# Dynamic Characteristics of Four Systems of Difference Equations with Higher Order

Authors: Abdul Qadeer Khan; Shahid Mahmood Qureshi; Imtiaz Ahmed
Journal: Advances in Mathematical Physics
 (2021)

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## 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.


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## Body

## 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, Og˘ul et al. [3] investigated the dynamics of the following higher-order system:
(1)xn+1=xn−15±1±xn−3xn−7xn−11xn−15.Kulenović et al. [4] investigated the dynamic behavior of the difference equation:
(2)xn+1=αxn+βxn−1A+xn−1.Zhang et al. [5] investigated the dynamic behavior of the difference equation system:
(3)xn+1=A+xn∑i=1kyn−i,yn+1=B+yn∑i=1kxn−i.Kalabus^ić et al. [6] investigated the dynamic behavior of the difference equation system:
(4)xn+1=α1+β1xnyn,yn+1=α2+γ2ynA2+xn.Kalabus^ić et al. [7] investigated the dynamic behavior of the difference equation system:
(5)xn+1=α1+β1xnA1+yn,yn+1=γ2ynA2+B2xn+yn.Kalabus^ić et al. [8] investigated the dynamic behavior of the difference equation system:
(6)xn+1=β1xnB1xn+yn,yn+1=α2+γ2ynA2+xn.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:
(7)xn+1=Axn+1xn−2.Abu-Saris and DeVault [14] studied global attractivity of the difference equation:
(8)xn+1=A+xnxn−k.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]:
(9)xn+1=xn−1α1+α2∏i=01xn−i,where αℓℓ=1,2 and x−ℓℓ=1,0 are real constants. By extending the work of [21], Zhang et al. [22] have explored the dynamical properties of the system:
(10)xn+1=xn−2α2+∏i=02yn−i,yn+1=yn−2α1+∏i=02xn−i,where αℓℓ=1,2 and x−ℓ,y−ℓℓ=2,1,0 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]:
(11)xn+1=α10yn−kα11+α12∏i=0kzn−i,yn+1=α13zn−kα14+α15∏i=0kxn−i,zn+1=α16xn−kα17+α18∏i=0kyn−i,(12)xn+1=α19zn−kα20+α21∏i=0kxn−i,yn+1=α22xn−kα23+α24∏i=0kyn−i,zn+1=α25yn−kα26+α27∏i=0kzn−i,(13)xn+1=α28yn−kα29+α30∏i=0kyn−i,yn+1=α31zn−kα32+α33∏i=0kzn−i,zn+1=α34xn−kα35+α36∏i=0kxn−i,(14)xn+1=α37zn−kα38+α39∏i=0kzn−i,yn+1=α40xn−kα41+α42∏i=0kxn−i,zn+1=α43yn−kα44+α45∏i=0kyn−i,where αℓℓ=10,⋯,45 and x−ℓ,y−ℓ,z−ℓℓ=−k,−k+1,⋯,1,0 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.


## 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, Og˘ul et al. [3] investigated the dynamics of the following higher-order system:
(1)xn+1=xn−15±1±xn−3xn−7xn−11xn−15.Kulenović et al. [4] investigated the dynamic behavior of the difference equation:
(2)xn+1=αxn+βxn−1A+xn−1.Zhang et al. [5] investigated the dynamic behavior of the difference equation system:
(3)xn+1=A+xn∑i=1kyn−i,yn+1=B+yn∑i=1kxn−i.Kalabus^ić et al. [6] investigated the dynamic behavior of the difference equation system:
(4)xn+1=α1+β1xnyn,yn+1=α2+γ2ynA2+xn.Kalabus^ić et al. [7] investigated the dynamic behavior of the difference equation system:
(5)xn+1=α1+β1xnA1+yn,yn+1=γ2ynA2+B2xn+yn.Kalabus^ić et al. [8] investigated the dynamic behavior of the difference equation system:
(6)xn+1=β1xnB1xn+yn,yn+1=α2+γ2ynA2+xn.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:
(7)xn+1=Axn+1xn−2.Abu-Saris and DeVault [14] studied global attractivity of the difference equation:
(8)xn+1=A+xnxn−k.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]:
(9)xn+1=xn−1α1+α2∏i=01xn−i,where αℓℓ=1,2 and x−ℓℓ=1,0 are real constants. By extending the work of [21], Zhang et al. [22] have explored the dynamical properties of the system:
(10)xn+1=xn−2α2+∏i=02yn−i,yn+1=yn−2α1+∏i=02xn−i,where αℓℓ=1,2 and x−ℓ,y−ℓℓ=2,1,0 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]:
(11)xn+1=α10yn−kα11+α12∏i=0kzn−i,yn+1=α13zn−kα14+α15∏i=0kxn−i,zn+1=α16xn−kα17+α18∏i=0kyn−i,(12)xn+1=α19zn−kα20+α21∏i=0kxn−i,yn+1=α22xn−kα23+α24∏i=0kyn−i,zn+1=α25yn−kα26+α27∏i=0kzn−i,(13)xn+1=α28yn−kα29+α30∏i=0kyn−i,yn+1=α31zn−kα32+α33∏i=0kzn−i,zn+1=α34xn−kα35+α36∏i=0kxn−i,(14)xn+1=α37zn−kα38+α39∏i=0kzn−i,yn+1=α40xn−kα41+α42∏i=0kxn−i,zn+1=α43yn−kα44+α45∏i=0kyn−i,where αℓℓ=10,⋯,45 and x−ℓ,y−ℓ,z−ℓℓ=−k,−k+1,⋯,1,0 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,P0=0,0,0 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
(15)β11,βn11,⋯,βn−k11,γ11,γn11,⋯,γn−k11,δ11,δn11,⋯,δn−k11↦Γ^=xn+1,xn,⋯,xn−k+1,yn+1,yn,⋯,yn−k+1,zn+1,zn,⋯,zn−k+1,where
(16)β11=α10yn−kα11+α12∏i=0kzn−i,βn11=xn,⋯,βn−k11=xn−k+1,γ11=α13zn−kα14+α15∏i=0kxn−i,γn11=yn,⋯,γn−k11=yn−k+1,δ11=α16xn−kα17+α18∏i=0kyn−i,δn11=zn,⋯,δn−k11=zn−k+1.The linearized form of (11) at Λ under (15) is
(17)Γn+1∧11=JΛΓn∧11,whereΓn∧11=xn,xn−1,⋯,xn−k,yn,yn−1,⋯,yn−k,zn,zn−1,⋯,zn−kT,
(18)J∧=00⋯0010⋯00⋮⋮⋱⋮⋮00⋯10−α13α15zxkα14+α15xk+12−α13α15zxkα14+α15xk+12⋯−α13α15zxkα14+α15xk+12−α13α15zxkα14+α15xk+1200⋯00⋮⋮⋱⋮⋮00⋯0000⋯0α16α17+α18yk+100⋯00⋮⋮⋱⋮⋮00⋯0000⋯0α10α11+α12zk+100⋯00⋮⋮⋱⋮⋮00⋯0000⋯0010⋯00⋮⋮⋱⋮⋮00⋯10−α16α18xykα17+α18yk+12−α16α18xykα17+α18yk+12⋯−α16α18xykα17+α18yk+12−α16α18xykα17+α18yk+1200⋯00⋮⋮⋱⋮⋮00⋯00−α10α12yzkα11+α12zk+12−α10α12yzkα11+α12zk+12⋯−α10α12yzkα11+α12zk+12−α10α12yzkα11+α12zk+1200⋯00⋮⋮⋱⋮⋮00⋯0000⋯0α13α14+α15xk+100⋯00⋮⋮⋱⋮⋮00⋯0000⋯0010⋯00⋮⋮⋱⋮⋮00⋯10,(19)Λ=x,y,z.


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

Linearized form of discrete system (12) at Λ under the map:
(20)β12,βn12,⋯,βn−k12,γ12,γn12,⋯,γn−k12,δ12,δn12,⋯,δn−k12↦Γ^,is
(21)Γn+1∧12=JΛΓn∧12,where
(22)J∧=−α19α21zxkα20+α21xk+12−α19α21zxkα20+α21xk+12⋯−α19α21zxkα20+α21xk+12−α19α21zxkα20+α21xk+1210⋯00⋮⋮⋱⋮⋮00⋯1000⋯0α22α23+α24yk+100⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯00−α22α24xykα23+α24yk+12−α22α24xykα23+α24yk+12⋯−α22α24xykα23+α24yk+12−α22α24xykα23+α24yk+1210⋯00⋮⋮⋱⋮⋮00⋯1000⋯0α25α26+α27zk+100⋯00⋮⋮⋱⋮⋮00⋯0000⋯0α19α20+α21zk+100⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯00−α25α27yzkα26+α27zk+12−α25α27yzkα26+α27zk+12⋯−α25α27yzkα26+α27zk+12−α25α27yzkα26+α27zk+1210⋯00⋮⋮⋱⋮⋮00⋯10,and
(23)β12=α19zn−kα20+α21∏i=0kxn−i,βn12=xn,⋯,βn−k12=xn−k+1,γ12=α22xn−kα23+α24∏i=0kyn−i,γn12=yn,⋯,γn−k12=yn−k+1,δ12=α25yn−kα26+α27∏i=0kzn−i,δn12=zn,⋯,δn−k12=zn−k+1.


