Research

In this paper


Motivation and Literature Review.
In an observed evolutionary phenomenon, diference 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 signifcance in mathematical models. One of the important perspectives of these equations is that they are used to study discretization method for diferential equations. Many results obtained through the theory of diference equations are more or less natural discrete form of corresponding fndings of diferential equations, and more specifcally this phenomenon is true in the case of stability theory. However, diference equation's theory is more productive than the corresponding theory of diferential equations. For example, frst-order diferential equations which lead to the origination of simple diference equations may have a term known as appearance of ghost solution or existence of chaotic orbit which appear in the case of higher-order diferential equations. Accordingly, this theory seems to be interesting at present and will assume much greater signifcance in the coming era. In addition, the theory of diference equations is rapidly applicable in diferent disciplines like control theory, computer science, numerical analysis, and fnite mathematics. In the light of above facts, the theory of diference equation is studied as a richly deserved feld. For instance, Tai et al. [1] have studied boundedness and persistence, global behavior, and convergence rate of the following systems of exponential diference equations: with positive B i (i � 1, . . . , 6) and β i , c i (i � −1, 0). Bešo et al. [2] have studied dynamical characteristics of the following diference equations: with positive B i (i � 1, 2) and β i , c i (i � −1, 0). Taşdemir [3] has investigated global dynamics of the following diference equations system with quadratic terms: with positive B i (i � 1, 2) and β i (i � −1, 0). Gümüş and Abo-zeid [4] have studied dynamical characteristics of the following diference equation: with positive B i (i � 1, 2) and β i (i � −1, 0). Okumus and Soykan [5] have studied dynamical characteristics of the following diference equation system: with positive B 1 and β i , c i , δ i (i � −1, 0). Kalabušić et al. [6] have studied dynamical characteristics of the following diference equation: with positive B i (i � 1, . . . , 4) and β i (i � −1, 0). Moranjkić and Nurkanović [7] have studied dynamical characteristics of the following diference equation: with positive B i (i � 1, . . . , 6) and β i (i � −1, 0).

Contributions.
Motivated from the aforementioned studies, the purpose of the present study is to explore dynamical characteristics including the study of fxed points, global analysis, and verifcation of theoretical results of the diference equation system numerically: which is an alternative form of the following diference equation system: by β n � x n /A, c n � y n /C, where Δ 1 � B/AC > 0 and Δ 2 � D/AC > 0 with positive A, B, C, and D and considering β i , c i (i � −1, 0) may be positive or negative.
1.3. Structure of the Paper. Te paper is organized as follows: fxed 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 briefy studied in Sections 4 and 5, respectively. Numerical simulations are presented in Section 6. Te conclusion of the paper and future work are given in Section 7.
In the following theorem, it should be concluded that solution (β n , c n ) ∞ n�−1 of (8) is bounded and persists.

Mathematical Problems in Engineering
Ten, Also, From (39), one gets Equations (36) and (37) yields Mathematical Problems in Engineering 5 Finally, from (40) and (41), one gets the proof of required statement as Proof. For Γ 2 , from (18), one gets where from (20), one gets

Mathematical Problems in Engineering
Now, it is recall that if P is a diagonal matrix with (36) holds and Ten, Moreover, if (40) and (46) holds true, then Mathematical Problems in Engineering 7 Finally, from (40) and (48), one gets the proof of required statement as □ Theorem 6. If Δ 1 , Δ 2 ∈ (0, 1/2), then Γ 1 of (8) is a global attractor.

Convergence Rate
Theorem 7. If (β n , c n ) ∞ n�−1 is the solution of (8) such that lim n⟶∞ β n � β and lim n⟶∞ c n � c, then satisfying the following mathematical relation where λJ Γ are roots of J Γ at Γ.

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

Data Availability
All the data utilized in this article have been included, and the sources from where they were adopted were cited accordingly.

Conflicts of Interest
Te author declares that he has no conficts of interest regarding the publication of this paper.