# Global Dynamics of a Nonsymmetric System of Difference Equations

Authors: Abdul Qadeer Khan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we study the existence of fixed points, boundedness and persistence, local dynamics at fixed points, global dynamics, and convergence rate of a nonsymmetric system of difference equations. Finally, theoretical results will be verified numerically.


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## Body

## 1. Introduction

### 1.1. Motivation and Literature Review

In an observed evolutionary phenomenon, difference equations appear as a natural description of this process because most of the measurements related to the variables those ensure time evolving phenomenon are discrete, and such these equations have their own significance in mathematical models. One of the important perspectives of these equations is that they are used to study discretization method for differential equations. Many results obtained through the theory of difference equations are more or less natural discrete form of corresponding findings of differential equations, and more specifically this phenomenon is true in the case of stability theory. However, difference equation’s theory is more productive than the corresponding theory of differential equations. For example, first-order differential equations which lead to the origination of simple difference equations may have a term known as appearance of ghost solution or existence of chaotic orbit which appear in the case of higher-order differential equations. Accordingly, this theory seems to be interesting at present and will assume much greater significance in the coming era. In addition, the theory of difference equations is rapidly applicable in different disciplines like control theory, computer science, numerical analysis, and finite mathematics. In the light of above facts, the theory of difference equation is studied as a richly deserved field. For instance, Thai et al. [1] have studied boundedness and persistence, global behavior, and convergence rate of the following systems of exponential difference equations:(1)βn+1=B1+B2e−βn−1B3+γn,γn+1=B4+B5e−γn−1B6+βn,βn+1=B1+B2e−γn−1β3+βn,γn+1=B4+B5e−βn−1B6+γn,with positive Bii=1,…,6 and βi,γii=−1,0. Bešo et al. [2] have studied dynamical characteristics of the following difference equations:(2)βn+1=B1+B2βnβn−12,with positive Bii=1,2 and βi,γii=−1,0. Taşdemir [3] has investigated global dynamics of the following difference equations system with quadratic terms:(3)βn+1=B1+B2γnγn−12,γn+1=B1+B2βnβn−12,with positive Bii=1,2 and βii=−1,0. Gümüş and Abo-zeid [4] have studied dynamical characteristics of the following difference equation:(4)βn+1=B11−βnβn−1,with positive Bii=1,2 and βii=−1,0. Okumus and Soykan [5] have studied dynamical characteristics of the following difference equation system:(5)βn+1=B1+βn−1δn,γn+1=B1+γn−1δn,δn+1=B1+δn−1γn,with positive B1 and βi,γi,δii=−1,0. Kalabušić et al. [6] have studied dynamical characteristics of the following difference equation:(6)βn+1=B1βn−1B2βn2+B3βnβn−1+B4βn−1,with positive Bii=1,…,4 and βii=−1,0. Moranjkić and Nurkanović [7] have studied dynamical characteristics of the following difference equation:(7)βn+1=B1βnβn−1+B2βn−12+B3B4βnβn−1+B5βn−12+B6,with positive Bii=1,…,6 and βii=−1,0.


### 1.2. Contributions

Motivated from the aforementioned studies, the purpose of the present study is to explore dynamical characteristics including the study of fixed points, global analysis, and verification of theoretical results of the difference equation system numerically:(8)βn+1=1+Δ1γnγn−12,γn+1=1+Δ2βnβn−12,which is an alternative form of the following difference equation system:(9)xn+1=A+Bynyn−12,yn+1=C+Dxnxn−12,by βn=xn/A,γn=yn/C, where Δ1=B/AC>0andΔ2=D/AC>0 with positive A, B, C, and D and considering βi,γii=−1,0 may be positive or negative.


### 1.3. Structure of the Paper

The paper is organized as follows: fixed points and linearized form of system (8) are studied in Section 2. In Section 3, we studied boundedness and persistence of system (8), whereas global dynamic behavior and convergence rate are briefly studied in Sections 4 and 5, respectively. Numerical simulations are presented in Section 6. The conclusion of the paper and future work are given in Section 7.