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

Linearized form of discrete system (13) at Λ under the map:
(24)β13,βn13,⋯,βn−k13,γ13,γn13,⋯,γn−k13,δ13,δn13,⋯,δn−k13↦Γ^,is
(25)Γn+1∧13=JΛΓn∧13,where
(26)J∧=00⋯0010⋯00⋮⋮⋱⋮⋮00⋯1000⋯0000⋯00⋮⋮⋱⋮⋮00⋯00−α34α36xk+1α35+α36xk+12−α34α36xk+1α35+α36xk+12⋯−α34α36xk+1α35+α36xk+12α34α35α35+α36xk+1200⋯00⋮⋮⋱⋮⋮00⋯00−α28α30yk+1α29+α30yk+12−α28α30yk+1α29+α30yk+12⋯−α28α30yk+1α29+α30yk+12α28α29α29+α30yk+1200⋯00⋮⋮⋱⋮⋮00⋯0000⋯0010⋯00⋮⋮⋱⋮⋮00⋯1000⋯0000⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯00−α31α33zk+1α32+α33zk+12−α31α33zk+1α32+α33zk+12⋯−α31α33zk+1α32+α33zk+12α31α32α32+α33zk+1200⋯00⋮⋮⋱⋮⋮00⋯0000⋯0010⋯00⋮⋮⋱⋮⋮00⋯10,β13=α28yn−kα29+α30∏i=0kyn−i,βn13=xn,⋯,βn−k13=xn−k+1,γ13=α31zn−kα32+α33∏i=0kzn−i,γn13=yn,⋯,γn−k13=yn−k+1,δ13=α34xn−kα35+α36∏i=0kxn−i,δn13=zn,⋯,δn−k13=zn−k+1.


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

Linearized form of discrete system (14) at Λ under the map:
(27)β14,βn14,⋯,βn−k14,γ14,γn14,⋯,γn−k14,δ14,δn14,⋯,δn−k14↦Γ^,is
(28)Γn+1∧14=JΛΓn∧14,where
(29)JΛ=00⋯0010⋯00⋮⋮⋱⋮⋮00⋯10−α40α42xk+1α41+α42xk+12−α40α42xk+1α41+α42xk+12⋯−α40α42xk+1α41+α42xk+12α40α41α41+α42xk+1200⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯0000⋯0010⋯00⋮⋮⋱⋮⋮00⋯10−α43α45yk+1α44+α45yk+12−α43α45yk+1α44+α45yk+12⋯−α43α45yk+1α44+α45yk+12α43α44α44+α45yk+1200⋯00⋮⋮⋱⋮⋮00⋯00−α37α39zk+1α38+α39zk+12−α37α39zk+1α38+α39zk+12⋯−α37α39zk+1α38+α39zk+12α37α38α38+α39zk+1200⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯0000⋯0010⋯00⋮⋮⋱⋮⋮00⋯10,β14=α37zn−kα38+α39∏i=0kzn−i,βn14=xn,⋯,βn−k14=xn−k+1,γ14=α40xn−kα41+α42∏i=0kxn−i,γn14=yn,⋯,γn−k14=yn−k+1,δ14=α43yn−kα43+α45∏i=0kyn−i,δn14=zn,⋯,δn−k14=zn−k+1.


## 2.1. Fixed Point

Obviously,P0=0,0,0 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
(15)β11,βn11,⋯,βn−k11,γ11,γn11,⋯,γn−k11,δ11,δn11,⋯,δn−k11↦Γ^=xn+1,xn,⋯,xn−k+1,yn+1,yn,⋯,yn−k+1,zn+1,zn,⋯,zn−k+1,where
(16)β11=α10yn−kα11+α12∏i=0kzn−i,βn11=xn,⋯,βn−k11=xn−k+1,γ11=α13zn−kα14+α15∏i=0kxn−i,γn11=yn,⋯,γn−k11=yn−k+1,δ11=α16xn−kα17+α18∏i=0kyn−i,δn11=zn,⋯,δn−k11=zn−k+1.The linearized form of (11) at Λ under (15) is
(17)Γn+1∧11=JΛΓn∧11,whereΓn∧11=xn,xn−1,⋯,xn−k,yn,yn−1,⋯,yn−k,zn,zn−1,⋯,zn−kT,
(18)J∧=00⋯0010⋯00⋮⋮⋱⋮⋮00⋯10−α13α15zxkα14+α15xk+12−α13α15zxkα14+α15xk+12⋯−α13α15zxkα14+α15xk+12−α13α15zxkα14+α15xk+1200⋯00⋮⋮⋱⋮⋮00⋯0000⋯0α16α17+α18yk+100⋯00⋮⋮⋱⋮⋮00⋯0000⋯0α10α11+α12zk+100⋯00⋮⋮⋱⋮⋮00⋯0000⋯0010⋯00⋮⋮⋱⋮⋮00⋯10−α16α18xykα17+α18yk+12−α16α18xykα17+α18yk+12⋯−α16α18xykα17+α18yk+12−α16α18xykα17+α18yk+1200⋯00⋮⋮⋱⋮⋮00⋯00−α10α12yzkα11+α12zk+12−α10α12yzkα11+α12zk+12⋯−α10α12yzkα11+α12zk+12−α10α12yzkα11+α12zk+1200⋯00⋮⋮⋱⋮⋮00⋯0000⋯0α13α14+α15xk+100⋯00⋮⋮⋱⋮⋮00⋯0000⋯0010⋯00⋮⋮⋱⋮⋮00⋯10,(19)Λ=x,y,z.


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

Linearized form of discrete system (12) at Λ under the map:
(20)β12,βn12,⋯,βn−k12,γ12,γn12,⋯,γn−k12,δ12,δn12,⋯,δn−k12↦Γ^,is
(21)Γn+1∧12=JΛΓn∧12,where
(22)J∧=−α19α21zxkα20+α21xk+12−α19α21zxkα20+α21xk+12⋯−α19α21zxkα20+α21xk+12−α19α21zxkα20+α21xk+1210⋯00⋮⋮⋱⋮⋮00⋯1000⋯0α22α23+α24yk+100⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯00−α22α24xykα23+α24yk+12−α22α24xykα23+α24yk+12⋯−α22α24xykα23+α24yk+12−α22α24xykα23+α24yk+1210⋯00⋮⋮⋱⋮⋮00⋯1000⋯0α25α26+α27zk+100⋯00⋮⋮⋱⋮⋮00⋯0000⋯0α19α20+α21zk+100⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯00−α25α27yzkα26+α27zk+12−α25α27yzkα26+α27zk+12⋯−α25α27yzkα26+α27zk+12−α25α27yzkα26+α27zk+1210⋯00⋮⋮⋱⋮⋮00⋯10,and
(23)β12=α19zn−kα20+α21∏i=0kxn−i,βn12=xn,⋯,βn−k12=xn−k+1,γ12=α22xn−kα23+α24∏i=0kyn−i,γn12=yn,⋯,γn−k12=yn−k+1,δ12=α25yn−kα26+α27∏i=0kzn−i,δn12=zn,⋯,δn−k12=zn−k+1.


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

Linearized form of discrete system (13) at Λ under the map:
(24)β13,βn13,⋯,βn−k13,γ13,γn13,⋯,γn−k13,δ13,δn13,⋯,δn−k13↦Γ^,is
(25)Γn+1∧13=JΛΓn∧13,where
(26)J∧=00⋯0010⋯00⋮⋮⋱⋮⋮00⋯1000⋯0000⋯00⋮⋮⋱⋮⋮00⋯00−α34α36xk+1α35+α36xk+12−α34α36xk+1α35+α36xk+12⋯−α34α36xk+1α35+α36xk+12α34α35α35+α36xk+1200⋯00⋮⋮⋱⋮⋮00⋯00−α28α30yk+1α29+α30yk+12−α28α30yk+1α29+α30yk+12⋯−α28α30yk+1α29+α30yk+12α28α29α29+α30yk+1200⋯00⋮⋮⋱⋮⋮00⋯0000⋯0010⋯00⋮⋮⋱⋮⋮00⋯1000⋯0000⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯00−α31α33zk+1α32+α33zk+12−α31α33zk+1α32+α33zk+12⋯−α31α33zk+1α32+α33zk+12α31α32α32+α33zk+1200⋯00⋮⋮⋱⋮⋮00⋯0000⋯0010⋯00⋮⋮⋱⋮⋮00⋯10,β13=α28yn−kα29+α30∏i=0kyn−i,βn13=xn,⋯,βn−k13=xn−k+1,γ13=α31zn−kα32+α33∏i=0kzn−i,γn13=yn,⋯,γn−k13=yn−k+1,δ13=α34xn−kα35+α36∏i=0kxn−i,δn13=zn,⋯,δn−k13=zn−k+1.


