Global Dynamics of a Nonsymmetric System of Difference Equations

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.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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:$$\begin{array}{c}\text{(1)}& {\beta }_{n+1}=\frac{{\mathfrak{B}}_1+{\mathfrak{B}}_2{e}^{-{\beta }_{n-1}}}{{\mathfrak{B}}_3+{\gamma }_n}\text{,}{\gamma }_{n+1}=\frac{{\mathfrak{B}}_4+{\mathfrak{B}}_5{e}^{-{\gamma }_{n-1}}}{{\mathfrak{B}}_6+{\beta }_n}\text{,}\\ {\beta }_{n+1}=\frac{{\mathfrak{B}}_1+{\mathfrak{B}}_2{e}^{-{\gamma }_{n-1}}}{{\beta }_3+{\beta }_n}\text{,}\\ {\gamma }_{n+1}=\frac{{\mathfrak{B}}_4+{\mathfrak{B}}_5{e}^{-{\beta }_{n-1}}}{{\mathfrak{B}}_6+{\gamma }_n}\text{,}\end{array}$$with positive ${\mathfrak{B}}_{\mathfrak{i}}i=1,\dots ,6$ and ${\beta }_i,{\gamma }_{i}i=-\mathrm{1,0}$. Bešo et al. [2] have studied dynamical characteristics of the following difference equations:$$\begin{array}{}\text{(2)}& {\beta }_{n+1}={\mathfrak{B}}_1+{\mathfrak{B}}_2\frac{{\beta }_n}{{\beta }_{n-1}^2}\text{,}\end{array}$$with positive ${\mathfrak{B}}_{\mathfrak{i}}i=\mathrm{1,2}$ and ${\beta }_i,{\gamma }_{i}i=-\mathrm{1,0}$. Taşdemir [3] has investigated global dynamics of the following difference equations system with quadratic terms:$$\begin{array}{c}\text{(3)}& {\beta }_{n+1}={\mathfrak{B}}_1+{\mathfrak{B}}_2\frac{{\gamma }_n}{{\gamma }_{n-1}^2}\text{,}{\gamma }_{n+1}={\mathfrak{B}}_1+{\mathfrak{B}}_2\frac{{\beta }_n}{{\beta }_{n-1}^2}\text{,}\end{array}$$with positive ${\mathfrak{B}}_{\mathfrak{i}}i=\mathrm{1,2}$ and ${\beta }_{i}i=-\mathrm{1,0}$. Gümüş and Abo-zeid [4] have studied dynamical characteristics of the following difference equation:$$\begin{array}{}\text{(4)}& {\beta }_{n+1}=\frac{{\mathfrak{B}}_1}{1-{\beta }_n{\beta }_{n-1}}\text{,}\end{array}$$with positive ${\mathfrak{B}}_{\mathfrak{i}}i=\mathrm{1,2}$ and ${\beta }_{i}i=-\mathrm{1,0}$. Okumus and Soykan [5] have studied dynamical characteristics of the following difference equation system:$$\begin{array}{c}\text{(5)}& {\beta }_{n+1}={\mathfrak{B}}_1+\frac{{\beta }_{n-1}}{{\delta }_n}\text{,}{\gamma }_{n+1}={\mathfrak{B}}_1+\frac{{\gamma }_{n-1}}{{\delta }_n}\text{,}\\ {\delta }_{n+1}={\mathfrak{B}}_1+\frac{{\delta }_{n-1}}{{\gamma }_n}\text{,}\end{array}$$with positive ${\mathfrak{B}}_1$ and ${\beta }_i,{\gamma }_i,{\delta }_{i}i=-\mathrm{1,0}$. Kalabušić et al. [6] have studied dynamical characteristics of the following difference equation:$$\begin{array}{}\text{(6)}& {\beta }_{n+1}=\frac{{\mathfrak{B}}_1{\beta }_{n-1}}{{\mathfrak{B}}_2{\beta }_n^2+{\mathfrak{B}}_3{\beta }_n{\beta }_{n-1}+{\mathfrak{B}}_4{\beta }_{n-1}}\text{,}\end{array}$$with positive ${\mathfrak{B}}_{\mathfrak{i}}i=1,\dots ,4$ and ${\beta }_{i}i=-\mathrm{1,0}$. Moranjkić and Nurkanović [7] have studied dynamical characteristics of the following difference equation:$$\begin{array}{}\text{(7)}& {\beta }_{n+1}=\frac{{\mathfrak{B}}_1{\beta }_n{\beta }_{n-1}+{\mathfrak{B}}_2{\beta }_{n-1}^2+{\mathfrak{B}}_3}{{\mathfrak{B}}_4{\beta }_n{\beta }_{n-1}+{\mathfrak{B}}_5{\beta }_{n-1}^2+{\mathfrak{B}}_6}\text{,}\end{array}$$with positive ${\mathfrak{B}}_{\mathfrak{i}}i=1,\dots ,6$ and ${\beta }_{i}i=-\mathrm{1,0}$.