## 1.1. Motivation and Literature Review

In an observed evolutionary phenomenon, difference equations appear as a natural description of this process because most of the measurements related to the variables those ensure time evolving phenomenon are discrete, and such these equations have their own significance in mathematical models. One of the important perspectives of these equations is that they are used to study discretization method for differential equations. Many results obtained through the theory of difference equations are more or less natural discrete form of corresponding findings of differential equations, and more specifically this phenomenon is true in the case of stability theory. However, difference equation’s theory is more productive than the corresponding theory of differential equations. For example, first-order differential equations which lead to the origination of simple difference equations may have a term known as appearance of ghost solution or existence of chaotic orbit which appear in the case of higher-order differential equations. Accordingly, this theory seems to be interesting at present and will assume much greater significance in the coming era. In addition, the theory of difference equations is rapidly applicable in different disciplines like control theory, computer science, numerical analysis, and finite mathematics. In the light of above facts, the theory of difference equation is studied as a richly deserved field. For instance, Thai et al. [1] have studied boundedness and persistence, global behavior, and convergence rate of the following systems of exponential difference equations:(1)βn+1=B1+B2e−βn−1B3+γn,γn+1=B4+B5e−γn−1B6+βn,βn+1=B1+B2e−γn−1β3+βn,γn+1=B4+B5e−βn−1B6+γn,with positive Bii=1,…,6 and βi,γii=−1,0. Bešo et al. [2] have studied dynamical characteristics of the following difference equations:(2)βn+1=B1+B2βnβn−12,with positive Bii=1,2 and βi,γii=−1,0. Taşdemir [3] has investigated global dynamics of the following difference equations system with quadratic terms:(3)βn+1=B1+B2γnγn−12,γn+1=B1+B2βnβn−12,with positive Bii=1,2 and βii=−1,0. Gümüş and Abo-zeid [4] have studied dynamical characteristics of the following difference equation:(4)βn+1=B11−βnβn−1,with positive Bii=1,2 and βii=−1,0. Okumus and Soykan [5] have studied dynamical characteristics of the following difference equation system:(5)βn+1=B1+βn−1δn,γn+1=B1+γn−1δn,δn+1=B1+δn−1γn,with positive B1 and βi,γi,δii=−1,0. Kalabušić et al. [6] have studied dynamical characteristics of the following difference equation:(6)βn+1=B1βn−1B2βn2+B3βnβn−1+B4βn−1,with positive Bii=1,…,4 and βii=−1,0. Moranjkić and Nurkanović [7] have studied dynamical characteristics of the following difference equation:(7)βn+1=B1βnβn−1+B2βn−12+B3B4βnβn−1+B5βn−12+B6,with positive Bii=1,…,6 and βii=−1,0.


## 1.2. Contributions

Motivated from the aforementioned studies, the purpose of the present study is to explore dynamical characteristics including the study of fixed points, global analysis, and verification of theoretical results of the difference equation system numerically:(8)βn+1=1+Δ1γnγn−12,γn+1=1+Δ2βnβn−12,which is an alternative form of the following difference equation system:(9)xn+1=A+Bynyn−12,yn+1=C+Dxnxn−12,by βn=xn/A,γn=yn/C, where Δ1=B/AC>0andΔ2=D/AC>0 with positive A, B, C, and D and considering βi,γii=−1,0 may be positive or negative.


## 1.3. Structure of the Paper

The paper is organized as follows: fixed points and linearized form of system (8) are studied in Section 2. In Section 3, we studied boundedness and persistence of system (8), whereas global dynamic behavior and convergence rate are briefly studied in Sections 4 and 5, respectively. Numerical simulations are presented in Section 6. The conclusion of the paper and future work are given in Section 7.