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

Linearized form of discrete system (14) at Λ under the map:
(27)β14,βn14,⋯,βn−k14,γ14,γn14,⋯,γn−k14,δ14,δn14,⋯,δn−k14↦Γ^,is
(28)Γn+1∧14=JΛΓn∧14,where
(29)JΛ=00⋯0010⋯00⋮⋮⋱⋮⋮00⋯10−α40α42xk+1α41+α42xk+12−α40α42xk+1α41+α42xk+12⋯−α40α42xk+1α41+α42xk+12α40α41α41+α42xk+1200⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯0000⋯0010⋯00⋮⋮⋱⋮⋮00⋯10−α43α45yk+1α44+α45yk+12−α43α45yk+1α44+α45yk+12⋯−α43α45yk+1α44+α45yk+12α43α44α44+α45yk+1200⋯00⋮⋮⋱⋮⋮00⋯00−α37α39zk+1α38+α39zk+12−α37α39zk+1α38+α39zk+12⋯−α37α39zk+1α38+α39zk+12α37α38α38+α39zk+1200⋯00⋮⋮⋱⋮⋮00⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯0000⋯0010⋯00⋮⋮⋱⋮⋮00⋯10,β14=α37zn−kα38+α39∏i=0kzn−i,βn14=xn,⋯,βn−k14=xn−k+1,γ14=α40xn−kα41+α42∏i=0kxn−i,γn14=yn,⋯,γn−k14=yn−k+1,δ14=α43yn−kα43+α45∏i=0kyn−i,δn14=zn,⋯,δn−k14=zn−k+1.


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

Hereafter by Theorem 1.1 of [16], local dynamical characteristics about P0 of discrete systems (11)–(14) is explored.


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

Theorem 1.
P0 of discrete system (11) is a sink if
(30)α10α11<1,α13α14<1,α16α17<1.Proof.
AtP0, (17) becomes
(31)Γn+1∧11=JP0Γn∧11.
From (18), one has
(32)B1=JP0=00⋯0000⋯0α10α1100⋯0010⋯0000⋯0000⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯1000⋯0000⋯0000⋯0000⋯0000⋯0α13α1400⋯0010⋯0000⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯0000⋯1000⋯0000⋯0α16α1700⋯0000⋯0000⋯0000⋯0010⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯0000⋯0000⋯10.
Now, ifσ1,σ2,σ3,⋯,σk+1,⋯,σ2k+1,⋯,σ3k+3 represent characteristic roots of P0 and diagonal matrix: Ω^=diagϱ1,ϱ2,⋯,ϱk+1,⋯,ϱ2k+1,⋯,ϱ3k+3 where
(33)ϱ1=ϱk+2=ϱ2k+3=1,ϱ1+ς=ϱk+2+ς=ϱ2k+3+ς=1−ςε,1≤ς≤kwhere0<ε<1,(34)0<ε<min1k1−α10α11,1k1−α13α14,1k1−α16α17.
Now,(35)Ω^B1Ω∧−1=00⋯0000⋯0ϱ1ϱ2k+2−1α10α11ϱ2ϱ1−10⋯0000⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯ϱk+1ϱk−1000⋯0000⋯0000⋯0000⋯00ϱk+3ϱk+2−10⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯0000⋯ϱ2k+2ϱ2k+1−1000⋯0ϱ2k+3ϱk+1−1α16α1700⋯0000⋯0000⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯0000⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯0000⋯0ϱk+2ϱ3k+3−1α13α1400⋯00⋮⋮⋱⋮⋮00⋯00000⋯0ϱ3k+2ϱ3k+1−10⋯00⋮⋮⋱⋮⋮00⋯ϱ3k+3ϱ3k+2−10.
So,(36)0<ϱk+1<⋯<ϱ2<ϱ1,0<ϱ2k+2<⋯<ϱk+2,0<ϱ3k+3<⋯<ϱ2k+3.
From (36), one obtains
(37)ϱ2ϱ1−1<1,⋯,ϱk+1ϱk−1<1,ϱk+3ϱk+2−1<1,⋯,ϱ2k+2ϱ2k+1−1<1,ϱ2k+4ϱ2k+3−1<1,⋯,ϱ3k+3ϱ3k+2−1<1.
From (33) and (34), one gets
(38)ϱ1ϱ2k+2−1α10α11=ϱ2k+2−1α10α11=α10α1111−kε<1,ϱk+2ϱ3k+3−1α13α14=ϱ3k+3−1α13α14=α13α1411−kε<1,ϱ2k+3ϱk+1−1α16α17=ϱk+1−1α16α17=α16α1711−kε<1.
Finally, from (37) and (38), one gets
(39)max1≤υ≤3k+3ρυ=Ω^B1Ω∧−1=maxϱ1ϱ2k+2−1α10α11,ϱ2ϱ1−1,⋯,ϱk+1ϱk−1,ϱk+3ϱk+2−1,⋯,ϱ2k+2ϱ2k+1−1,ϱ3k+2ϱ3k+1−1,⋯,ϱ3k+3ϱ3k+2−1,ϱk+2ϱ3k+3−1α13α14,ϱ2k+3ϱk+1−1α16α17<1.
From (39), we get the required result.In a similar way, dynamical characteristics of discrete systems (12)–(14) around P0 can be investigated.


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

Theorem 2.
For local dynamical characteristics aboutP0 of discrete systems (12)–(14), the following statements hold:
(i)
P0 of (12) is a sink if(40)α19α20<1,α22α23<1,α25α26<1(ii)
P0 of (13) is a sink if(41)α28α29<1,α31α32<1,α34α35<1(iii)
P0 of (14) is a sink if(42)α37α38<1,α40α41<1,α43α44<1Proof.
Its proof is the same as the proof of Theorem1.


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

Theorem 1.
P0 of discrete system (11) is a sink if
(30)α10α11<1,α13α14<1,α16α17<1.Proof.
AtP0, (17) becomes
(31)Γn+1∧11=JP0Γn∧11.
From (18), one has
(32)B1=JP0=00⋯0000⋯0α10α1100⋯0010⋯0000⋯0000⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯1000⋯0000⋯0000⋯0000⋯0000⋯0α13α1400⋯0010⋯0000⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯0000⋯1000⋯0000⋯0α16α1700⋯0000⋯0000⋯0000⋯0010⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯0000⋯0000⋯10.
Now, ifσ1,σ2,σ3,⋯,σk+1,⋯,σ2k+1,⋯,σ3k+3 represent characteristic roots of P0 and diagonal matrix: Ω^=diagϱ1,ϱ2,⋯,ϱk+1,⋯,ϱ2k+1,⋯,ϱ3k+3 where
(33)ϱ1=ϱk+2=ϱ2k+3=1,ϱ1+ς=ϱk+2+ς=ϱ2k+3+ς=1−ςε,1≤ς≤kwhere0<ε<1,(34)0<ε<min1k1−α10α11,1k1−α13α14,1k1−α16α17.
Now,(35)Ω^B1Ω∧−1=00⋯0000⋯0ϱ1ϱ2k+2−1α10α11ϱ2ϱ1−10⋯0000⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯ϱk+1ϱk−1000⋯0000⋯0000⋯0000⋯00ϱk+3ϱk+2−10⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯0000⋯ϱ2k+2ϱ2k+1−1000⋯0ϱ2k+3ϱk+1−1α16α1700⋯0000⋯0000⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯0000⋯0000⋯0000⋯00⋮⋮⋱⋮⋮00⋯0000⋯0ϱk+2ϱ3k+3−1α13α1400⋯00⋮⋮⋱⋮⋮00⋯00000⋯0ϱ3k+2ϱ3k+1−10⋯00⋮⋮⋱⋮⋮00⋯ϱ3k+3ϱ3k+2−10.
So,(36)0<ϱk+1<⋯<ϱ2<ϱ1,0<ϱ2k+2<⋯<ϱk+2,0<ϱ3k+3<⋯<ϱ2k+3.
From (36), one obtains
(37)ϱ2ϱ1−1<1,⋯,ϱk+1ϱk−1<1,ϱk+3ϱk+2−1<1,⋯,ϱ2k+2ϱ2k+1−1<1,ϱ2k+4ϱ2k+3−1<1,⋯,ϱ3k+3ϱ3k+2−1<1.
From (33) and (34), one gets
(38)ϱ1ϱ2k+2−1α10α11=ϱ2k+2−1α10α11=α10α1111−kε<1,ϱk+2ϱ3k+3−1α13α14=ϱ3k+3−1α13α14=α13α1411−kε<1,ϱ2k+3ϱk+1−1α16α17=ϱk+1−1α16α17=α16α1711−kε<1.
Finally, from (37) and (38), one gets
(39)max1≤υ≤3k+3ρυ=Ω^B1Ω∧−1=maxϱ1ϱ2k+2−1α10α11,ϱ2ϱ1−1,⋯,ϱk+1ϱk−1,ϱk+3ϱk+2−1,⋯,ϱ2k+2ϱ2k+1−1,ϱ3k+2ϱ3k+1−1,⋯,ϱ3k+3ϱ3k+2−1,ϱk+2ϱ3k+3−1α13α14,ϱ2k+3ϱk+1−1α16α17<1.
From (39), we get the required result.In a similar way, dynamical characteristics of discrete systems (12)–(14) around P0 can be investigated.