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:$$\begin{array}{c}\text{(8)}& {\beta }_{n+1}=1+{\Delta }_1\frac{{\gamma }_n}{{\gamma }_{n-1}^2}\text{,}{\gamma }_{n+1}=1+{\Delta }_2\frac{{\beta }_n}{{\beta }_{n-1}^2}\text{,}\end{array}$$which is an alternative form of the following difference equation system:$$\begin{array}{c}\text{(9)}& x_{n+1}=A+B\frac{y_n}{y_{n-1}^2}\text{,}{y}_{n+1}=C+D\frac{x_n}{x_{n-1}^2}\text{,}\end{array}$$by ${\beta }_n=x_n/A,{\gamma }_n=y_n/C\text{,}$ where ${\Delta }_1=B/AC>0\text{\hspace{0.17em}and\hspace{0.17em}}{\Delta }_2=D/AC>0$ with positive $A$, $B$, $C$, and $D$ and considering ${\beta }_i,{\gamma }_{i}i=-\mathrm{1,0}$ 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 ${\mathrm{\Gamma }}_1={\mathcal{l}}_{11},{\mathcal{l}}_{12}$ and ${\mathrm{\Gamma }}_2={\mathcal{l}}_{21},{\mathcal{l}}_{22}$ where$$\begin{array}{}\text{(10)}& \begin{array}{c}{\mathcal{l}}_{11}=\frac{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}{2},\\ {\mathcal{l}}_{12}=\frac{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}{2},\\ {\mathcal{l}}_{21}=\frac{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}{2},\\ {\mathcal{l}}_{22}=\frac{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}{2}\text{.}\end{array}\end{array}$$

Proof.

If $\mathrm{\Gamma }=\overline{\beta },\overline{\gamma }$ be the fixed point of (8), then$$\begin{array}{c}\text{(11)}& \overline{\beta }=1+{\Delta }_1\frac{\overline{\gamma }}{{\overline{\gamma }}^2}\text{,}\overline{\gamma }=1+{\Delta }_2\frac{\overline{\beta }}{{\overline{\beta }}^2}\text{.}\end{array}$$

From the $1^{\text{st}}$ equation of (11), one gets$$\begin{array}{}\text{(12)}& \overline{\gamma }=\frac{{\Delta }_1}{\overline{\beta }-1}\text{.}\end{array}$$

From the $2^{\text{nd}}$ equation of (11), one gets$$\begin{array}{}\text{(13)}& \overline{\beta }=\frac{{\Delta }_2}{\overline{\gamma }-1}\text{.}\end{array}$$

Using (12) in (13), one gets$$\begin{array}{}\text{(14)}& {\overline{\beta }}^2-\overline{\beta }1+{\Delta }_1-{\Delta }_2-{\Delta }_2=0\text{,}\end{array}$$whose roots are$$\begin{array}{}\text{(15)}& \overline{\beta }=\frac{1+{\Delta }_1-{\Delta }_2\pm \sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}{2}\text{.}\end{array}$$

Next, using (13) in (12) and then after manipulation, one gets$$\begin{array}{}\text{(16)}& {\overline{\gamma }}^2-\overline{\gamma }1+{\Delta }_2-{\Delta }_1-{\Delta }_1=0\text{,}\end{array}$$whose roots are$$\begin{array}{}\text{(17)}& \overline{\gamma }=\frac{1-{\Delta }_1+{\Delta }_2\pm \sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}{2}\text{.}\end{array}$$

In view of (15) and (17), one can conclude that (8) has fixed points ${\mathrm{\Gamma }}_1={\mathcal{l}}_{11},{\mathcal{l}}_{12}$ and ${\mathrm{\Gamma }}_2={\mathcal{l}}_{21},{\mathcal{l}}_{22}\text{,}$ where ${\mathcal{l}}_{11}$, ${\mathcal{l}}_{12}$, ${\mathcal{l}}_{21}\text{,}$ and ${\mathcal{l}}_{22}$ are depicted in (10).

Next, the linearized form of (8) about $\mathrm{\Gamma }=\overline{\beta },\overline{\gamma }$ under the map $f_1,f_2\mapsto {\beta }_{n+1},{\gamma }_{n+1}$ is$$\begin{array}{}\text{(18)}& {\varpi }_{n+1}≔=J_{\Gamma }{\varpi }_n\text{,}\end{array}$$where$$\begin{array}{c}\text{(19)}& {\varpi }_n=\begin{array}{c}{\beta }_n\\ {\beta }_{n-1}\\ {\gamma }_n\\ {\gamma }_{n-1}\end{array}\text{,}\text{(20)}& J_{\Gamma }=\begin{array}{cccc}0& 0& \frac{{\Delta }_1}{{\overline{\gamma }}^2}& -\frac{2{\Delta }_1}{{\overline{\gamma }}^2}\\ 1& 0& 0& 0\\ \frac{{\Delta }_2}{{\overline{\beta }}^2}& -\frac{2{\Delta }_2}{{\overline{\beta }}^2}& 0& 0\\ 0& 0& 1& 0\end{array}\text{,}\text{(21)}& f_1≔=1+{\Delta }_1\frac{{\gamma }_n}{{\gamma }_{n-1}^2}\text{,}{f}_2≔=1+{\Delta }_2\frac{{\beta }_n}{{\beta }_{n-1}^2}\text{.}\end{array}$$

3. Boundedness and Persistence

In the following theorem, it should be concluded that solution ${{\beta }_n,{\gamma }_n}_{n=-1}^{\infty }$ of (8) is bounded and persists.

Theorem 2.

If$$\begin{array}{}\text{(22)}& {\Delta }_1{\Delta }_2<1\text{,}\end{array}$$

then ${{\beta }_n,{\gamma }_n}_{n=-1}^{\infty }$ of (8) is bounded and persists.

Proof.

