |
Since (3.15) is satisfied, from (3.16), it follows that
Lα=0, Rα=0, limn→∞D(xn,x∗)=limn→∞supα∈(0,1]{max{|Ln,α−Lα|,|Rn,α−Rα|}}=0. | (3.19) |
Therefore, by virtue of Proposition 3.3, the fixed point x∗=0 is globally asymptotically stable.
Theorem 3.3. Consider discrete-time laser model (1.1), where A,B,C,H and initial value x0 are positive fuzzy numbers, there exists positive number NA, ∀α∈(0,1], Ar,α<NA<1. If (3.14) holds true, and
Bl,α>Hl,α(1−Al,α), Br,α>Hr,α(1−Ar,α), ∀α∈(0,1], | (3.20) |
then the following statements are true.
(i) Every positive solution of (1.1) is bounded.
(ii) Equation (1.1) has a unique positive fixed point x∗ which is asymptotically stable.
Proof. (i) Since A,B,C,H and x0 are positive fuzzy numbers, there exist positive constants MA,NA,MB, NB,MC,NC,MH,NH,M0,N0 such that
{[A]α=[Al,α,Ar,α]⊂¯⋃α∈(0,1][Al,α,Ar,α]⊂[MA,NA][B]α=[Bl,α,Br,α]⊂¯⋃α∈(0,1][Bl,α,Br,α]⊂[MB,NB][C]α=[Cl,α,Cr,α]⊂¯⋃α∈(0,1][Cl,α,Cr,α]⊂[MC,NC][H]α=[Hl,α,Hr,α]⊂¯⋃α∈(0,1][Hl,α,Hr,α]⊂[MH,NH][x0]α=[L0,α,R0,α]⊂¯⋃α∈(0,1][L0,α,R0,α]⊂[M0,N0] | (3.21) |
Using (i) of Lemma 3.1, we can get that
0<Ln≤Bl,αCl,α(1−Al,α)+L0,α, 0<Rn≤Br,αCr,α(1−Ar,α)+R0,α. | (3.22) |
From (3.21) and (3.22), we have that for α∈(0,1]
[Ln,α,Rn,α]⊂[0,N], n≥1. | (3.23) |
where N=NBMC(1−NA)+N0. From (3.22), we have for n≥1,⋃α∈(0,1][Ln,α,Rn,α]⊂(0,N], so ¯⋃α∈(0,1][Ln,α,Rn,α]⊂(0,N]. Thus the proof of (i) is completed.
(ii) We consider system (3.18), then the positive solution of (3.18) is given by
Lα=Bl,αCl,α(1−Al,α)−Hl,αCl,α, Rα=Br,αCr,α(1−Ar,α)−Hr,αCr,α,α∈(0,1]. | (3.24) |
Let xn be a positive solution of (1.1) such that [xn]α=[Ln,α,Rn,α],α∈(0,1],n=0,1,2,⋯. Then applying Proposition 3.2 to system (3.16), we have
limn→∞Ln,α=Lα, limn→∞Rn,α=Rα | (3.25) |
From (3.23) and (3.25), we have, for 0<α1<α2<1,
0<Lα1≤Lα2≤Rα2≤Rα1. | (3.26) |
Since Al,α,Ar,α,Bl,α,Br,α,Cl,α,Cr,α,Hl,α,Hr,α are left continuous. It follows from (3.24) that Lα,Rα are also left continuous.