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

Theorem 2.
For local dynamical characteristics aboutP0 of discrete systems (12)–(14), the following statements hold:
(i)
P0 of (12) is a sink if(40)α19α20<1,α22α23<1,α25α26<1(ii)
P0 of (13) is a sink if(41)α28α29<1,α31α32<1,α34α35<1(iii)
P0 of (14) is a sink if(42)α37α38<1,α40α41<1,α43α44<1Proof.
Its proof is the same as the proof of Theorem1.


## 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.
IfXn=xn,yn,zn is the positive solution of discrete system (11), then for μ≥0, the following holds:
(43)0<xn≤α10α11μ+1α13α14μα16α17μy−k,ifn=3+3kμ+1,α10α11μ+1α13α14μα16α17μy1−k,ifn=3+3kμ+2,⋮α10α11μ+1α13α14μα16α17μy−1,ifn=3+3kμ+k,α10α11μ+1α13α14μα16α17μy0,ifn=3+3kμ+k+1,α10α11μ+1α13α14μ+1α16α17μz−k,ifn=3+3kμ+k+2,α10α11μ+1α13α14μ+1α16α17μz1−k,ifn=3+3kμ+k+3,⋮α10α11μ+1α13α14μ+1α16α17μz−1,ifn=3+3kμ+2k+1,α10α11μ+1α13α14μ+1α16α17μz0,ifn=3+3kμ+2k+2,α10α11α13α14α16α17μ+1x−k,ifn=3+3kμ+2k+3,α10α11α13α14α16α17μ+1x1−k,ifn=3+3kμ+2k+4,⋮α10α11α13α14α16α17μ+1x−1,ifn=3+3kμ+3k+2,α10α11α13α14α16α17μ+1x0,ifn=3+3kμ+3k+3.(44)0<yn≤α10α11μα13α14μ+1α16α17μz−k,ifn=3+3kμ+1,α10α11μα13α14μ+1α16α17μz1−k,ifn=3+3kμ+2,⋮α10α11μα13α14μ+1α16α17μz−1ifn=3+3kμ+k,α10α11μα13α14μ+1α16α17μz0,ifn=3+3kμ+k+1,α10α11μα13α14μ+1α16α17μ+1x−k,ifn=3+3kμ+k+2,α10α11α13α14μ+1α16α17μ+1x1−k,ifn=3+3kμ+k+3,⋮α10α11μα13α14μ+1α16α17μx−1,ifn=3+3kμ+2k+1,α10α11μα13α14μ+1α16α17μx0,ifn=3+3kμ+2k+2,α10α11α13α14α16α17μ+1y−k,ifn=3+3kμ+2k+3,α10α11α13α14α16α17μ+1y1−k,ifn=3+3kμ+2k+4,⋮α10α11α13α14α16α17μ+1y−1,ifn=3+3kμ+3k+2,α10α11α13α14α16α17μ+1y0,ifn=3+3kμ+3k+3.
Finally,(45)0<zn≤α10α11μα13α14μα16α17μ+1x−k,ifn=3+3kμ+1,α10α11μα13α14μα16α17μ+1x1−k,ifn=3+3kμ+2,⋮α10α11μα13α14μα16α17μ+1x−1,ifn=3+3kμ+k,α10α11μα13α14μα16α17μ+1x0,ifn=3+3kμ+k+1,α10α11μ+1α13α14μα16α17μ+1y−k,ifn=3+3kμ+k+2,α10α11μ+1α13α14μα16α17μ+1y1−k,ifn=3+3kμ+k+3,⋮α10α11μ+1α13α14μα16α17μ+1y−1,ifn=3+3kμ+2k+1,α10α11μ+1α13α14μα16α17μ+1y0,ifn=3+3kμ+2k+2,α10α11α13α14α16α17μ+1z−k,ifn=3+3kμ+2k+3,α10α11α13α14α16α17μ+1z1−k,ifn=3+3kμ+2k+4,⋮α10α11α13α14α16α17μ+1z−1,ifn=3+3kμ+3k+2,α10α11α13α14α16α17μ+1z0,ifn=3+3kμ+3k+3.Proof.
Noticeably, (43), (44), and (45) hold if μ=0. If (43), (44), and (45) are true for μ=σ, i.e.,
(46)0<xn≤α10α11σ+1α13α14σα16α17σy−k,ifn=3+3kσ+1,α10α11σ+1α13α14σα16α17σy1−k,ifn=3+3kσ+2,⋮α10α11σ+1α13α14σα16α17σy−1,ifn=3+3kσ+k,α10α11σ+1α13α14σα16α17σy0,ifn=3+3kσ+k+1,α10α11σ+1α13α14σ+1α16α17σz−k,ifn=3+3kσ+k+2,α10α11σ+1α13α14σ+1α16α17σz1−k,ifn=3+3kσ+k+3,⋮α10α11σ+1α13α14σ+1α16α17σz−1,ifn=3+3kσ+2k+1,α10α11σ+1α13α14σ+1α16α17σz0,ifn=3+3kσ+2k+2,α10α11α13α14α16α17σ+1x−k,ifn=3+3kσ+2k+3,α10α11α13α14α16α17σ+1x1−k,ifn=3+3kσ+2k+4,⋮α10α11α13α14α16α17σ+1x−1,ifn=3+3kσ+3k+2,α10α11α13α14α16α17σ+1x0,ifn=3+3kσ+3k+3.0<yn≤α10α11σα13α14σ+1α16α17σz−k,ifn=3+3kσ+1,α10α11σα13α14σ+1α16α17σz1−k,ifn=3+3kσ+2,⋮α10α11σα13α14σ+1α16α17σz−1,ifn=3+3kσ+k,α10α11σα13α14σ+1α16α17σz0,ifn=3+3kσ+k+1,α10α11σα13α14σ+1α16α17σ+1x−k,ifn=3+3kσ+k+2,α10α11α13α14σ+1α16α17σ+1x1−k,ifn=3+3kσ+k+3,⋮α10α11σα13α14σ+1α16α17σx−1,ifn=3+3kσ+2k+1,α10α11σα13α14σ+1α16α17σx0,ifn=3+3kσ+2k+2,α10α11α13α14α16α17σ+1y−k,ifn=3+3kσ+2k+3,α10α11α13α14α16α17σ+1y1−k,ifn=3+3kσ+2k+4,⋮α10α11α13α14α16α17σ+1y−1,ifn=3+3kσ+3k+2,α10α11α13α14α16α17σ+1y0,ifn=3+3kσ+3k+3.
Finally,(47)0<zn≤α10α11σα13α14σα16α17σ+1x−k,ifn=3+3kσ+1,α10α11σα13α14σα16α17σ+1x1−k,ifn=3+3kσ+2,⋮α10α11σα13α14σα16α17σ+1x−1,ifn=3+3kσ+k,α10α11σα13α14σα16α17σ+1x0,ifn=3+3kσ+k+1,α10α11σ+1α13α14σα16α17σ+1y−k,ifn=3+3kσ+k+2,α10α11σ+1α13α14σα16α17σ+1y1−k,ifn=3+3kσ+k+3,⋮α10α11σ+1α13α14σα16α17σ+1y−1,ifn=3+3kσ+2k+1,α10α11σ+1α13α14σα16α17σ+1y0,ifn=3+3kσ+2k+2,α10α11α13α14α16α17σ+1z−k,ifn=3+3kσ+2k+3,α10α11α13α14α16α17σ+1z1−k,ifn=3+3kσ+2k+4,⋮α10α11α13α14α16α17σ+1z−1,ifn=3+3kσ+3k+2,α10α11α13α14α16α17σ+1z0,ifn=3+3kσ+3k+3.
Than forμ=σ+1, we have