If ${{\beta }_n,{\gamma }_n}_{n=-1}^{\infty }$ is the solution of (8), then$$\begin{array}{}\text{(23)}& {\beta }_n\ge 1,{\gamma }_n\ge 1\text{.}\end{array}$$

Moreover, from (8) and (23), one gets$$\begin{array}{}\text{(24)}& {\beta }_{n+1}\le 1+{\Delta }_1+{\Delta }_1{\Delta }_2{\beta }_{n-1},{\gamma }_{n+1}\le 1+{\Delta }_2+{\Delta }_1{\Delta }_2{\gamma }_{n-1}\text{.}\end{array}$$

From the $1^{\text{st}}$ inequality of (24), one gets$$\begin{array}{}\text{(25)}& u_{n+1}=1+{\Delta }_1+{\Delta }_1{\Delta }_2{u}_{n-1}\text{,}\end{array}$$whose solution is$$\begin{array}{}\text{(26)}& u_n=\frac{1+{\Delta }_1}{1-{\Delta }_1{\Delta }_2}+c_1{{\Delta }_1{\Delta }_2}^{n/2}+c_2{-1}^n{{\Delta }_1{\Delta }_2}^{n/2}\text{,}\end{array}$$where constants $c_{i}i=\mathrm{1,2}$ depends on $u_{i}i=-\mathrm{1,0}$. From the $2^{\text{nd}}$ inequality of (24), one gets$$\begin{array}{}\text{(27)}& v_{n+1}=1+{\Delta }_2+{\Delta }_1{\Delta }_2{v}_{n-1}\text{,}\end{array}$$whose solution is$$\begin{array}{}\text{(28)}& v_n=\frac{1+{\Delta }_2}{1-{\Delta }_1{\Delta }_2}+c_3{{\Delta }_1{\Delta }_2}^{n/2}+c_4{-1}^n{{\Delta }_1{\Delta }_2}^{n/2}\text{,}\end{array}$$where constants $c_{i}i=\mathrm{3,4}$ depends on $v_{i}i=-\mathrm{1,0}$. Now, if one considers the solution in which $u_{-1}={\beta }_{-1}$, $u_0={\beta }_0$, $v_{-1}={\gamma }_{-1}$, $v_0={\gamma }_0$ and additionally if ${\Delta }_1{\Delta }_2<1$ holds true, then from (24), (26), and (28), one gets$$\begin{array}{}\text{(29)}& {\beta }_n\le \frac{1+{\Delta }_1}{1-{\Delta }_1{\Delta }_2},{\gamma }_n\le \frac{1+{\Delta }_2}{1-{\Delta }_1{\Delta }_2}\text{.}\end{array}$$

Finally, from (23) and (29), one gets$$\begin{array}{}\text{(30)}& 1\le {\beta }_n\le \frac{1+{\Delta }_1}{1-{\Delta }_1{\Delta }_2},1\le {\gamma }_n\le \frac{1+{\Delta }_2}{1-{\Delta }_1{\Delta }_2}\text{.}\end{array}$$

Theorem 3.

The set $1,1+{\Delta }_1/1-{\Delta }_1{\Delta }_2\times 1,1+{\Delta }_2/1-{\Delta }_1{\Delta }_2$ is an invariant rectangle.

Proof.

If ${{\beta }_n,{\gamma }_n}_{n=-1}^{\infty }$ is the solution of (8) with ${\beta }_{-1},{\beta }_0\in 1,1+{\Delta }_1/1-{\Delta }_1{\Delta }_2$ and ${\gamma }_{-1},{\gamma }_0\in 1,1+{\Delta }_2/1-{\Delta }_1{\Delta }_2$, then$$\begin{array}{c}\text{(31)}& 1\le {\beta }_1=1+{\Delta }_1\frac{{\gamma }_0}{{\gamma }_{-1}^2}\le \frac{1+{\Delta }_1}{1-{\Delta }_1{\Delta }_2}\text{,}1\le {\gamma }_1=1+{\Delta }_2\frac{{\beta }_0}{{\beta }_{-1}^2}\le \frac{1+{\Delta }_2}{1-{\Delta }_1{\Delta }_2}\text{.}\end{array}$$

From (31), it can be concluded that ${\beta }_1\in 1,1+{\Delta }_1/1-{\Delta }_1{\Delta }_2$ and ${\gamma }_1\in 1,1+{\Delta }_2/1-{\Delta }_1{\Delta }_2$. Finally, by induction, it can also be concluded that ${\beta }_{k+1}\in 1,1+{\Delta }_1/1-{\Delta }_1{\Delta }_2$ and ${\gamma }_{k+1}\in 1,1+{\Delta }_2/1-{\Delta }_1{\Delta }_2$ if ${\beta }_k\in 1,1+{\Delta }_1/1-{\Delta }_1{\Delta }_2$ and ${\gamma }_k\in 1,1+{\Delta }_2/1-{\Delta }_1{\Delta }_2$.

4. Global Dynamic Behavior

In the present section, local dynamic behavior about ${\mathrm{\Gamma }}_{\mathrm{1,2}}$ will be explored and motivated from the existing literature [8–16].

Theorem 4.

${\mathrm{\Gamma }}_1$ of system (8) is stable if$$\begin{array}{c}\text{(32)}& \frac{8{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_1<1}\text{,}\frac{8{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_2<1}\text{.}\end{array}$$

Proof.