From (3.21) and (3.24), it follows that
c=MBNC(1−MA)−NHMC≤Lα≤Rα≤NBMA(1−NA)−MHNC=d. | (3.27) |
Therefore (3.27) implies that [Lα,Rα]⊂[c,d], and so ⋃α∈(0,1][Lα,Rα]⊂[c,d]. It is clear that
⋃α∈(0,1][Lα,Rα] is compact and ⋃α∈(0,1][Lα,Rα]⊂(0,∞). | (3.28) |
So from Definition 2.2, (3.24), (3.26), (3.28) and since Lα,Rα,α∈(0,1] determine a fuzzy number x∗ such that
x∗=Ax∗+Bx∗Cx∗+H, [x∗]α=[Lα,Rα], α∈(0,1]. | (3.29) |
Suppose that there exists another positive fixed point ˉx of (1.1), then there exist functions ¯Lα,¯Rα:(0,1)→(0,∞) such that
ˉx=Aˉx+BˉxCˉx+H, [x]α=[¯Lα,¯Rα], α∈(0,1]. |
From which we have
¯Lα=Al,α¯Lα+Bl,α¯LαCl,α¯Lα+Hl,α, ¯Rα=Ar,α¯Rα+Br,α¯RαCr,α¯Rα+Hr,α. |
So ¯Lα=Lα,¯Rα=Rα,α∈(0,1]. Hence ˉx=x∗, namely x∗ is a unique positive fixed point of (1.1).
From (3.25), we have
limn→∞D(xn,x∗)=limn→∞supα∈(0,1]max{|Ln,α−Lα|,|Rn,α−Rα|}=0. | (3.30) |
Namely, every positive solution xn of (1.1) converges the unique fixed point x∗ with respect to D as n→∞. Applying Proposition 3.2, it can obtain that the positive fixed point x∗ is globally asymptotically stable.
Secondly, if Case II holds true, it follows that for n∈{0,1,2,⋯},α∈(0,1]
Ln+1,α=Al,αLn,α+Br,αRn,αCr,αRn,α+Hr,α, Rn+1,α==Ar,αRn,α+Bl,αLn,αCl,αLn,α+Hl,α | (3.31) |
We need the following lemma.
Lemma 3.2. Consider the system of difference equations
yn+1=a1yn+b2znc2zn+h2, zn+1=a2zn+b1ync1yn+h1, n=0,1,⋯, | (3.32) |
where ai∈(0,1),bi,ci,hi∈(0,+∞)(i=1,2),y0,z0∈(0,+∞). If
a1+a2<1. | (3.33) |
and
b1b2>h1h2(1−a1)(1−a2). | (3.34) |
Then the following statements are true.
(i) Every positive solution (yn,zn) of (3.32) satisfy
0<yn≤b2(1−a1)c2+y0, 0<zn≤b1(1−a2)c1+z0. | (3.35) |
(ii) System (3.32) has fixed point (y,z)=(0,0) which is asymptotically stable.
(iii) System (3.32) has a unique fixed point
y=(1−a2)Kb1c2+h2c1(1−a2),z=(1−a1)Kb2c1+h1c2(1−a1), | (3.36) |
where K=b1b2−h1h2(1−a1)(1−a2)(1−a1)(1−a2).
Proof. (i) Let (yn,zn) be a positive solution of (3.32). It follows from (3.32) that, for n≥0,
0<yn+1=a1yn+b2znc2zn+h2≤a1yn+b2c2, 0<zn+1=a2zn+b1ync1yn+h1≤a2zn+b1c1. |
From which, we have
{0<yn≤b2c2(1−a1)+(y0−b2c2(1−a1))an1≤b2c2(1−a1)+y00<zn≤b1c1(1−a2)+(z0−b1c1(1−a2))an2≤b1c1(1−a2)+z0. |
This completes the proof of (i).
(ii) It is clear that (0,0) is a fixed point of (3.32). We can obtain that the linearized system of (3.32) about the fixed point (0,0) is
Xn+1=D1Xn, | (3.37) |
where Xn=(xn,yn)T and
D1=(a1b2c2b1h1a2). |
Thus the characteristic equation of (3.37) is
λ2−(a1+a2)λ+a1a2−b1b2h1h2=0. | (3.38) |
Since (3.33) and (3.34) hold true, we have
a1+a2+a1a2−b1b2h1h2<a1+q2+a1a2−(1−a1)(1−a2)<1 | (3.39) |
By virtue of Theorem 1.3.7 [28], we obtain that the fixed point (0,0) is asymptotically stable.