## 2. Fixed Points and Linearized Form of the System

The results regarding existence of fixed points can be interpreted as Theorem (8).Theorem 1.
Discrete system (8) has two fixed points Γ1=l11,l12 and Γ2=l21,l22 where(10)l11=1+Δ1−Δ2+4Δ2+1+Δ1−Δ222,l12=1−Δ1+Δ2+4Δ2+1+Δ1−Δ222,l21=1+Δ1−Δ2−4Δ2+1+Δ1−Δ222,l22=1−Δ1+Δ2−4Δ2+1+Δ1−Δ222.Proof.
IfΓ=β¯,γ¯ be the fixed point of (8), then(11)β¯=1+Δ1γ¯γ¯2,γ¯=1+Δ2β¯β¯2.
From the1st equation of (11), one gets(12)γ¯=Δ1β¯−1.
From the2nd equation of (11), one gets(13)β¯=Δ2γ¯−1.
Using (12) in (13), one gets(14)β¯2−β¯1+Δ1−Δ2−Δ2=0,whose roots are(15)β¯=1+Δ1−Δ2±4Δ2+1+Δ1−Δ222.
Next, using (13) in (12) and then after manipulation, one gets(16)γ¯2−γ¯1+Δ2−Δ1−Δ1=0,whose roots are(17)γ¯=1−Δ1+Δ2±4Δ2+1+Δ1−Δ222.
In view of (15) and (17), one can conclude that (8) has fixed points Γ1=l11,l12 and Γ2=l21,l22, where l11, l12, l21, and l22 are depicted in (10).
Next, the linearized form of (8) about Γ=β¯,γ¯ under the map f1,f2↦βn+1,γn+1 is(18)ϖn+1≔=JΓϖn,where(19)ϖn=βnβn−1γnγn−1,(20)JΓ=00Δ1γ¯2−2Δ1γ¯21000Δ2β¯2−2Δ2β¯2000010,(21)f1≔=1+Δ1γnγn−12,f2≔=1+Δ2βnβn−12.


## 3. Boundedness and Persistence

In the following theorem, it should be concluded that solutionβn,γnn=−1∞ of (8) is bounded and persists.Theorem 2.
If(22)Δ1Δ2<1,
thenβn,γnn=−1∞ of (8) is bounded and persists.Proof.
Ifβn,γnn=−1∞ is the solution of (8), then(23)βn≥1,γn≥1.
Moreover, from (8) and (23), one gets(24)βn+1≤1+Δ1+Δ1Δ2βn−1,γn+1≤1+Δ2+Δ1Δ2γn−1.
From the1st inequality of (24), one gets(25)un+1=1+Δ1+Δ1Δ2un−1,whose solution is(26)un=1+Δ11−Δ1Δ2+c1Δ1Δ2n/2+c2−1nΔ1Δ2n/2,where constants cii=1,2 depends on uii=−1,0. From the 2nd inequality of (24), one gets(27)vn+1=1+Δ2+Δ1Δ2vn−1,whose solution is(28)vn=1+Δ21−Δ1Δ2+c3Δ1Δ2n/2+c4−1nΔ1Δ2n/2,where constants cii=3,4 depends on vii=−1,0. Now, if one considers the solution in which u−1=β−1, u0=β0, v−1=γ−1, v0=γ0 and additionally if Δ1Δ2<1 holds true, then from (24), (26), and (28), one gets(29)βn≤1+Δ11−Δ1Δ2,γn≤1+Δ21−Δ1Δ2.
Finally, from (23) and (29), one gets(30)1≤βn≤1+Δ11−Δ1Δ2,1≤γn≤1+Δ21−Δ1Δ2.Theorem 3.
The set1,1+Δ1/1−Δ1Δ2×1,1+Δ2/1−Δ1Δ2 is an invariant rectangle.Proof.
Ifβn,γnn=−1∞ is the solution of (8) with β−1,β0∈1,1+Δ1/1−Δ1Δ2 and γ−1,γ0∈1,1+Δ2/1−Δ1Δ2, then(31)1≤β1=1+Δ1γ0γ−12≤1+Δ11−Δ1Δ2,1≤γ1=1+Δ2β0β−12≤1+Δ21−Δ1Δ2.
From (31), it can be concluded that β1∈1,1+Δ1/1−Δ1Δ2 and γ1∈1,1+Δ2/1−Δ1Δ2. Finally, by induction, it can also be concluded that βk+1∈1,1+Δ1/1−Δ1Δ2 and γk+1∈1,1+Δ2/1−Δ1Δ2 if βk∈1,1+Δ1/1−Δ1Δ2 and γk∈1,1+Δ2/1−Δ1Δ2.