For ${\mathrm{\Gamma }}_1$, (18) gives$$\begin{array}{}\text{(33)}& {\varpi }_{n+1}≔=J{|}_{{\Gamma }_1}{\varpi }_n\text{,}\end{array}$$where from (20), one gets$$\begin{array}{c}\text{(34)}& J_{{\Gamma }_1}=\begin{array}{cc}0& 0\\ 1& 0\\ \frac{4{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}& \frac{-8{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ 0& 0\end{array}\begin{array}{cc}\frac{4{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}& \frac{-8{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ 0& 0\\ 0& 0\\ 1& 0\end{array}\text{.}\end{array}$$

Now, if $J_{{\mathrm{\Gamma }}_1}$ has eigenvalues ${\lambda }_{l}l=1,\dots ,4$ and the diagonal matrix,$$\begin{array}{}\text{(35)}& P=\begin{array}{cccc}{\zeta }_1& 0& 0& 0\\ 0& {\zeta }_2& 0& 0\\ 0& 0& {\zeta }_3& 0\\ 0& 0& 0& {\zeta }_4\end{array},\end{array}$$with$$\begin{array}{c}\text{(36)}& {\zeta }_1={\zeta }_3=1\text{,}{\zeta }_2={\zeta }_4=1-\epsilon \text{for}0<\epsilon <1\text{,}\end{array}$$and$$\begin{array}{c}\text{(37)}& 0<\epsilon <\min 1-\frac{8{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_1},1-\frac{8{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_2}\text{.}\end{array}$$

Then,$$\begin{array}{c}\text{(38)}& PJ{|}_{{\Gamma }_1}{P}^{-1}=\begin{array}{cc}0& 0\\ {\zeta }_2{\zeta }_1^{-1}& 0\\ \frac{4{\Delta }_2{\zeta }_3{\zeta }_1^{-1}}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}& \frac{-8{\Delta }_2{\zeta }_3{\zeta }_2^{-1}}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ 0& 0\end{array}\begin{array}{cc}\frac{4{\Delta }_1{\zeta }_1{\zeta }_3^{-1}}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}& \frac{-8{\Delta }_1{\zeta }_1{\zeta }_4^{-1}}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ 0& 0\\ 0& 0\\ {\zeta }_4{\zeta }_3^{-1}& 0\end{array}\text{.}\end{array}$$

Also,$$\begin{array}{c}\text{(39)}& {\zeta }_1>{\zeta }_2>0\text{,}{\zeta }_3>{\zeta }_4>0\text{.}\end{array}$$

From (39), one gets$$\begin{array}{c}\text{(40)}& {\zeta }_2{\zeta }_1^{-1}<1\text{,}{\zeta }_4{\zeta }_3^{-1}<1\text{.}\end{array}$$

Equations (36) and (37) yields$$\begin{array}{c}\text{(41)}& \frac{4{\Delta }_1{\zeta }_1{\zeta }_3^{-1}}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}+\frac{8{\Delta }_1{\zeta }_1{\zeta }_4^{-1}}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}=\frac{4{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\times 1+\frac{2}{1-\epsilon }<1\text{,}\frac{4{\Delta }_2{\zeta }_3{\zeta }_1^{-1}}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}+\frac{8{\Delta }_2{\zeta }_3{\zeta }_2^{-1}}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}=\frac{4{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\times 1+\frac{2}{1-\epsilon }<1\text{.}\end{array}$$

Finally, from (40) and (41), one gets the proof of required statement as$$\begin{array}{}\text{(42)}& \underset{1\le k\le 4}{\max }{\lambda }_l=PJ{\Gamma }_1{P}^{-1}=\max {\zeta }_2{\zeta }_1^{-1},\frac{4{\Delta }_1{\zeta }_1{\zeta }_3^{-1}}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}+\frac{8{\Delta }_1{\zeta }_1{\zeta }_4^{-1}}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2},{\zeta }_4{\zeta }_3^{-1},\frac{4{\Delta }_2{\zeta }_3{\zeta }_1^{-1}}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}+\frac{8{\Delta }_2{\zeta }_3{\zeta }_2^{-1}}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}<1\text{.}\end{array}$$

Theorem 5.

${\mathrm{\Gamma }}_2$ of system (8) is a stable if$$\begin{array}{c}\text{(43)}& \frac{8{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_1}<1\text{,}\frac{8{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_2}<1\text{.}\end{array}$$

Proof.

For ${\mathrm{\Gamma }}_2$, from (18), one gets$$\begin{array}{}\text{(44)}& {\varpi }_{n+1}≔=J{|}_{{\Gamma }_2}{\varpi }_n\text{,}\end{array}$$where from (20), one gets$$\begin{array}{c}\text{(45)}& J{\Gamma }_2=\begin{array}{cc}0& 0\\ 1& 0\\ \frac{4{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}& \frac{-8{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ 0& 0\end{array}\begin{array}{cc}\frac{4{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}& \frac{-8{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ 0& 0\\ 0& 0\\ 1& 0\end{array}\text{.}\end{array}$$

Now, it is recall that if $P$ is a diagonal matrix with (36) holds and$$\begin{array}{c}\text{(46)}& 0<\epsilon <\min 1-\frac{8{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_1}\text{,}1-\frac{8{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_2}\text{.}\end{array}$$

Then,$$\begin{array}{c}\text{(47)}& PJ{\Gamma }_2{P}^{-1}=\begin{array}{cc}0& 0\\ {\zeta }_2{\zeta }_1^{-1}& 0\\ \frac{4{\Delta }_2{\zeta }_3{\zeta }_1^{-1}}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}& \frac{-8{\Delta }_2{\zeta }_3{\zeta }_2^{-1}}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ 0& 0\end{array}\begin{array}{cc}\frac{4{\Delta }_1{\zeta }_1{\zeta }_3^{-1}}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}& \frac{-8{\Delta }_1{\zeta }_1{\zeta }_4^{-1}}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ 0& 0\\ 0& 0\\ {\zeta }_4{\zeta }_3^{-1}& 0\end{array}\text{.}\end{array}$$

