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.
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:with positive and . Bešo et al. [2] have studied dynamical characteristics of the following difference equations:with positive and . Taşdemir [3] has investigated global dynamics of the following difference equations system with quadratic terms:with positive and . Gümüş and Abo-zeid [4] have studied dynamical characteristics of the following difference equation:with positive and . Okumus and Soykan [5] have studied dynamical characteristics of the following difference equation system:with positive and . Kalabušić et al. [6] have studied dynamical characteristics of the following difference equation:with positive and . Moranjkić and Nurkanović [7] have studied dynamical characteristics of the following difference equation:with positive and .
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:which is an alternative form of the following difference equation system:by where with positive , , , and and considering 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 and where
Proof. If be the fixed point of (8), thenFrom the equation of (11), one getsFrom the equation of (11), one getsUsing (12) in (13), one getswhose roots areNext, using (13) in (12) and then after manipulation, one getswhose roots areIn view of (15) and (17), one can conclude that (8) has fixed points and where , , and are depicted in (10).
Next, the linearized form of (8) about under the map iswhere
3. Boundedness and Persistence
In the following theorem, it should be concluded that solution of (8) is bounded and persists.
Theorem 2. Ifthen of (8) is bounded and persists.
Proof. If is the solution of (8), thenMoreover, from (8) and (23), one getsFrom the inequality of (24), one getswhose solution iswhere constants depends on . From the inequality of (24), one getswhose solution iswhere constants depends on . Now, if one considers the solution in which , , , and additionally if holds true, then from (24), (26), and (28), one getsFinally, from (23) and (29), one gets
Theorem 3. The set is an invariant rectangle.
Proof. If is the solution of (8) with and , thenFrom (31), it can be concluded that and . Finally, by induction, it can also be concluded that and if and .
4. Global Dynamic Behavior
In the present section, local dynamic behavior about will be explored and motivated from the existing literature [8–16].
Theorem 4. of system (8) is stable if
Proof. For , (18) giveswhere from (20), one getsNow, if has eigenvalues and the diagonal matrix,withandThen,Also,From (39), one getsEquations (36) and (37) yieldsFinally, from (40) and (41), one gets the proof of required statement as
Theorem 5. of system (8) is a stable if
Proof. For , from (18), one getswhere from (20), one getsNow, it is recall that if is a diagonal matrix with (36) holds andThen,Moreover, if (40) and (46) holds true, thenFinally, from (40) and (48), one gets the proof of required statement as
Theorem 6. If then of (8) is a global attractor.
Proof. If is the solution of discrete system (8) such that , , and thenIn view of (8) and (50), one getsFrom the and inequalities of (51) and (52), one getsSimilarly, from the and inequalities of (51) and (52), one getsFrom (53) and (54), one getswhich implies thatIf then from (56), one can observe thatFinally, from (56), one gets and .
5. Convergence Rate
Theorem 7. If is the solution of (8) such that and thensatisfying the following mathematical relationwhere are roots of at .
Proof. If and then in order for error terms, one getsSetFrom (60) and (61), one getswhereFrom (63), one getsThat is,where as . In view of existing literature [17], one getswhereTherefore, for the error system becomeswhich is the same as at . Particularly about , (69) becomeswhich are same as about , respectively.
6. Numerical Simulations
In order to verify theoretical results numerically, the following cases are presented:
Case 1. then from (22), one gets , and therefore, from (30), the condition for the existence of boundedness solution, i.e., and holds true. Additionally, parametric conditions, which are depicted in (32), under which fixed point of discrete system (8) is stable also holds true, i.e., and . Hence, Figures 1(a) and 1(b) imply that interior fixed point 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.
(a)
(b)
(c)
Case 2. If then from (22), one gets , and therefore from (30), the condition for the existence of boundedness solution, i.e., and holds true. Additionally, parametric conditions, which are depicted in (32), under which fixed point of the discrete system (8) is stable also hold true, i.e., and . Hence, Figures 2(a) and 2(b) imply that interior fixed point 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.
(a)
(b)
(c)
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 of the discrete system (8) is unstable. If then from (22), one gets . Additionally, it is also important here to be noted that conditions (32) under which of discrete-time difference equation system, which is depicted in (8), is stable does not hold true, i.e., and . Hence, Figures 3(a) and 3(b) imply that of system (8) is unstable.
(a)
(b)
Case 4. If then Figures 4(a) and 4(b) imply that of discrete system (8) is stable, i.e., and holds true, and therefore, simulation agrees with the theoretical results obtained in Theorem 5.
(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 and where , , and are depicted in (10). It is also proved that if then solution of discrete system (8) is bounded and persisting, and the set is an invariant rectangle. Further, it is investigated that if and then of system (8) is stable, and more importantly, it is global attractor if . It is also shown that if and then 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.
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
The author declares that he has no conflicts of interest regarding the publication of this paper.