## 4. Global Dynamic Behavior

In the present section, local dynamic behavior aboutΓ1,2 will be explored and motivated from the existing literature [8–16].Theorem 4.
Γ1 of system (8) is stable if(32)8Δ11−Δ1+Δ2+4Δ2+1+Δ1−Δ222−4Δ1<1,8Δ21+Δ1−Δ2+4Δ2+1+Δ1−Δ222−4Δ2<1.Proof.
ForΓ1, (18) gives(33)ϖn+1≔=J|Γ1ϖn,where from (20), one gets(34)JΓ1=00104Δ21+Δ1−Δ2+4Δ2+1+Δ1−Δ222−8Δ21+Δ1−Δ2+4Δ2+1+Δ1−Δ222004Δ11−Δ1+Δ2+4Δ2+1+Δ1−Δ222−8Δ11−Δ1+Δ2+4Δ2+1+Δ1−Δ222000010.
Now, ifJΓ1 has eigenvalues λll=1,…,4 and the diagonal matrix,(35)P=ζ10000ζ20000ζ30000ζ4,with(36)ζ1=ζ3=1,ζ2=ζ4=1−εfor0<ε<1,and(37)0<ε<min1−8Δ11−Δ1+Δ2+4Δ2+1+Δ1−Δ222−4Δ1,1−8Δ21+Δ1−Δ2+4Δ2+1+Δ1−Δ222−4Δ2.
Then,(38)PJ|Γ1P−1=00ζ2ζ1−104Δ2ζ3ζ1−11+Δ1−Δ2+4Δ2+1+Δ1−Δ222−8Δ2ζ3ζ2−11+Δ1−Δ2+4Δ2+1+Δ1−Δ222004Δ1ζ1ζ3−11−Δ1+Δ2+4Δ2+1+Δ1−Δ222−8Δ1ζ1ζ4−11−Δ1+Δ2+4Δ2+1+Δ1−Δ2220000ζ4ζ3−10.
Also,(39)ζ1>ζ2>0,ζ3>ζ4>0.
From (39), one gets(40)ζ2ζ1−1<1,ζ4ζ3−1<1.
Equations (36) and (37) yields(41)4Δ1ζ1ζ3−11−Δ1+Δ2+4Δ2+1+Δ1−Δ222+8Δ1ζ1ζ4−11−Δ1+Δ2+4Δ2+1+Δ1−Δ222=4Δ11−Δ1+Δ2+4Δ2+1+Δ1−Δ222×1+21−ε<1,4Δ2ζ3ζ1−11+Δ1−Δ2+4Δ2+1+Δ1−Δ222+8Δ2ζ3ζ2−11+Δ1−Δ2+4Δ2+1+Δ1−Δ222=4Δ21+Δ1−Δ2+4Δ2+1+Δ1−Δ222×1+21−ε<1.
Finally, from (40) and (41), one gets the proof of required statement as(42)max1≤k≤4λl=PJΓ1P−1=maxζ2ζ1−1,4Δ1ζ1ζ3−11−Δ1+Δ2+4Δ2+1+Δ1−Δ222+8Δ1ζ1ζ4−11−Δ1+Δ2+4Δ2+1+Δ1−Δ222,ζ4ζ3−1,4Δ2ζ3ζ1−11+Δ1−Δ2+4Δ2+1+Δ1−Δ222+8Δ2ζ3ζ2−11+Δ1−Δ2+4Δ2+1+Δ1−Δ222<1.Theorem 5.
Γ2 of system (8) is a stable if(43)8Δ11−Δ1+Δ2−4Δ2+1+Δ1−Δ222−4Δ1<1,8Δ21+Δ1−Δ2−4Δ2+1+Δ1−Δ222−4Δ2<1.Proof.
ForΓ2, from (18), one gets(44)ϖn+1≔=J|Γ2ϖn,where from (20), one gets(45)JΓ2=00104Δ21+Δ1−Δ2−4Δ2+1+Δ1−Δ222−8Δ21+Δ1−Δ2−4Δ2+1+Δ1−Δ222004Δ11−Δ1+Δ2−4Δ2+1+Δ1−Δ222−8Δ11−Δ1+Δ2−4Δ2+1+Δ1−Δ222000010.