Moreover, if (40) and (46) holds true, then$$\begin{array}{c}\text{(48)}& \frac{4{\Delta }_1{\zeta }_1{\zeta }_3^{-1}}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}+\frac{8{\Delta }_1{\zeta }_1{\zeta }_4^{-1}}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}=\frac{4{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\times 1+\frac{2}{1-\epsilon }<1\text{,}\\ \frac{4{\Delta }_2{\zeta }_3{\zeta }_1^{-1}}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ +\frac{8{\Delta }_2{\zeta }_3{\zeta }_2^{-1}}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}=\frac{4{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\times 1+\frac{2}{1-\epsilon }<1\text{.}\end{array}$$

Finally, from (40) and (48), one gets the proof of required statement as$$\begin{array}{}\text{(49)}& \underset{1\le k\le 4}{\max }{\lambda }_2=PJ{\Gamma }_2{P}^{-1}=\max {\zeta }_2{\zeta }_1^{-1},\frac{4{\Delta }_1{\zeta }_1{\zeta }_3^{-1}}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}+\frac{8{\Delta }_1{\zeta }_1{\zeta }_4^{-1}}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2},{\zeta }_4{\zeta }_3^{-1},\frac{4{\Delta }_2{\zeta }_3{\zeta }_1^{-1}}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}+\frac{8{\Delta }_2{\zeta }_3{\zeta }_2^{-1}}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}<1\text{.}\end{array}$$

Theorem 6.

If ${\Delta }_1,{\Delta }_2\in 0,1/2\text{,}$ then ${\mathrm{\Gamma }}_1$ of (8) is a global attractor.

Proof.

If ${{\beta }_n,{\gamma }_n}_{n=-1}^{\infty }$ is the solution of discrete system (8) such that ${\lim }_{n⟶\infty }\inf {\beta }_n=l_1$, ${\lim }_{n⟶\infty }\inf {\gamma }_n=l_2$, ${\lim }_{n⟶\infty }\sup {\beta }_n={\beta }_1\text{,}$ and ${\lim }_{n⟶\infty }\sup {\gamma }_n={\beta }_2\text{,}$ then$$\begin{array}{c}\text{(50)}& 1<L_1=\underset{n⟶\infty }{\lim }\inf {\beta }_n,1<L_2=\underset{n⟶\infty }{\lim }\inf {\gamma }_n\text{,}{\beta }_1=\underset{n⟶\infty }{\lim }\sup {\beta }_n<\infty ,{\beta }_2=\underset{n⟶\infty }{\lim }\sup {\gamma }_n<\infty \text{.}\end{array}$$

In view of (8) and (50), one gets$$\begin{array}{}\text{(51)}& {\beta }_1\le 1+{\Delta }_1\frac{{\beta }_2}{L_2^2},L_1\ge 1+{\Delta }_1\frac{L_2}{{\beta }_2^2}\text{,}\text{(52)}& {\beta }_2\le 1+{\Delta }_2\frac{{\beta }_1}{L_1^2},L_2\ge 1+{\Delta }_2\frac{L_1}{{\beta }_1^2}\text{.}\end{array}$$

From the $1^{st}$ and $2^{nd}$ inequalities of (51) and (52), one gets$$\begin{array}{}\text{(53)}& {\beta }_1+{\Delta }_2\frac{L_1}{{\beta }_1}\le {\beta }_1{L}_2\le L_2+{\Delta }_1\frac{{\beta }_2}{L_2}\text{.}\end{array}$$

Similarly, from the $2^{nd}$ and $1^{st}$ inequalities of (51) and (52), one gets$$\begin{array}{}\text{(54)}& {\beta }_2+{\Delta }_1\frac{L_2}{{\beta }_2}\le {\beta }_2{L}_1\le L_1+{\Delta }_2\frac{{\beta }_1}{L_1}\text{.}\end{array}$$

From (53) and (54), one gets$$\begin{array}{}\text{(55)}& {\beta }_1+{\Delta }_2\frac{L_1}{{\beta }_1}+{\beta }_2+{\Delta }_1\frac{L_2}{{\beta }_2}\le L_1+{\Delta }_2\frac{{\beta }_1}{L_1}+L_2+{\Delta }_1\frac{{\beta }_2}{L_2}\text{,}\end{array}$$which implies that$$\begin{array}{}\text{(56)}& {\beta }_1-L_{1}1-{\Delta }_2\frac{1}{L_1}+\frac{\beta 1}{{\beta }_1}+{\beta }_2-L_{2}1-{\Delta }_1\frac{1}{{\beta }_2}+\frac{1}{L_2}\le 0\text{.}\end{array}$$

If ${\Delta }_1,{\Delta }_2\in 0,1/2\text{,}$ then from (56), one can observe that$$\begin{array}{}\text{(57)}& 1-{\Delta }_2\frac{1}{{\beta }_1}+\frac{1}{L_1}>\mathrm{0,1}-{\Delta }_1\frac{1}{{\beta }_2}+\frac{1}{L_2}>0\text{.}\end{array}$$

Finally, from (56), one gets ${\beta }_1=L_1$ and ${\beta }_2=L_2$.

5. Convergence Rate

Theorem 7.

