Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2020 / Article
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Stability and Bifurcation Analysis of Discrete Dynamical Systems 2020

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Research Article | Open Access

Volume 2020 |Article ID 3696874 | https://doi.org/10.1155/2020/3696874

A. Q. Khan, S. M. Qureshi, "Global Dynamical Properties of Rational Higher-Order System of Difference Equations", Discrete Dynamics in Nature and Society, vol. 2020, Article ID 3696874, 15 pages, 2020. https://doi.org/10.1155/2020/3696874

Global Dynamical Properties of Rational Higher-Order System of Difference Equations

Academic Editor: Rigoberto Medina
Received17 Apr 2020
Accepted05 Jun 2020
Published15 Jul 2020

Abstract

In this paper, global dynamical properties of rational higher-order system are explored in the interior of . It is explored that under certain parametric conditions, the discrete-time system has at most eight equilibria. By the method of linearization, local dynamics has been explored. It is explored that positive solution of the system is bounded, and moreover fixed point is globally stable if , . It is also investigated that the positive solution of the system under consideration converges to . Lastly, theoretical results are confirmed by numerical simulation. The presented work is significantly extended and improves current results in the literature.

1. Introduction

It is a well-known fact that difference equations arise naturally as discrete analogues and as numerical solutions of differential as well as delay differential equations having applications in many fields like physics, biology, economy, and ecology. Recently, a lot of studies have been conducted concerning the global dynamics of difference equations and their systems [319]. It is really not easy to understand global dynamics of difference equations along their systems; particularly, investigating the global behavior of higher-order equations is a challenging job in recent years. Therefore, investigating the global dynamics of such difference equations along their systems is worth further consideration. For illustration, Gibbons et al. [20] explored global dynamics of the following equation:where and are positive constants. Çinar [21] investigated the dynamics of the following equations:where are positive constants. Shojaei et al. [22] investigated global dynamics of the following difference equations:where and are positive constants. Bajo and Liz [23] investigated the dynamics of the following difference equation:where and are positive constants. Zhang et al. [24] have extended the work explored by numerous authors [2123] to investigate the dynamics of the following rational system:where and are positive constants. Recently, Qureshi and Khan [25] investigated the global dynamics of the following rational system, which is extension of the work [2124]:where and are positive constants. Inspired from aforesaid studies, we will extend the work studied by numerous authors [2125] to investigate global dynamics of the following -dimensional system:where and are positive constants.

The organization of this paper is as follows. In Section 2, existence of equilibria in and corresponding linearized form are investigated. Section 3 deals with the study of local dynamics about equilibrium points. Boundedness of positive solution for the discrete-time system is studied in Section 4. Further, global dynamics about is explored in Section 5. In Section 6, we studied the rate of convergence which converges to of the system. Theoretical results are numerically verified in Section 7, while concluding remarks are given in Section 8.

2. Equilibria and Linearized Form of System (7)

The existence of equilibrium solution in the interior of and linearized form about of system (7) are investigated in this section. So, existence of equilibrium solution can be summarized as the following lemma.

Lemma 1. In the interior of , discrete-time system (7) has at most eight equilibria. More precisely,(i), is the unique boundary point of discrete-time system (7).(ii) is the boundary equilibrium point of system (7) if .(iii) is the boundary equilibrium point of system (7) if .(iv) is the boundary equilibrium point of system (7) if .(v) is the boundary equilibrium point of discrete-time system (7) if .(vi) is the boundary equilibrium point of system (7) if .(vii) is the boundary equilibrium point of discrete-time system (7) if .(viii) is the unique positive equilibrium point of system (7) if .

Hereafter, we establish the corresponding linearized form of (7). For this, one has the following map in order to construct the corresponding linearized form:where

Finally, about under map (8) becomes

3. Local Dynamics about Equilibria

By Theorem 1.5 of [1], the detailed local stability analysis about boundary equilibria , , , , , , and and the positive equilibrium point will be investigated in this section.

3.1. Local Dynamics about Boundary Points

Theorem 1. of system (7) is a sink if

Proof. About the equilibrium point , (10) takes the following form:Now, if denotes characteristic roots of , the diagonal matrix , whereNow,So,From (16), one obtainsFrom (13) and (14), one obtainsIn view of (17) and (18), one obtainsFrom (19), one gets the required statement.

Theorem 2. of (7) is unstable.

Proof. About the equilibrium point , (10) takes the following form:whereNow, if denotes characteristic roots of , the diagonal matrix , where (13) holds. Moreover,where . From (16) and (17), we obtainFrom (23), one gets . But , and hence of (7) is unstable.
Similarly, local dynamics about and of system under consideration can be summarized as follows.

Theorem 3. (i) of system (7) is unstable.(ii) of system (7) is unstable.

Proof. Same as proof of Theorems 1 and 2.

Theorem 4. of system (7) is unstable.

Proof. About the equilibrium point , (10) takes the following form:Now, if denotes characteristic roots of , the diagonal matrix , where (13) holds. Moreover,From (16) and (17), we obtainwhich gives of (7) is unstable.
Similarly, local dynamics about of system under consideration can be summarized as follows.

Theorem 5. (i) of system (7) is unstable.(ii) of system (7) is unstable.

Proof. Same as proof of Theorems 1, 2, and 4.

3.2. Local Dynamics about Positive Point

Theorem 6. of (7) is unstable.

Proof. About the equilibrium point , (10) takes the following form:whereand is depicted in (21). Moreover,where . From (16) and (17), we obtainFrom (30), one can conclude that of (7) is unstable.

4. Boundedness

The boundedness of positive solution of (7) is investigated in this section, as follows.

Theorem 7. If is a positive solution of (7), then following holds for :

Proof. For , (31)–(33) are true trivially. Suppose that (31)–(33) are true for , that is,Finally for and using (7), one obtains<