Now, it is recall that ifP is a diagonal matrix with (36) holds and(46)0<ε<min1−8Δ11−Δ1+Δ2−4Δ2+1+Δ1−Δ222−4Δ1,1−8Δ21+Δ1−Δ2−4Δ2+1+Δ1−Δ222−4Δ2.
Then,(47)PJΓ2P−1=00ζ2ζ1−104Δ2ζ3ζ1−11+Δ1−Δ2−4Δ2+1+Δ1−Δ222−8Δ2ζ3ζ2−11+Δ1−Δ2−4Δ2+1+Δ1−Δ222004Δ1ζ1ζ3−11−Δ1+Δ2−4Δ2+1+Δ1−Δ222−8Δ1ζ1ζ4−11−Δ1+Δ2−4Δ2+1+Δ1−Δ2220000ζ4ζ3−10.
Moreover, if (40) and (46) holds true, then(48)4Δ1ζ1ζ3−11−Δ1+Δ2−4Δ2+1+Δ1−Δ222+8Δ1ζ1ζ4−11−Δ1+Δ2−4Δ2+1+Δ1−Δ222=4Δ11−Δ1+Δ2−4Δ2+1+Δ1−Δ222×1+21−ε<1,4Δ2ζ3ζ1−11+Δ1−Δ2−4Δ2+1+Δ1−Δ222+8Δ2ζ3ζ2−11+Δ1−Δ2−4Δ2+1+Δ1−Δ222=4Δ21+Δ1−Δ2−4Δ2+1+Δ1−Δ222×1+21−ε<1.
Finally, from (40) and (48), one gets the proof of required statement as(49)max1≤k≤4λ2=PJΓ2P−1=maxζ2ζ1−1,4Δ1ζ1ζ3−11−Δ1+Δ2−4Δ2+1+Δ1−Δ222+8Δ1ζ1ζ4−11−Δ1+Δ2−4Δ2+1+Δ1−Δ222,ζ4ζ3−1,4Δ2ζ3ζ1−11+Δ1−Δ2−4Δ2+1+Δ1−Δ222+8Δ2ζ3ζ2−11+Δ1−Δ2−4Δ2+1+Δ1−Δ222<1.Theorem 6.
IfΔ1,Δ2∈0,1/2, then Γ1 of (8) is a global attractor.Proof.
Ifβn,γnn=−1∞ is the solution of discrete system (8) such that limn⟶∞infβn=l1, limn⟶∞infγn=l2, limn⟶∞supβn=β1, and limn⟶∞supγn=β2, then(50)1<L1=limn⟶∞infβn,1<L2=limn⟶∞infγn,β1=limn⟶∞supβn<∞,β2=limn⟶∞supγn<∞.
In view of (8) and (50), one gets(51)β1≤1+Δ1β2L22,L1≥1+Δ1L2β22,(52)β2≤1+Δ2β1L12,L2≥1+Δ2L1β12.
From the1st and 2nd inequalities of (51) and (52), one gets(53)β1+Δ2L1β1≤β1L2≤L2+Δ1β2L2.
Similarly, from the2nd and 1st inequalities of (51) and (52), one gets(54)β2+Δ1L2β2≤β2L1≤L1+Δ2β1L1.
From (53) and (54), one gets(55)β1+Δ2L1β1+β2+Δ1L2β2≤L1+Δ2β1L1+L2+Δ1β2L2,which implies that(56)β1−L11−Δ21L1+β1β1+β2−L21−Δ11β2+1L2≤0.
IfΔ1,Δ2∈0,1/2, then from (56), one can observe that(57)1−Δ21β1+1L1>0,1−Δ11β2+1L2>0.
Finally, from (56), one gets β1=L1 and β2=L2.