(48)x3k+3σ+1+1=α10y3k+3σ+2k+3α11+α12∏i=0kz3k+3σ+3k+3−i,≤α10α11y3k+3σ+2k+3≤α10α11σ+2α13α14σ+1α16α17σ+1y−k,x3k+3σ+1+2=α10y3k+3σ+2k+4α11+α12∏i=0kz3k+3σ+3k+4−i,≤α10α11y3k+3σ+2k+4≤α10α11σ+2α13α14σ+1α16α17σ+1y−k+1,⋮x3k+3σ+1+k=α10y3k+3σ+3k+2α11+α12∏i=0kz3k+3σ+4k+2−i,≤α10α11y3k+3σ+3k+2≤α10α11σ+2α13α14σ+1α16α17σ+1y−1,x3k+3σ+1+k+1=α10y3k+3σ+3k+3α11+α12∏i=0kz3k+3σ+4k+3−i,≤α10α11y3k+3σ+3k+3≤α10α11σ+2α13α14σ+1α16α17σ+1y0,y3k+3σ+1+1=α13z3k+3σ+2k+3α14+α15∏i=0kx3k+3σ+3k+3−i,≤α13α14z3k+3σ+2k+3≤α10α11σ+1α13α14σ+2α16α17σ+1z−k,y3k+3σ+1+2=α13z3k+3σ+2k+4α14+α15∏i=0kx3k+3σ+3k+4−i,≤α13α14z3k+3σ+2k+4≤α10α11σ+1α13α14σ+2α16α17σ+1z−k+1,⋮y3k+3σ+1+k=α13z3k+3σ+3k+2α14+α15∏i=0kx3k+3σ+4k+2−i,≤α13α14z3k+3σ+3k+2≤α10α11σ+1α13α14σ+2α16α17σ+1z−1,y3k+3σ+1+k+1=α13z3k+3σ+3k+3α14+α15∏i=0kx3k+3σ+4k+3−i,≤α13α14z3k+3σ+3k+3≤α10α11σ+1α13α14σ+2α16α17σ+1z0,z3k+3σ+1+1=α16x3k+3σ+2k+3α17+α18∏i=0ky3k+3σ+3k+3−i,≤α16α17x3k+3σ+2k+3≤α10α11σ+1α13α14σ+1α16α17σ+2x−k,z3k+3σ+1+2=α16x3k+3σ+2k+4α17+α18∏i=0ky3k+3σ+3k+4−i,≤α16α17x3k+3σ+2k+4≤α10α11σ+1α13α14σ+1α16α17σ+2x−k+1,⋮z3k+3σ+1+k=α16x3k+3σ+3k+2α17+α18∏i=0ky3k+3σ+4k+2−i,≤α16α17x3k+3σ+3k+2≤α10α11σ+1α13α14σ+1α16α17σ+2x−1,z3k+3σ+1+k+1=α16x3k+3σ+3k+3α17+α18∏i=0ky3k+3σ+4k+3−i,≤α16α17x3k+3σ+3k+3≤α10α11σ+1α13α14σ+1α16α17σ+2x0.
And(49)x3k+3σ+1+k+2=α10y3k+3σ+1+1α11+α12∏i=0kz3k+3σ+4k+4−i,≤α10α11y3k+3σ+1+1≤α10α11σ+2α13α14σ+2α16α17σ+1z−k,x3k+3σ+1+k+3=α10y3k+3σ+1+2α11+α12∏i=0kz3k+3σ+4k+5−i,≤α10α11y3k+3σ+1+2≤α10α11σ+2α13α14σ+2α16α17σ+1z−k+1,⋮x3k+3σ+1+2k+1=α10y3k+3σ+1+kα11+α12∏i=0kz3k+3σ+5k+3−i,≤α10α11y3k+3σ+1+k≤α10α11σ+2α13α14σ+2α16α17σ+1z−1,x3k+3σ+1+2k+2=α10y3k+3σ+1+k+1α11+α12∏i=0kz3k+3σ+5k+4−i,≤α10α11y3k+3σ+1+2≤α10α11σ+2α13α14σ+2α16α17σ+1z0,y3k+3σ+1+k+2=α13z3k+3σ+1+1α14+α15∏i=0kx3k+3σ+4k+4−i,≤α13α14z3k+3σ+1+1≤α10α11σ+1α13α14σ+2α16α17σ+2x−k,y3k+3σ+1+k+3=α13z3k+3σ+1+2α14+α15∏i=0kx3k+3σ+4k+5−i,≤α13α14z3k+3σ+1+2≤α10α11σ+1α13α14σ+2α16α17σ+2x−k+1,⋮y3k+3σ+1+2k+1=α13z3k+3σ+1+kα14+α15∏i=0kx3k+3σ+5k+3−i,≤α13α14z3k+3σ+1+k≤α10α11σ+1α13α14σ+2α16α17σ+2x−1,y3k+3σ+1+2k+2=α13z3k+3σ+1+k+1α14+α15∏i=0kx3k+3σ+5k+4−i,≤α13α14z3k+3σ+1+k+1≤α10α11σ+1α13α14σ+2α16α17σ+2x0,z3k+3σ+1+k+2=α16x3k+3σ+1+1α17+α18∏i=0ky3k+3σ+4k+4−i,≤α16α17x3k+3σ+1+1≤α10α11σ+2α13α14σ+1α16α17σ+2y−k,z3k+3σ+1+k+3=α16x3k+3σ+1+2α17+α18∏i=0ky3k+3σ+4k+5−i,≤α16α17x3k+3σ+1+2≤α10α11σ+2α13α14σ+1α16α17σ+2y−k+1,⋮z3k+3σ+1+2k+1=α16x3k+3σ+1+kα17+α18∏i=0ky3k+3σ+5k+3−i,≤α16α17x3k+3σ+1+k≤α10α11σ+2α13α14σ+1α16α17σ+2y−1,z3k+3σ+1+2k+2=α16x3k+3σ+1+k+1α17+α18∏i=0ky3k+3σ+5k+4−i,≤α16α17x3k+3σ+1+k+1≤α10α11σ+2α13α14σ+1α16α17σ+2y0.
Finally,(50)x3k+3σ+1+2k+3=α10y3k+3σ+1+k+2α11+α12∏i=0kz3k+3σ+5k+5−i,≤α10α11y3k+3σ+1+k+2≤α10α11α13α14α16α17σ+2x−k,x3k+3σ+1+2k+4=α10y3k+3σ+1+k+3α11+α12∏i=0kz3k+3σ+5k+6−i,≤α10α11y3k+3σ+1+k+3≤α10α11α13α14α16α17σ+2x−k+1,⋮x3k+3σ+1+3k+2=α10y3k+3σ+1+2k+1α11+α12∏i=0kz3k+3σ+6k+4−i,≤α10α11y3k+3σ+1+2k+1≤α10α11α13α14α16α17σ+2x−1,x3k+3σ+1+3k+3=α10y3k+3σ+1+2k+2α11+α12∏i=0kz3k+3σ+6k+5−i,≤α10α11y3k+3σ+1+2k+2≤α10α11α13α14α16α17σ+2x0,y3k+3σ+1+2k+3=α13z3k+3σ+1+k+2α14+α15∏i=0kx3k+3σ+5k+5−i,≤α13α14z3k+3σ+1+k+2≤α10α11α13α14α16α17σ+2y−k,y3k+3σ+1+2k+4=α13z3k+3σ+1+k+3α14+α15∏i=0kx3k+3σ+5k+6−i,≤α13α14z3k+3σ+1+k+3≤α10α11α13α14α16α17σ+2y−k+1,⋮y3k+3σ+1+3k+2=α13z3k+3σ+1+2k+1α14+α15∏i=0kx3k+3σ+6k+4−i,≤α13α14z3k+3σ+1+2k+1≤α10α11α13α14α16α17σ+2y−1,y3k+3σ+1+3k+3=α13z3k+3σ+1+2k+2α14+α15∏i=0kx3k+3σ+6k+5−i,≤α13α14z3k+3σ+1+2k+2≤α10α11α13α14α16α17σ+2y0,z3k+3σ+1+2k+3=α16x3k+3σ+1+k+2α17+α18∏i=0ky3k+3σ+5k+5−i,≤α16α17x3k+3σ+1+k+2≤α10α11α13α14α16α17σ+2z−k,z3k+3σ+1+2k+4=α16x3k+3σ+1+k+3α17+α18∏i=0ky3k+3σ+5k+6−i,≤α16α17x3k+3σ+1+k+3≤α10α11α13α14α16α17σ+2z−k+1,⋮z3k+3σ+1+3k+2=α16x3k+3σ+1+2k+1α17+α18∏i=0ky3k+3σ+6k+4−i,≤α16α17x3k+3σ+1+2k+1≤α10α11α13α14α16α17σ+2z−1,z3k+3σ+1+3k+3=α16x3k+3σ+1+2k+2α17+α18∏i=0ky3k+3σ+6k+5−i,≤α16α17x3k+3σ+1+2k+2≤α10α11α13α14α16α17σ+2z0.Corollary 4.
Xn of (11) is bounded if (30) holds.Proof.
It is a direct result of Theorem3.
Hereafter, we will present boundedness for discrete systems (12)–(14).Theorem 5.
(i) IfXn is the positive solution of (12), then for μ≥0, the following holds:
(51)0<xn≤α19α20μ+1α22α23μα25α26μz−k,ifn=3+3kμ+1,α19α20μ+1α22α23μα25α26μz−k+1,ifn=3+3kμ+2,⋮α19α20μ+1α22α23μα25α26μz−1,ifn=3+3kμ+k,α19α20μ+1α22α23μα25α26μz0,ifn=3+3kμ+k+1,α19α20μ+1α22α23μ+1α25α26μy−k,ifn=3+3kμ+k+2,α19α20μ+1α22α23μ+1α25α26μy−k+1,ifn=3+3kμ+k+3,⋮α19α20μ+1α22α23μ+1α25α26μy−1,ifn=3+3kμ+2k+1,α19α20μ+1α22α23μ+1α25α26μy0,ifn=3+3kμ+2k+2,α19α20α22α23α25α26μ+1x−k,ifn=3+3kμ+2k+3,α19α20α22α23α25α26μ+1x−k+1,ifn=3+3kμ+2k+3,⋮α19α20α22α23α25α26μ+1x−1,ifn=3+3kμ+3k+2,α19α20α22α23α25α26μ+1x0,ifn=3+3kμ+3k+3.0<yn≤α19α20μα22α23μ+1α25α26μx−k,ifn=3+3kμ+1,α19α20μα22α23μ+1α25α26μx−k+1,ifn=3+3kμ+2,⋮α19α20μα22α23μ+1α25α26μx−1,ifn=3+3kμ+k,α19α20μα22α23μ+1α25α26μx0,ifn=3+3kμ+k+1,α19α20μα22α23μ+1α25α26μz−k,ifn=3+3kμ+k+2,α19α20μα22α23μ+1α25α26μz−k+1,ifn=3+3kμ+k+3,⋮α19α20μα22α23μ+1α25α26μz−1,ifn=3+3kμ+2k+1,α19α20μα22α23μ+1α25α26μz0,ifn=3+3kμ+2k+2,α19α20α22α23α25α26μ+1y−k,ifn=3k+3μ+2k+3,α19α20α22α23α25α26μ+1y−k+1,ifn=3+3kμ+2k+3,⋮α19α20α22α23α25α26μ+1y−1,ifn=3+3kμ+3k+2,α19α20α22α23α25α26μ+1y0,ifn=3+3kμ+3k+3.