If ${{\beta }_n,{\gamma }_n}_{n=-1}^{\infty }$ is the solution of (8) such that ${\lim }_{n⟶\infty }{\beta }_n=\overline{\beta }$ and ${\lim }_{n⟶\infty }{\gamma }_n=\overline{\gamma }\text{,}$ then$$\begin{array}{}\text{(58)}& {\varphi }_n=\begin{array}{c}{\varphi }_n^1\\ {\varphi }_{n-1}^1\\ {\varphi }_n^2\\ {\varphi }_{n-1}^2\end{array}\text{,}\end{array}$$satisfying the following mathematical relation$$\begin{array}{}\text{(59)}& \underset{n\to \infty }{\lim }\sqrt[n]{{\varphi }_n}=\lambda J\Gamma ,\underset{n\to \infty }{\lim }\frac{{\varphi }_{n+1}}{{\varphi }_n}=\lambda J\Gamma \text{,}\end{array}$$where $\lambda J_{\mathrm{\Gamma }}$ are roots of $J_{\mathrm{\Gamma }}$ at $\mathrm{\Gamma }$.

Proof.

If ${\lim }_{n⟶\infty }{\beta }_n=\overline{\beta }$ and ${\lim }_{n⟶\infty }{\gamma }_n=\overline{\gamma }\text{,}$ then in order for error terms, one gets$$\begin{array}{c}\text{(60)}& {\beta }_{n+1}-\overline{\beta }={\Delta }_1\frac{{\gamma }_n}{{\gamma }_{n-1}^2}-{\Delta }_1\frac{1}{\overline{\gamma }}\text{,}=\frac{{\Delta }_1}{{\gamma }_{n-1}^2}{\gamma }_n-\overline{\gamma }-\frac{{\Delta }_1{\gamma }_{n-1}+\overline{\gamma }}{\overline{\gamma }{\gamma }_{n-1}^2}{\gamma }_{n-1}-\overline{\gamma }\text{,}\\ {\gamma }_{n+1}-\overline{\gamma }={\Delta }_2\frac{{\beta }_n}{{\beta }_{n-1}^2}-{\Delta }_2\frac{1}{\overline{\beta }}\text{,}\\ =\frac{{\Delta }_2}{{\beta }_{n-1}^2}{\beta }_n-\overline{\beta }-\frac{{\Delta }_2{\beta }_{n-1}+\overline{\beta }}{\overline{\beta }{\beta }_{n-1}^2}{\beta }_{n-1}-\overline{\beta }\text{.}\end{array}$$

Set$$\begin{array}{}\text{(61)}& {\varphi }_n^1={\beta }_n-\overline{\beta },{\varphi }_n^2={\gamma }_n-\overline{\gamma }\text{.}\end{array}$$

From (60) and (61), one gets$$\begin{array}{c}\text{(62)}& {\varphi }_{n+1}^1={\alpha }_{11}{\varphi }_n^2+{\alpha }_{12}{\varphi }_{n-1}^2\text{,}{\varphi }_{n+1}^2={\alpha }_{21}{\varphi }_n^1+{\alpha }_{22}{\varphi }_{n-1}^1\text{,}\end{array}$$where$$\begin{array}{c}\text{(63)}& {\alpha }_{11}=\frac{{\Delta }_1}{{\gamma }_{n-1}^2}\text{,}{\alpha }_{12}=-\frac{{\Delta }_1{\gamma }_{n-1}+\overline{\gamma }}{\overline{\gamma }{\gamma }_{n-1}^2}\text{,}\\ {\alpha }_{21}=\frac{{\Delta }_2}{{\beta }_{n-1}^2}\text{,}\\ {\alpha }_{22}=-\frac{{\Delta }_2{\beta }_{n-1}+\overline{\beta }}{\overline{\beta }{\beta }_{n-1}^2}\text{.}\end{array}$$

From (63), one gets$$\begin{array}{c}\text{(64)}& \underset{n⟶\infty }{\lim }{\alpha }_{11}=\frac{{\Delta }_1}{{\overline{\gamma }}^2}\text{,}\underset{n⟶\infty }{\lim }{\alpha }_{12}=-\frac{2{\Delta }_1}{{\overline{\gamma }}^2}\text{,}\\ \underset{n⟶\infty }{\lim }{\alpha }_{21}=\frac{{\Delta }_2}{{\overline{\beta }}^2}\text{,}\\ \underset{n⟶\infty }{\lim }{\alpha }_{22}=-\frac{2{\Delta }_2}{{\overline{\beta }}^2}\text{.}\end{array}$$

That is,$$\begin{array}{c}\text{(65)}& {\alpha }_{11}=\frac{{\Delta }_1}{{\overline{\gamma }}^2}+{\sigma }_{11}\text{,}{\alpha }_{12}=-\frac{2{\Delta }_1}{{\overline{\gamma }}^2}+{\sigma }_{12}\text{,}\\ {\alpha }_{21}=\frac{{\Delta }_2}{{\overline{\beta }}^2}+{\sigma }_{21}\text{,}\\ {\alpha }_{22}=-\frac{2{\Delta }_2}{{\overline{\beta }}^2}+{\sigma }_{22}\text{,}\end{array}$$where ${\sigma }_{11},{\sigma }_{12},{\sigma }_{21},{\sigma }_{22}⟶0$ as $n⟶\infty$. In view of existing literature [17], one gets$$\begin{array}{}\text{(66)}& {\varphi }_{n+1}=A+B_n{\varphi }_n\text{,}\end{array}$$where$$\begin{array}{}\text{(67)}& A=\begin{array}{cccc}0& 0& \frac{{\Delta }_1}{{\overline{\gamma }}^2}& -\frac{2{\Delta }_1}{{\overline{\gamma }}^2}\\ 1& 0& 0& 0\\ \frac{{\Delta }_2}{{\overline{\beta }}^2}& -\frac{2{\Delta }_2}{{\overline{\beta }}^2}& 0& 0\\ 0& 0& 1& 0\end{array}\text{,}\text{(68)}& B_n=\begin{array}{cccc}0& 0& {\sigma }_{11}& {\sigma }_{12}\\ 1& 0& 0& 0\\ {\sigma }_{21}& {\sigma }_{22}& 0& 0\\ 0& 0& 1& 0\end{array}\text{.}\end{array}$$

