#### Abstract

In this paper, we explore the global dynamical characteristics, boundedness, and rate of convergence of certain higher-order discrete systems of difference equations. More precisely, it is proved that for all involved respective parameters, discrete systems have a trivial fixed point. We have studied local and global dynamical characteristics at trivial fixed point and proved that trivial fixed point of the discrete systems is globally stable under respective definite parametric conditions. We have also studied boundedness and rate of convergence for under consideration discrete systems. Finally, theoretical results are confirmed numerically. Our findings in this paper are considerably extended and improve existing results in the literature.

#### 1. Introduction

##### 1.1. Motivation and Literature Review

No one can deny the significance of difference equations these days. These equation models not only are the discrete physical phenomenon but also are the integral part of numerical schemes used to solve differential equations. These equations are widely applied in many branches of scientific field. Difference equations describe the phenomenon of discrete dynamical systems and have applications in various branches of science such as statistical problems, resource management, neural networks, ecology, economics, queuing problems, number theory, sociology, physics, engineering, psychology, quanta in radiation, genetics in biology, combinatorial analysis, probability theory, geometry, population dynamics, electrical networks, stochastic time series, and queuing problems [1, 2]. Studying the global characteristics of higher-order nonlinear difference equations or systems of difference equations is a difficult but rewarding task. These findings pave the way for the construction of a basic theory of higher-order difference equations. Recently, many mathematicians have explored the dynamics of difference equation along their system. For instance, Oul et al. [3] investigated the dynamics of the following higher-order system:

Kulenović et al. [4] investigated the dynamic behavior of the difference equation:

Zhang et al. [5] investigated the dynamic behavior of the difference equation system:

Kalabuić et al. [6] investigated the dynamic behavior of the difference equation system:

Kalabuić et al. [7] investigated the dynamic behavior of the difference equation system:

Kalabuić et al. [8] investigated the dynamic behavior of the difference equation system:

Garić-Demirović et al. [9] investigated the dynamic behavior of four distinct difference equation systems. Elsayed [10, 11] studied solutions form of difference equations and their systems. Further, Khan and Qureshi [12] studied the dynamic behavior of a competitive system. DeVault et al. [13] investigated the dynamic behavior of the difference equation:

Abu-Saris and DeVault [14] studied global attractivity of the difference equation:

On the other hand, in recent years, many works have been published that discussed dynamic behavior of difference equation along their systems [15–20]. In continuation of existing study, it is important to mention that dynamical characteristics of the following difference equation have been investigated by Bajo and Liz [21]: where and are real constants. By extending the work of [21], Zhang et al. [22] have explored the dynamical properties of the system: where and are real constants.

##### 1.2. Objective, Contributions, and Novelties

Motivated from the aforementioned studies, the objective of the present study is to investigate the behavior of certain rational systems by extending the work done by [21, 22]: where and are real constants. More precisely, our main finding in this paper includes the following: (i)Exploration of trivial fixed point of discrete systems (11)–(14)(ii)Construction of the corresponding linearized system(iii)Investigation of global dynamics by stability theory(iv)Study of boundedness of positive solution and convergence rate of discrete systems (11)–(14)(v)Validation of obtained results numerically

##### 1.3. Paper Structure

The rest of the paper is structured as follows: In the subsequent section, we explore trivial fixed point and linearized form of discrete systems (11)–(14). Local dynamical characteristics of systems (11)–(14) are investigated in Section 3 while the boundedness for (11)–(14) is explored in Section 4. In Section 5, global dynamics is investigated. Section 6 includes the investigation of rate of converges for said discrete systems. Obtained results are numerically confirmed in Section 7. The conclusion and future work are given in Section 8.

#### 2. Linearized Form and Trivial Fixed Point of Systems (11)–(14)

Linearized form and trivial fixed points of discrete systems (11)–(14) are studied in this section.

##### 2.1. Fixed Point

Obviously, is the trivial fixed point of discrete systems (11)–(14). Now, in the rest of the section, linearized form for discrete systems (11)–(14) is explored.

##### 2.2. Linearized Form of Discrete System (11)

The map for the linearized form of discrete system (11) is where

The linearized form of (11) at under (15) is

where ,

##### 2.3. Linearized Form of Discrete System (12)

Linearized form of discrete system (12) at under the map: is where and

##### 2.4. Linearized Form of Discrete System (13)

Linearized form of discrete system (13) at under the map: is where

##### 2.5. Linearized Form of Discrete System (14)

Linearized form of discrete system (14) at under the map: is where

#### 3. Local Dynamical Characteristics of Discrete Systems (11)–(14)

Hereafter by Theorem 1.1 of [16], local dynamical characteristics about of discrete systems (11)–(14) is explored.

##### 3.1. Local Dynamical Characteristics of Discrete System (11)

Theorem 1. * of discrete system (11) is a sink if
*

*Proof. *At , (17) becomes
From (18), one has
Now, if represent characteristic roots of and diagonal matrix: where
Now,
So,
From (36), one obtains
From (33) and (34), one gets
Finally, from (37) and (38), one gets
From (39), we get the required result.

In a similar way, dynamical characteristics of discrete systems (12)–(14) around can be investigated.

##### 3.2. Local Dynamical Characteristics of Discrete Systems (12)–(14)

Theorem 2. *For local dynamical characteristics about of discrete systems (12)–(14), the following statements hold:
*(i)* of (12) is a sink if**(ii)** of (13) is a sink if**(iii)** of (14) is a sink if*

*Proof. *Its proof is the same as the proof of Theorem 1.

#### 4. Boundedness of Discrete Systems (11)–(14)

The boundedness of discrete systems (11)–(14) is explored in this section.

##### 4.1. Boundedness of Discrete System (11)

Theorem 3. *If is the positive solution of discrete system (11), then for , the following holds:
**Finally,
*

*Proof. *Noticeably, (43), (44), and (45) hold if . If (43), (44), and (45) are true for , i.e.,