(iii) Let (yn,zn)=(y,z) be fixed point of (3.32). We consider the following system
y=a1y+b1zc1z+h1, z=a2z+b2yc2y+h2. | (3.40) |
It is clear that the positive fixed point (y,z) can be written by (3.36).
Theorem 3.4. Consider the difference Eq (1.1), where A,B,C,H are positive fuzzy numbers. There exists positive number NA,∀α∈(0,1],Ar,α<NA<1. If
Bl,αLn,αBr,αRn,α≥Cl,αLn,α+Hl,αCr,αRn,α+Hr,α,∀α∈(0,1], | (3.41) |
Al,α+Ar,α<1,∀α∈(0,1], | (3.42) |
and
Bl,αBr,α>Hl,αHr,α(1−Al,α)(1−Ar,α),∀α∈(0,1]. | (3.43) |
Then the following statements are true
(i) Every positive solution of (1.1) is bounded.
(ii) The Eq (1.1) has a fixed point 0 which is globally asymptotically stable.
(iii) The Eq (1.1) has a unique positive fixed point x such that
[x]α=[Lα,Rα], Lα=(1−Ar,α)KαBl,αCr,α+Hr,αCl,α(1−Ar,α),Rα=(1−Al,α)KαBr,αCl,α+Hl,αCr,α(1−Al,α), |
where
Kα=Bl,αBr,α−Hl,αHr,α(1−Al,α)(1−Ar,α)(1−Al,α)(1−Ar,α). | (3.44) |
Proof. (i) Let xn be a positive solution of (1.1). Applying (i) of Lemma 3.2, we have
0<Ln≤Br,α(1−Al,α)Cr,α+L0,α, 0<Rn≤Bl,α(1−Ar,α)Cl,α+R0,α. | (3.45) |
Next, the proof is similar to (i) of Theorem 3.3. So we omit it.
(ii) The proof is similar to those of Theorem 3.2. We omit it.
(iii) Let xn=x be a fixed point of (1.1), then
x=Ax+BxCx+H. | (3.46) |
Taking α-cuts on both sides of (3.46), since (3.41) holds true, one gets the following system
Lα=Al,αLα+Br,αRαCr,αRα+Hr,α, Rα=Ar,αRα+Bl,αLαCl,αLα+Hl,α, α∈(0,1]. | (3.47) |
From which we obtain that
Lα=(1−Ar,α)KαBl,αCr,α+Hr,αCl,α(1−Ar,α),Rα=(1−Al,α)KαBr,αCl,α+Hl,αCr,α(1−Al,α), | (3.48) |
Next, we can show that Lα,Rα constitute a positive fuzzy number x such that [x]α=[Lα,Rα],α∈(0,1]. The proof is similar to (ii) of Theorem 3.3. We omit it.
Remark 3.1. In dynamical system model, the parameters of model derived from statistic data with vagueness or uncertainty. It corresponds to reality to use fuzzy parameters in dynamical system model. Compared with classic discrete time laser model, the solution of discrete time fuzzy laser model is within a range of value (approximate value), which are taken into account fuzzy uncertainties. Furthermore the global asymptotic behaviour of discrete time laser model are obtained in fuzzy context. The results obtained is generation of discrete time Beverton-Holt population model with fuzzy environment [25].
In this section, we give two numerical examples to verify the effectiveness of theoretic results obtained.