Finally,(52)0<zn≤α19α20μα22α23μα25α26μ+1y−k,ifn=3+3kμ+1,α19α20μα22α23μα25α26μ+1y−k+1,ifn=3+3kμ+2,⋮α19α20μα22α23μα25α26μ+1y−1,ifn=3+3kμ+k,α19α20μα22α23μα25α26μ+1y0,ifn=3+3kμ+k+1,α19α20μ+1α22α23μα25α26μ+1x−k,ifn=3+3kμ+k+2,α19α20μ+1α22α23μα25α26μ+1x−k+1,ifn=3+3kμ+k+3,⋮α19α20μ+1α22α23μα25α26μ+1x−1,ifn=3+3kμ+2k+1,α19α20μ+1α22α23μα25α26μ+1x0,ifn=3+3kμ+2k+2,α19α20α22α23α25α26μ+1z−k,ifn=3+3kμ+2k+3,α19α20α22α23α25α26μ+1z−k+1,ifn=3+3kμ+2k+3,⋮α19α20α22α23α25α26μ+1z−1,ifn=3+3kμ+3k+2,α19α20α22α23α25α26μ+1z0,ifn=3+3kμ+3k+3.
(ii) IfXn is the positive solution of (13), then for μ≥0, the following holds:
(53)0<xn≤α28α29μ+1α31α32μα34α35μy−k,ifn=3+3kμ+1,α28α29μ+1α31α32μα34α35μy−k+1,ifn=3+3kμ+2,⋮α28α29μ+1α31α32μα34α35μy−1,ifn=3+3kμ+k,α28α29μ+1α31α32μα34α35μy0,ifn=3+3kμ+k+1,α28α29μ+1α31α32μ+1α34α35μz−k,ifn=3+3kμ+k+2,α28α29μ+1α31α32μ+1α34α35μz−k+1,ifn=3+3kμ+k+3,⋮α28α29μ+1α31α32μ+1α34α35μz−1,ifn=3+3kμ+2k+1,α28α29μ+1α31α32μ+1α34α35μz0,ifn=3+3kμ+2k+2,α28α29α31α32α34α35μ+1x−k,ifn=3+3kμ+2k+3,α28α29α31α32α34α35μ+1x−k+1,ifn=3+3kμ+2k+4,⋮α28α29α31α32α34α35μ+1x−1,ifn=3+3kμ+3k+2,α28α29α31α32α34α35μ+1x0,ifn=3+3kμ+3k+3.
And(54)0<yn≤α28α29μα31α32μ+1α34α35μz−k,ifn=3+3kμ+1,α28α29μα31α32μ+1α34α35μz−k+1,ifn=3+3kμ+2⋮α28α29μα31α32μ+1α34α35μz−1,ifn=3+3kμ+k,α28α29μα31α32μ+1α34α35μz0,ifn=3+3kμ+k+1,α28α29μα31α32μ+1α34α35μx−k,ifn=3+3kμ+k+2,α28α29μα31α32μ+1α34α35μx−k+1,ifn=3+3kμ+k+3,⋮α28α29μα31α32μ+1α34α35μx−1,ifn=3+3kμ+2k+1,α28α29μα31α32μ+1α34α35μx0,ifn=3+3kμ+2k+2,α28α29α31α32α34α35μ+1y−k,ifn=3+3kμ+2k+3,α28α29α31α32α34α35μ+1y−k+1,ifn=3+3kμ+2k+4,⋮α28α29α31α32α34α35μ+1y−1,ifn=3+3kμ+3k+2,α28α29α31α32α34α35μ+1y0,ifn=3+3kμ+3k+3.
Finally,(55)0<zn≤α28α29μα31α32μα34α35μ+1x−k,ifn=3+3kμ+1,α28α29μα31α32μα34α35μ+1x−k+1,ifn=3+3kμ+2,⋮α28α29μα31α32μα34α35μ+1x−1,ifn=3+3kμ+k,α28α29μα31α32μα34α35μ+1x0,ifn=3+3kμ+k+1,α28α29μ+1α31α32μα34α35μ+1y−k,ifn=3+3kμ+k+2,α28α29μ+1α31α32μα34α35μ+1y−k+1,ifn=3+3kμ+k+3,⋮α28α29μ+1α31α32μα34α35μ+1y−1,ifn=3+3kμ+2k+1,α28α29μ+1α31α32μα34α35μ+1y0,ifn=3+3kμ+2k+2,α28α29α31α32α34α35μ+1z−k,ifn=3+3kμ+2k+3,α28α29α31α32α34α35μ+1z−k+1,ifn=3+3kμ+2k+4,⋮α28α29α31α32α34α35μ+1z−1,ifn=3+3kμ+3k+2,α28α29α31α32α34α35μ+1z0,ifn=3+3kμ+3k+3.
(iii) IfXn is the positive solution of (14), then for μ≥0, the following holds:
(56)0<xn≤α37α38μ+1α40α41μα43α44μz−k,ifn=3+3kμ+1,α37α38μ+1α40α41μα43α44μz−k+1,ifn=3+3kμ+2,⋮α37α38μ+1α40α41μα43α44μz−1,ifn=3+3kμ+k,α37α38μ+1α40α41μα43α44μz0,ifn=3+3kμ+k+1,α37α38μ+1α40α41μ+1α43α44μy−k,ifn=3+3kμ+k+2,α37α38μ+1α40α41μ+1α43α44μy−k+1,ifn=3+3kμ+k+3,⋮α37α38μ+1α40α41μ+1α43α44μy−1,ifn=3+3kμ+2k+1,α37α38μ+1α40α41μ+1α43α44μy0,ifn=3+3kμ+2k+2,α37α38α40α41α43α44μ+1x−k,ifn=3+3kμ+2k+3,α37α38α40α41α43α44μ+1x−k+1,ifn=3+3kμ+2k+4,⋮α37α38α40α41α43α44μ+1x−1,ifn=3+3kμ+3k+2,α37α38α40α41α43α44μ+1x0,ifn=3+3kμ+3k+3.
And(57)0<yn≤α37α38μα40α41μ+1α43α44μx−k,ifn=3+3kμ+1,α37α38μα40α41μ+1α43α44μx−k+1,ifn=3+3kμ+2,⋮α37α38μα40α41μ+1α43α44μx−1,ifn=3+3kμ+k,α37α38μα40α41μ+1α43α44μx0,ifn=3+3kμ+k+1,α37α38μα40α41μ+1α43α44μz−k,ifn=3+3kμ+k+2,α37α38μα40α41μ+1α43α44μz−k+1,ifn=3+3kμ+k+3,⋮α37α38μα40α41μ+1α43α44μz−1,ifn=3+3kμ+2k+1,α37α38μα40α41μ+1α43α44μz0,ifn=3+3kμ+2k+2,α37α38α40α41α43α44μ+1y−k,ifn=3+3kμ+2k+3,α37α38α40α41α43α44μ+1y−k+1,ifn=3+3kμ+2k+4,⋮α37α38α40α41α43α44μ+1y−1,ifn=3+3kμ+3k+2,α37α38α40α41α43α44m+1y0,ifn=3+3kμ+3k+3.
Finally,(58)0<zn≤α37α38μα40α41μα43α44μ+1y−k,ifn=3+3kμ+1,α37α38μα40α41μα43α44μ+1y−k+1,ifn=3+3kμ+2,⋮α37α38μα40α41μα43α44μ+1y−1,ifn=3+3kμ+k,α37α38μα40α41μα43α44μ+1y0,ifn=3+3kμ+k+1,α37α38μ+1α40α41μα43α44μ+1x−k,ifn=3+3kμ+k+2,α37α38μ+1α40α41μα43α44μ+1x−k+1,ifn=3+3kμ+k+3,⋮α37α38μ+1α40α41μα43α44μ+1x−1,ifn=3+3kμ+2k+1,α37α38μ+1α40α41μα43α44μ+1x0,ifn=3+3kμ+2k+2α37α38α40α41α43α44μ+1z−k,ifn=3+3kμ+2k+3,α37α38α40α41α43α44μ+1z1−k,ifn=3+3kμ+2k+4,⋮α37α38α40α41α43α44μ+1z−1,ifn=3+3kμ+3k+2,α37α38α40α41α43α44μ+1z0,ifn=3+3kμ+3k+3.Proof.
Same as the proof of Theorem3.Corollary 6.
The following statements are true for (12)–(14):
(i)
If (40) holds, then Xn of discrete system (12) is bounded(ii)
If (41) holds, then Xn of discrete system (13) is bounded(iii)
If (42) holds, then Xn of discrete system (14) is boundedProof.
Its consequence of Theorem5.