## 5. Convergence Rate

Theorem 7.
Ifβn,γnn=−1∞ is the solution of (8) such that limn⟶∞βn=β¯ and limn⟶∞γn=γ¯, then(58)ϕn=ϕn1ϕn−11ϕn2ϕn−12,satisfying the following mathematical relation(59)limn→∞ϕnn=λJΓ,limn→∞ϕn+1ϕn=λJΓ,where λJΓ are roots of JΓ at Γ.Proof.
Iflimn⟶∞βn=β¯ and limn⟶∞γn=γ¯, then in order for error terms, one gets(60)βn+1−β¯=Δ1γnγn−12−Δ11γ¯,=Δ1γn−12γn−γ¯−Δ1γn−1+γ¯γ¯γn−12γn−1−γ¯,γn+1−γ¯=Δ2βnβn−12−Δ21β¯,=Δ2βn−12βn−β¯−Δ2βn−1+β¯β¯βn−12βn−1−β¯.
Set(61)ϕn1=βn−β¯,ϕn2=γn−γ¯.
From (60) and (61), one gets(62)ϕn+11=α11ϕn2+α12ϕn−12,ϕn+12=α21ϕn1+α22ϕn−11,where(63)α11=Δ1γn−12,α12=−Δ1γn−1+γ¯γ¯γn−12,α21=Δ2βn−12,α22=−Δ2βn−1+β¯β¯βn−12.
From (63), one gets(64)limn⟶∞α11=Δ1γ¯2,limn⟶∞α12=−2Δ1γ¯2,limn⟶∞α21=Δ2β¯2,limn⟶∞α22=−2Δ2β¯2.
That is,(65)α11=Δ1γ¯2+σ11,α12=−2Δ1γ¯2+σ12,α21=Δ2β¯2+σ21,α22=−2Δ2β¯2+σ22,where σ11,σ12,σ21,σ22⟶0 as n⟶∞. In view of existing literature [17], one gets(66)ϕn+1=A+Bnϕn,where(67)A=00Δ1γ¯2−2Δ1γ¯21000Δ2β¯2−2Δ2β¯2000010,(68)Bn=00σ11σ121000σ21σ22000010.
Therefore, forΓ, the error system becomes(69)ϕn+11ϕn1ϕn+12ϕn2=00Δ1γ¯2−2Δ1γ¯21000Δ2β¯2−2Δ2β¯2000010ϕn1ϕn−11ϕn2ϕn−12,which is the same as JΓ at Γ. Particularly about Γ1,2, (69) becomes(70)ϕn+11ϕn1ϕn+12ϕn2=00104Δ21+Δ1−Δ2+4Δ2+1+Δ1−Δ222−8Δ21+Δ1−Δ2+4Δ2+1+Δ1−Δ222004Δ11−Δ1+Δ2+4Δ2+1+Δ1−Δ222−8Δ11−Δ1+Δ2+4Δ2+1+Δ1−Δ222000010ϕn1ϕn−11ϕn2ϕn−12,ϕn+11ϕn1ϕn+12ϕn2=00104Δ21+Δ1−Δ2−4Δ2+1+Δ1−Δ222−8Δ21+Δ1−Δ2−4Δ2+1+Δ1−Δ222004Δ11−Δ1+Δ2−4Δ2+1+Δ1−Δ222−8Δ11−Δ1+Δ2−4Δ2+1+Δ1−Δ222000010ϕn1ϕn−11ϕn2ϕn−12,which are same as J|Γ1,2 about Γ1,2, respectively.