Therefore, for $\mathrm{\Gamma }\text{,}$ the error system becomes$$\begin{array}{}\text{(69)}& \begin{array}{c}{\varphi }_{n+1}^1\\ {\varphi }_n^1\\ {\varphi }_{n+1}^2\\ {\varphi }_n^2\end{array}=\begin{array}{cccc}0& 0& \frac{{\Delta }_1}{{\overline{\gamma }}^2}& -\frac{2{\Delta }_1}{{\overline{\gamma }}^2}\\ 1& 0& 0& 0\\ \frac{{\Delta }_2}{{\overline{\beta }}^2}& -\frac{2{\Delta }_2}{{\overline{\beta }}^2}& 0& 0\\ 0& 0& 1& 0\end{array}\begin{array}{c}{\varphi }_n^1\\ {\varphi }_{n-1}^1\\ {\varphi }_n^2\\ {\varphi }_{n-1}^2\end{array}\text{,}\end{array}$$which is the same as $J_{\mathrm{\Gamma }}$ at $\mathrm{\Gamma }$. Particularly about ${\mathrm{\Gamma }}_{\mathrm{1,2}}$, (69) becomes$$\begin{array}{c}\text{(70)}& \begin{array}{c}{\varphi }_{n+1}^1\\ {\varphi }_n^1\\ {\varphi }_{n+1}^2\\ {\varphi }_n^2\end{array}=\begin{array}{cc}0& 0\\ 1& 0\\ \frac{4{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}& \frac{-8{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ 0& 0\end{array}\begin{array}{cc}\frac{4{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}& \frac{-8{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ 0& 0\\ 0& 0\\ 1& 0\end{array}\begin{array}{c}{\varphi }_n^1\\ {\varphi }_{n-1}^1\\ {\varphi }_n^2\\ {\varphi }_{n-1}^2\end{array}\text{,}\begin{array}{c}{\varphi }_{n+1}^1\\ {\varphi }_n^1\\ {\varphi }_{n+1}^2\\ {\varphi }_n^2\end{array}=\begin{array}{cc}0& 0\\ 1& 0\\ \frac{4{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}& \frac{-8{\Delta }_2}{{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ 0& 0\\ \frac{4{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}& \frac{-8{\Delta }_1}{{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2}\\ 0& 0\\ 0& 0\\ 1& 0\end{array}\begin{array}{c}{\varphi }_n^1\\ {\varphi }_{n-1}^1\\ {\varphi }_n^2\\ {\varphi }_{n-1}^2\end{array}\text{,}\end{array}$$which are same as $J{|}_{{\mathrm{\Gamma }}_{\mathrm{1,2}}}$ about ${\mathrm{\Gamma }}_{1{,}_2}$, respectively.

6. Numerical Simulations

In order to verify theoretical results numerically, the following cases are presented:

Case 1.

$\text{If\hspace{0.17em}}{\Delta }_1=0.45,{\Delta }_2=0.47\in 0,1/2\text{,}$ then from (22), one gets ${\Delta }_1{\Delta }_2=0.2115<1$, and therefore, from (30), the condition for the existence of boundedness solution, i.e., $1<{\beta }_n<1+{\Delta }_1/1-{\Delta }_1{\Delta }_2=1.8389$ and $1<{\gamma }_n<1+{\Delta }_2/1-{\Delta }_1{\Delta }_2=1.8643$ holds true. Additionally, parametric conditions, which are depicted in (32), under which fixed point ${\mathrm{\Gamma }}_1=\mathrm{1.3327,1.3527}$ of discrete system (8) is stable also holds true, i.e., $8{\Delta }_1/{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_1=0.6523<1$ and $8{\Delta }_2/{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_2=0.7197<1$. Hence, Figures 1(a) and 1(b) imply that interior fixed point ${\mathrm{\Gamma }}_1=\mathrm{1.3327,1.3527}$ 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(s) omitted; refer to PDF]

Case 2.

If ${\Delta }_1=0.32,{\Delta }_2=0.2\in 0,1/2\text{,}$ then from (22), one gets ${\Delta }_1{\Delta }_2=0.064<1$, and therefore from (30), the condition for the existence of boundedness solution, i.e., $1<{\beta }_n<1+{\Delta }_1/1-{\Delta }_1{\Delta }_2=1.4103$ and $1<{\gamma }_n<1+{\Delta }_2/1-{\Delta }_1{\Delta }_2=1.2821$ holds true. Additionally, parametric conditions, which are depicted in (32), under which fixed point ${\mathrm{\Gamma }}_1=\mathrm{1.2767,1.1567}$ of the discrete system (8) is stable also hold true, i.e., $8{\Delta }_1/{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_1=0.6288<1$ and $8{\Delta }_2/{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_2=0.2797<1$. Hence, Figures 2(a) and 2(b) imply that interior fixed point ${\mathrm{\Gamma }}_1=\mathrm{1.2767,1.1567}$ 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(s) omitted; refer to PDF]

Case 3.