## 4.1. Boundedness of Discrete System (11)

Theorem 3.
IfXn=xn,yn,zn is the positive solution of discrete system (11), then for μ≥0, the following holds:
(43)0<xn≤α10α11μ+1α13α14μα16α17μy−k,ifn=3+3kμ+1,α10α11μ+1α13α14μα16α17μy1−k,ifn=3+3kμ+2,⋮α10α11μ+1α13α14μα16α17μy−1,ifn=3+3kμ+k,α10α11μ+1α13α14μα16α17μy0,ifn=3+3kμ+k+1,α10α11μ+1α13α14μ+1α16α17μz−k,ifn=3+3kμ+k+2,α10α11μ+1α13α14μ+1α16α17μz1−k,ifn=3+3kμ+k+3,⋮α10α11μ+1α13α14μ+1α16α17μz−1,ifn=3+3kμ+2k+1,α10α11μ+1α13α14μ+1α16α17μz0,ifn=3+3kμ+2k+2,α10α11α13α14α16α17μ+1x−k,ifn=3+3kμ+2k+3,α10α11α13α14α16α17μ+1x1−k,ifn=3+3kμ+2k+4,⋮α10α11α13α14α16α17μ+1x−1,ifn=3+3kμ+3k+2,α10α11α13α14α16α17μ+1x0,ifn=3+3kμ+3k+3.(44)0<yn≤α10α11μα13α14μ+1α16α17μz−k,ifn=3+3kμ+1,α10α11μα13α14μ+1α16α17μz1−k,ifn=3+3kμ+2,⋮α10α11μα13α14μ+1α16α17μz−1ifn=3+3kμ+k,α10α11μα13α14μ+1α16α17μz0,ifn=3+3kμ+k+1,α10α11μα13α14μ+1α16α17μ+1x−k,ifn=3+3kμ+k+2,α10α11α13α14μ+1α16α17μ+1x1−k,ifn=3+3kμ+k+3,⋮α10α11μα13α14μ+1α16α17μx−1,ifn=3+3kμ+2k+1,α10α11μα13α14μ+1α16α17μx0,ifn=3+3kμ+2k+2,α10α11α13α14α16α17μ+1y−k,ifn=3+3kμ+2k+3,α10α11α13α14α16α17μ+1y1−k,ifn=3+3kμ+2k+4,⋮α10α11α13α14α16α17μ+1y−1,ifn=3+3kμ+3k+2,α10α11α13α14α16α17μ+1y0,ifn=3+3kμ+3k+3.
Finally,(45)0<zn≤α10α11μα13α14μα16α17μ+1x−k,ifn=3+3kμ+1,α10α11μα13α14μα16α17μ+1x1−k,ifn=3+3kμ+2,⋮α10α11μα13α14μα16α17μ+1x−1,ifn=3+3kμ+k,α10α11μα13α14μα16α17μ+1x0,ifn=3+3kμ+k+1,α10α11μ+1α13α14μα16α17μ+1y−k,ifn=3+3kμ+k+2,α10α11μ+1α13α14μα16α17μ+1y1−k,ifn=3+3kμ+k+3,⋮α10α11μ+1α13α14μα16α17μ+1y−1,ifn=3+3kμ+2k+1,α10α11μ+1α13α14μα16α17μ+1y0,ifn=3+3kμ+2k+2,α10α11α13α14α16α17μ+1z−k,ifn=3+3kμ+2k+3,α10α11α13α14α16α17μ+1z1−k,ifn=3+3kμ+2k+4,⋮α10α11α13α14α16α17μ+1z−1,ifn=3+3kμ+3k+2,α10α11α13α14α16α17μ+1z0,ifn=3+3kμ+3k+3.Proof.
Noticeably, (43), (44), and (45) hold if μ=0. If (43), (44), and (45) are true for μ=σ, i.e.,
(46)0<xn≤α10α11σ+1α13α14σα16α17σy−k,ifn=3+3kσ+1,α10α11σ+1α13α14σα16α17σy1−k,ifn=3+3kσ+2,⋮α10α11σ+1α13α14σα16α17σy−1,ifn=3+3kσ+k,α10α11σ+1α13α14σα16α17σy0,ifn=3+3kσ+k+1,α10α11σ+1α13α14σ+1α16α17σz−k,ifn=3+3kσ+k+2,α10α11σ+1α13α14σ+1α16α17σz1−k,ifn=3+3kσ+k+3,⋮α10α11σ+1α13α14σ+1α16α17σz−1,ifn=3+3kσ+2k+1,α10α11σ+1α13α14σ+1α16α17σz0,ifn=3+3kσ+2k+2,α10α11α13α14α16α17σ+1x−k,ifn=3+3kσ+2k+3,α10α11α13α14α16α17σ+1x1−k,ifn=3+3kσ+2k+4,⋮α10α11α13α14α16α17σ+1x−1,ifn=3+3kσ+3k+2,α10α11α13α14α16α17σ+1x0,ifn=3+3kσ+3k+3.0<yn≤α10α11σα13α14σ+1α16α17σz−k,ifn=3+3kσ+1,α10α11σα13α14σ+1α16α17σz1−k,ifn=3+3kσ+2,⋮α10α11σα13α14σ+1α16α17σz−1,ifn=3+3kσ+k,α10α11σα13α14σ+1α16α17σz0,ifn=3+3kσ+k+1,α10α11σα13α14σ+1α16α17σ+1x−k,ifn=3+3kσ+k+2,α10α11α13α14σ+1α16α17σ+1x1−k,ifn=3+3kσ+k+3,⋮α10α11σα13α14σ+1α16α17σx−1,ifn=3+3kσ+2k+1,α10α11σα13α14σ+1α16α17σx0,ifn=3+3kσ+2k+2,α10α11α13α14α16α17σ+1y−k,ifn=3+3kσ+2k+3,α10α11α13α14α16α17σ+1y1−k,ifn=3+3kσ+2k+4,⋮α10α11α13α14α16α17σ+1y−1,ifn=3+3kσ+3k+2,α10α11α13α14α16α17σ+1y0,ifn=3+3kσ+3k+3.
Finally,(47)0<zn≤α10α11σα13α14σα16α17σ+1x−k,ifn=3+3kσ+1,α10α11σα13α14σα16α17σ+1x1−k,ifn=3+3kσ+2,⋮α10α11σα13α14σα16α17σ+1x−1,ifn=3+3kσ+k,α10α11σα13α14σα16α17σ+1x0,ifn=3+3kσ+k+1,α10α11σ+1α13α14σα16α17σ+1y−k,ifn=3+3kσ+k+2,α10α11σ+1α13α14σα16α17σ+1y1−k,ifn=3+3kσ+k+3,⋮α10α11σ+1α13α14σα16α17σ+1y−1,ifn=3+3kσ+2k+1,α10α11σ+1α13α14σα16α17σ+1y0,ifn=3+3kσ+2k+2,α10α11α13α14α16α17σ+1z−k,ifn=3+3kσ+2k+3,α10α11α13α14α16α17σ+1z1−k,ifn=3+3kσ+2k+4,⋮α10α11α13α14α16α17σ+1z−1,ifn=3+3kσ+3k+2,α10α11α13α14α16α17σ+1z0,ifn=3+3kσ+3k+3.
Than forμ=σ+1, we have
(48)x3k+3σ+1+1=α10y3k+3σ+2k+3α11+α12∏i=0kz3k+3σ+3k+3−i,≤α10α11y3k+3σ+2k+3≤α10α11σ+2α13α14σ+1α16α17σ+1y−k,x3k+3σ+1+2=α10y3k+3σ+2k+4α11+α12∏i=0kz3k+3σ+3k+4−i,≤α10α11y3k+3σ+2k+4≤α10α11σ+2α13α14σ+1α16α17σ+1y−k+1,⋮x3k+3σ+1+k=α10y3k+3σ+3k+2α11+α12∏i=0kz3k+3σ+4k+2−i,≤α10α11y3k+3σ+3k+2≤α10α11σ+2α13α14σ+1α16α17σ+1y−1,x3k+3σ+1+k+1=α10y3k+3σ+3k+3α11+α12∏i=0kz3k+3σ+4k+3−i,≤α10α11y3k+3σ+3k+3≤α10α11σ+2α13α14σ+1α16α17σ+1y0,y3k+3σ+1+1=α13z3k+3σ+2k+3α14+α15∏i=0kx3k+3σ+3k+3−i,≤α13α14z3k+3σ+2k+3≤α10α11σ+1α13α14σ+2α16α17σ+1z−k,y3k+3σ+1+2=α13z3k+3σ+2k+4α14+α15∏i=0kx3k+3σ+3k+4−i,≤α13α14z3k+3σ+2k+4≤α10α11σ+1α13α14σ+2α16α17σ+1z−k+1,⋮y3k+3σ+1+k=α13