## 6. Numerical Simulations

In order to verify theoretical results numerically, the following cases are presented:Case 1.
IfΔ1=0.45,Δ2=0.47∈0,1/2, then from (22), one gets Δ1Δ2=0.2115<1, and therefore, from (30), the condition for the existence of boundedness solution, i.e., 1<βn<1+Δ1/1−Δ1Δ2=1.8389 and 1<γn<1+Δ2/1−Δ1Δ2=1.8643 holds true. Additionally, parametric conditions, which are depicted in (32), under which fixed point Γ1=1.3327,1.3527 of discrete system (8) is stable also holds true, i.e., 8Δ1/1−Δ1+Δ2+4Δ2+1+Δ1−Δ222−4Δ1=0.6523<1 and 8Δ2/1+Δ1−Δ2+4Δ2+1+Δ1−Δ222−4Δ2=0.7197<1. Hence, Figures 1(a) and 1(b) imply that interior fixed point Γ1=1.3327,1.3527 of system (8) is a sink, whereas Figure 1(c) shows that it is a global attractor. Therefore, simulation agrees with the theoretical results obtained in Theorems 4 and 6.Figure 1
Dynamics of discrete system (8) with βi,γii=−1,0 are 0.7,1.9,10.4,2.1.
(a)(b)(c)Case 2.
IfΔ1=0.32,Δ2=0.2∈0,1/2, then from (22), one gets Δ1Δ2=0.064<1, and therefore from (30), the condition for the existence of boundedness solution, i.e., 1<βn<1+Δ1/1−Δ1Δ2=1.4103 and 1<γn<1+Δ2/1−Δ1Δ2=1.2821 holds true. Additionally, parametric conditions, which are depicted in (32), under which fixed point Γ1=1.2767,1.1567 of the discrete system (8) is stable also hold true, i.e., 8Δ1/1−Δ1+Δ2+4Δ2+1+Δ1−Δ222−4Δ1=0.6288<1 and 8Δ2/1+Δ1−Δ2+4Δ2+1+Δ1−Δ222−4Δ2=0.2797<1. Hence, Figures 2(a) and 2(b) imply that interior fixed point Γ1=1.2767,1.1567 of system (8) is a sink, whereas Figure 2(c) shows that it is a global attractor. Therefore, simulation agrees with the theoretical results obtained in Theorems 4 and 6.Figure 2
Dynamics of discrete system (8) with βi,γii=−1,0 are 0.7,0.9,18.4,2.1.
(a)(b)(c)Case 3.
Here, we a give counter example which do not satisfy the conclusion of Theorem4, and hence, this implies that the fixed point Γ1 of the discrete system (8) is unstable. If Δ1=1.1,Δ2=1.3∉0,1/2, then from (22), one gets Δ1Δ2=1.43>1. Additionally, it is also important here to be noted that conditions (32) under which Γ1=1.6083,1.8083 of discrete-time difference equation system, which is depicted in (8), is stable does not hold true, i.e., 8Δ1/1−Δ1+Δ2+4Δ2+1+Δ1−Δ222−4Δ1=1.0138>1 and 8Δ2/1+Δ1−Δ2+4Δ2+1+Δ1−Δ222−4Δ2=2.0208>1. Hence, Figures 3(a) and 3(b) imply that Γ1=1.6083,1.8083 of system (8) is unstable.Figure 3
Dynamics of discrete system (8) with βi,γii=−1,0 are 0.7,0.9,8.4,2.1.
(a)(b)Case 4.
IfΔ1=1.1,Δ2=1.3, then Figures 4(a) and 4(b) imply that Γ2=−0.8083,−0.6083 of discrete system (8) is stable, i.e., 8Δ1/1−Δ1+Δ2−4Δ2+1+Δ1−Δ222−4Δ1=−3.0138<1 and 8Δ2/1+Δ1−Δ2−4Δ2+1+Δ1−Δ222−4Δ2=−4.0208<1 holds true, and therefore, simulation agrees with the theoretical results obtained in Theorem 5.Figure 4
Dynamics of discrete system (8) with βi,γii=−1,0 are −0.007,−0.9,−19.4,−2.1.
(a)(b)


## 7. Conclusion

This work is about the global dynamics of a nonsymmetric system of difference equations. More specially, we have proved that system (8) has two fixed points Γ1=l11,l12 and Γ2=l21,l22, where l11, l12, l21, and l22 are depicted in (10). It is also proved that if Δ1Δ2<1, then solution βn,γnn=−1∞ of discrete system (8) is bounded and persisting, and the set 1,1+Δ1/1−Δ1Δ2×1,1+Δ2/1−Δ1Δ2 is an invariant rectangle. Further, it is investigated that if 8Δ1/1−Δ1+Δ2+4Δ2+1+Δ1−Δ222−4Δ1<1 and 8Δ2/1+Δ1−Δ2+4Δ2+1+Δ1−Δ222−4Δ2<1, then Γ1 of system (8) is stable, and more importantly, it is global attractor if Δ1,Δ2∈0,1/2. It is also shown that if 8Δ1/1−Δ1+Δ2−4Δ2+1+Δ1−Δ222−4Δ1<1 and 8Δ2/1+Δ1−Δ2−4Δ2+1+Δ1−Δ222−4Δ2<1, then Γ2 of system (8) is stable. Further, the convergence rate is also studied for under consideration discrete system. Finally, obtained results are confirmed numerically.


### 7.1. Future Work

The semicycle analysis and construction of forbidden set for under consideration discrete system (8) are our next aim to study.


## 7.1. Future Work

The semicycle analysis and construction of forbidden set for under consideration discrete system (8) are our next aim to study.



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