Here, we a give counter example which do not satisfy the conclusion of Theorem 4, and hence, this implies that the fixed point ${\mathrm{\Gamma }}_1$ of the discrete system (8) is unstable. If ${\Delta }_1=1.1,{\Delta }_2=1.3\notin 0,1/2\text{,}$ then from (22), one gets ${\Delta }_1{\Delta }_2=1.43>1$. Additionally, it is also important here to be noted that conditions (32) under which ${\mathrm{\Gamma }}_1=\mathrm{1.6083,1.8083}$ of discrete-time difference equation system, which is depicted in (8), is stable does not hold true, i.e., $8{\Delta }_1/{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_1=1.0138>1$ and $8{\Delta }_2/{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_2=2.0208>1$. Hence, Figures 3(a) and 3(b) imply that ${\mathrm{\Gamma }}_1=\mathrm{1.6083,1.8083}$ of system (8) is unstable.

[figure(s) omitted; refer to PDF]

Case 4.

If ${\Delta }_1=1.1,{\Delta }_2=1.3\text{,}$ then Figures 4(a) and 4(b) imply that ${\mathrm{\Gamma }}_2=-0.8083,-0.6083$ of discrete system (8) is stable, i.e., $8{\Delta }_1/{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_1=-3.0138<1$ and $8{\Delta }_2/{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_2=-4.0208<1$ holds true, and therefore, simulation agrees with the theoretical results obtained in Theorem 5.

[figure(s) omitted; refer to PDF]

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 ${\mathrm{\Gamma }}_1={\mathcal{l}}_{11},{\mathcal{l}}_{12}$ and ${\mathrm{\Gamma }}_2={\mathcal{l}}_{21},{\mathcal{l}}_{22}\text{,}$ where ${\mathcal{l}}_{11}$, ${\mathcal{l}}_{12}$, ${\mathcal{l}}_{21}\text{,}$ and ${\mathcal{l}}_{22}$ are depicted in (10). It is also proved that if ${\Delta }_1{\Delta }_2<1\text{,}$ then solution ${{\beta }_n,{\gamma }_n}_{n=-1}^{\infty }$ of discrete system (8) is bounded and persisting, and the set $1,1+{\Delta }_1/1-{\Delta }_1{\Delta }_2\times 1,1+{\Delta }_2/1-{\Delta }_1{\Delta }_2$ is an invariant rectangle. Further, it is investigated that if $8{\Delta }_1/{1-{\Delta }_1+{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_1<1$ and $8{\Delta }_2/{1+{\Delta }_1-{\Delta }_2+\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_2<1\text{,}$ then ${\mathrm{\Gamma }}_1$ of system (8) is stable, and more importantly, it is global attractor if ${\Delta }_1,{\Delta }_2\in 0,1/2$. It is also shown that if $8{\Delta }_1/{1-{\Delta }_1+{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_1<1$ and $8{\Delta }_2/{1+{\Delta }_1-{\Delta }_2-\sqrt{4{\Delta }_2+{1+{\Delta }_1-{\Delta }_2}^2}}^2-4{\Delta }_2<1\text{,}$ then ${\mathrm{\Gamma }}_2$ 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.

References

[1] T. H. Thai, N. A. Dai, P. T. Anh, "Global dynamics of some system of second-order difference equations," Electronic Research Archive, vol. 29 no. 6,DOI: 10.3934/era.2021077, 2021.

[2] E. Bešo, S. Kalabušić, N. Mujić, E. Pilav, "Boundedness of solutions and stability of certain second-order difference equation with quadratic term," Advances in Difference Equations, vol. 2020 no. 1,DOI: 10.1186/s13662-019-2490-9, 2020.

[3] E. Taşdemir, "On the global asymptotic stability of a system of difference equations with quadratic terms," Journal of Applied Mathematics and Computing, vol. 66 no. 1-2, pp. 423-437, DOI: 10.1007/s12190-020-01442-4, 2021.

[4] M. Gümüş, R. Abo-Zeid, "Global behavior of a rational second order difference equation," Journal of Applied Mathematics and Computing, vol. 62 no. 1-2, pp. 119-133, DOI: 10.1007/s12190-019-01276-9, 2020.

[5] I. Okumus, Y. Soykan, "Dynamical behavior of a system of three-dimensional nonlinear difference equations," Advances in Difference Equations, vol. 2018 no. 1,DOI: 10.1186/s13662-018-1667-y, 2018.

[6] S. Kalabušić, M. R. S. Kulenović, M. Mehuljić, "Global dynamics of monotone second order difference equation," Journal of Computational Analysis and Applications, vol. 29 no. 1, pp. 172-184, 2021.

[7] S. Moranjkić, Z. Nurkanović, "Local and global dynamics of certain second-order rational difference equations containing quadratic terms," Advances in Dynamical Systems and Applications, vol. 12 no. 2, pp. 123-157, 2017.

[8] R. P. Agarwal, Difference Equations and Inequalities Inc, 2000.

[9] S. N. Elaydi, An Introduction to Difference Equations, 1996.

[10] E. A. Grove, G. Ladas, Periodicities in Nonlinear Difference Equations, 2004.

[11] M. R. S. Kulenović, G. Ladas, Dynamics of Second-Order Rational Difference Equations with Open Problems and Conjectures, 2001.

[12] E. Camouzis, G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, 2007.

[13] S. Lynch, Dynamical Systems with Applications Using Mathematica, 2007.

[14] C. Robinson, Dynamical Systems, Stability, Symbolic Dynamics, and Chaos, 1999.

[15] H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, 2003.

[16] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, 1993.

[17] M. Pituk, "More on Poincaré’s and Perron’s theorems for difference equations ∗," Journal of Difference Equations and Applications, vol. 8 no. 3, pp. 201-216, DOI: 10.1080/10236190211954, 2002.