Example 4.1. Consider the following fuzzy discrete time laser model
xn+1=Axn+BxnCxn+H, n=0,1,⋯, | (4.1) |
we take A,B,C,H and the initial values x0 such that
A(x)={10x−4,0.4≤x≤0.5−10x+6,0.5≤x≤0.6,B(x)={5x−4,0.8≤x≤1−5x+6,1≤x≤1.2 | (4.2) |
C(x)={2x−2,1≤x≤1.5−2x+4,1.5≤x≤2,H(x)={2x−6,3≤x≤3.5−2x+8,3.5≤x≤4 | (4.3) |
x0(x)={x−6,6≤x≤7−x+8,7≤x≤8 | (4.4) |
From (4.2), we get
[A]α=[0.4+110α,0.6−110α], [B]α=[0.8+15α,1.2−15α], α∈(0,1]. | (4.5) |
From (4.3) and (4.4), we get
[C]α=[1+12α,2−12α], [H]α=[3+12α,4−12α],[x0]α=[6+α,8−α], α∈(0,1]. | (4.6) |
Therefore, it follows that
¯⋃α∈(0,1][A]α=[0.4,0.6], ¯⋃α∈(0,1][B]α=[0.8,1.2], ¯⋃α∈(0,1][C]α=[1,2],¯⋃α∈(0,1][H]α=[3,4].¯⋃α∈(0,1][x0]α=[6,8]. | (4.7) |
From (4.1), it results in a coupled system of difference equations with parameter α,
Ln+1,α=Al,αLn,α+Bl,αLn,αCl,αLn,α+Hl,α, Rn+1,α=Ar,αRn,α+Br,αRn,αCr,αRn,α+Hr,α, α∈(0,1]. | (4.8) |
Therefore, it is clear that Ar,α<1,∀α∈(0,1], (3.14) and (3.15) hold true. so from Theorem 3.2, we have that every positive solution xn of Eq (4.1) is bounded In addition, from Theorem 3.2, Eq (4.1) has a fixed point 0. Moreover every positive solution xn of Eq (4.1) converges the fixed point 0 with respect to D as n→∞. (see Figures 1–3).
Example 4.2. Consider the following fuzzy discrete time laser model (4.1). where A,C,H and the initial values x0 are same as Example 4.1.
B(x)={x−2,2≤x≤3−x+4,3≤x≤4 | (4.9) |
From (4.9), we get
[B]α=[2+α,4−α], | (4.10) |
Therefore, it follows that
¯⋃α∈(0,1][B]α=[2,4],α∈(0,1]. | (4.11) |
It is clear that (3.20) is satisfied, so from Theorem 3.3, Eq (4.1) has a unique positive equilibrium ¯x=(0.341,1.667,3). Moreover every positive solution xn of Eq (4.1) converges the unique equilibrium ¯x with respect to D as n→∞. (see Figures 4–6)
In this work, according to a generalization of division (g-division) of fuzzy number, we study the fuzzy discrete time laser model xn+1=Axn+BxnCxn+H. The existence of positive solution and qualitative behavior to (1.1) are investigated. The main results are as follows
(i) Under Case I, the positive solution is bounded if Bl,α<Hl,α(1−Al,α),Br,α<Hr,α(1−Ar,α),α∈(0,1]. Moreover system (1.1) has a fixed point 0 which is globally asymptotically stable. Otherwise, if Bl,α≥Hl,α(1−Al,α),Br,α≥Hr,α(1−Ar,α),α∈(0,1]. Then system (1.1) has a unique positive fixed point x∗ which is asymptotically stable.
(ii) Under Case II, if Al,α+Ar,α<1 and Bl,αBr,α>Hl,αHr,α(1−Al,α)(1−Ar,α),α∈(0,1], then the positive solution is bounded. Moreover system (1.1) has a unique positive fixed point x and fixed point 0 which is global asymptotically stable.
The authors would like to thank the Editor and the anonymous Reviewers for their helpful comments and valuable suggestions to improve the paper. The work is partially supported by National Natural Science Foundation of China (11761018), Scientific Research Foundation of Guizhou Provincial Department of Science and Technology([2020]1Y008, [2019]1051), Priority Projects of Science Foundation at Guizhou University of Finance and Economics (2018XZD02), and Scientific Climbing Programme of Xiamen University of Technology(XPDKQ20021).
The authors declare that they have no competing interests.
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