z3k+3σ+3k+2α14+α15∏i=0kx3k+3σ+4k+2−i,≤α13α14z3k+3σ+3k+2≤α10α11σ+1α13α14σ+2α16α17σ+1z−1,y3k+3σ+1+k+1=α13z3k+3σ+3k+3α14+α15∏i=0kx3k+3σ+4k+3−i,≤α13α14z3k+3σ+3k+3≤α10α11σ+1α13α14σ+2α16α17σ+1z0,z3k+3σ+1+1=α16x3k+3σ+2k+3α17+α18∏i=0ky3k+3σ+3k+3−i,≤α16α17x3k+3σ+2k+3≤α10α11σ+1α13α14σ+1α16α17σ+2x−k,z3k+3σ+1+2=α16x3k+3σ+2k+4α17+α18∏i=0ky3k+3σ+3k+4−i,≤α16α17x3k+3σ+2k+4≤α10α11σ+1α13α14σ+1α16α17σ+2x−k+1,⋮z3k+3σ+1+k=α16x3k+3σ+3k+2α17+α18∏i=0ky3k+3σ+4k+2−i,≤α16α17x3k+3σ+3k+2≤α10α11σ+1α13α14σ+1α16α17σ+2x−1,z3k+3σ+1+k+1=α16x3k+3σ+3k+3α17+α18∏i=0ky3k+3σ+4k+3−i,≤α16α17x3k+3σ+3k+3≤α10α11σ+1α13α14σ+1α16α17σ+2x0.
And(49)x3k+3σ+1+k+2=α10y3k+3σ+1+1α11+α12∏i=0kz3k+3σ+4k+4−i,≤α10α11y3k+3σ+1+1≤α10α11σ+2α13α14σ+2α16α17σ+1z−k,x3k+3σ+1+k+3=α10y3k+3σ+1+2α11+α12∏i=0kz3k+3σ+4k+5−i,≤α10α11y3k+3σ+1+2≤α10α11σ+2α13α14σ+2α16α17σ+1z−k+1,⋮x3k+3σ+1+2k+1=α10y3k+3σ+1+kα11+α12∏i=0kz3k+3σ+5k+3−i,≤α10α11y3k+3σ+1+k≤α10α11σ+2α13α14σ+2α16α17σ+1z−1,x3k+3σ+1+2k+2=α10y3k+3σ+1+k+1α11+α12∏i=0kz3k+3σ+5k+4−i,≤α10α11y3k+3σ+1+2≤α10α11σ+2α13α14σ+2α16α17σ+1z0,y3k+3σ+1+k+2=α13z3k+3σ+1+1α14+α15∏i=0kx3k+3σ+4k+4−i,≤α13α14z3k+3σ+1+1≤α10α11σ+1α13α14σ+2α16α17σ+2x−k,y3k+3σ+1+k+3=α13z3k+3σ+1+2α14+α15∏i=0kx3k+3σ+4k+5−i,≤α13α14z3k+3σ+1+2≤α10α11σ+1α13α14σ+2α16α17σ+2x−k+1,⋮y3k+3σ+1+2k+1=α13z3k+3σ+1+kα14+α15∏i=0kx3k+3σ+5k+3−i,≤α13α14z3k+3σ+1+k≤α10α11σ+1α13α14σ+2α16α17σ+2x−1,y3k+3σ+1+2k+2=α13z3k+3σ+1+k+1α14+α15∏i=0kx3k+3σ+5k+4−i,≤α13α14z3k+3σ+1+k+1≤α10α11σ+1α13α14σ+2α16α17σ+2x0,z3k+3σ+1+k+2=α16x3k+3σ+1+1α17+α18∏i=0ky3k+3σ+4k+4−i,≤α16α17x3k+3σ+1+1≤α10α11σ+2α13α14σ+1α16α17σ+2y−k,z3k+3σ+1+k+3=α16x3k+3σ+1+2α17+α18∏i=0ky3k+3σ+4k+5−i,≤α16α17x3k+3σ+1+2≤α10α11σ+2α13α14σ+1α16α17σ+2y−k+1,⋮z3k+3σ+1+2k+1=α16x3k+3σ+1+kα17+α18∏i=0ky3k+3σ+5k+3−i,≤α16α17x3k+3σ+1+k≤α10α11σ+2α13α14σ+1α16α17σ+2y−1,z3k+3σ+1+2k+2=α16x3k+3σ+1+k+1α17+α18∏i=0ky3k+3σ+5k+4−i,≤α16α17x3k+3σ+1+k+1≤α10α11σ+2α13α14σ+1α16α17σ+2y0.
Finally,(50)x3k+3σ+1+2k+3=α10y3k+3σ+1+k+2α11+α12∏i=0kz3k+3σ+5k+5−i,≤α10α11y3k+3σ+1+k+2≤α10α11α13α14α16α17σ+2x−k,x3k+3σ+1+2k+4=α10y3k+3σ+1+k+3α11+α12∏i=0kz3k+3σ+5k+6−i,≤α10α11y3k+3σ+1+k+3≤α10α11α13α14α16α17σ+2x−k+1,⋮x3k+3σ+1+3k+2=α10y3k+3σ+1+2k+1α11+α12∏i=0kz3k+3σ+6k+4−i,≤α10α11y3k+3σ+1+2k+1≤α10α11α13α14α16α17σ+2x−1,x3k+3σ+1+3k+3=α10y3k+3σ+1+2k+2α11+α12∏i=0kz3k+3σ+6k+5−i,≤α10α11y3k+3σ+1+2k+2≤α10α11α13α14α16α17σ+2x0,y3k+3σ+1+2k+3=α13z3k+3σ+1+k+2α14+α15∏i=0kx3k+3σ+5k+5−i,≤α13α14z3k+3σ+1+k+2≤α10α11α13α14α16α17σ+2y−k,y3k+3σ+1+2k+4=α13z3k+3σ+1+k+3α14+α15∏i=0kx3k+3σ+5k+6−i,≤α13α14z3k+3σ+1+k+3≤α10α11α13α14α16α17σ+2y−k+1,⋮y3k+3σ+1+3k+2=α13z3k+3σ+1+2k+1α14+α15∏i=0kx3k+3σ+6k+4−i,≤α13α14z3k+3σ+1+2k+1≤α10α11α13α14α16α17σ+2y−1,y3k+3σ+1+3k+3=α13z3k+3σ+1+2k+2α14+α15∏i=0kx3k+3σ+6k+5−i,≤α13α14z3k+3σ+1+2k+2≤α10α11α13α14α16α17σ+2y0,z3k+3σ+1+2k+3=α16x3k+3σ+1+k+2α17+α18∏i=0ky3k+3σ+5k+5−i,≤α16α17x3k+3σ+1+k+2≤α10α11α13α14α16α17σ+2z−k,z3k+3σ+1+2k+4=α16x3k+3σ+1+k+3α17+α18∏i=0ky3k+3σ+5k+6−i,≤α16α17x3k+3σ+1+k+3≤α10α11α13α14α16α17σ+2z−k+1,⋮z3k+3σ+1+3k+2=α16x3k+3σ+1+2k+1α17+α18∏i=0ky3k+3σ+6k+4−i,≤α16α17x3k+3σ+1+2k+1≤α10α11α13α14α16α17σ+2z−1,z3k+3σ+1+3k+3=α16x3k+3σ+1+2k+2α17+α18∏i=0ky3k+3σ+6k+5−i,≤α16α17x3k+3σ+1+2k+2≤α10α11α13α14α16α17σ+2z0.Corollary 4.
Xn of (11) is bounded if (30) holds.Proof.
It is a direct result of Theorem3.
Hereafter, we will present boundedness for discrete systems (12)–(14).Theorem 5.
(i) IfXn is the positive solution of (12), then for μ≥0, the following holds:
(51)0<xn≤α19α20μ+1α22α23μα25α26μz−k,ifn=3+3kμ+1,α19α20μ+1α22α23μα25α26μz−k+1,ifn=3+3kμ+2,⋮α19α20μ+1α22α23μα25α26μz−1,ifn=3+3kμ+k,α19α20μ+1α22α23μα25α26μz0,ifn=3+3kμ+k+1,α19α20μ+1α22α23μ+1α25α26μy−k,ifn=3+3kμ+k+2,α19α20μ+1α22α23μ+1α25α26μy−k+1,ifn=3+3kμ+k+3,⋮α19α20μ+1α22α23μ+1α25α26μy−1,ifn=3+3kμ+2k+1,α19α20μ+1α22α23μ+1α25α26μy0,ifn=3+3kμ+2k+2,α19α20α22α23α25α26μ+1x−k,ifn=3+3kμ+2k+3,α19α20α22α23α25α26μ+1x−k+1,ifn=3+3kμ+2k+3,⋮α19α20α22α23α25α26μ+1x−1,ifn=3+3kμ+3k+2,α19α20α22α23α25α26μ+1x0,ifn=3+3kμ+3k+3.0<yn≤α19α20μα22α23μ+1α25α26μx−k,ifn=3+3kμ+1,α19α20μα22α23μ+1α25α26μx−k+1,ifn=3+3kμ+2,⋮α19α20μα22α23μ+1α25α26μx−1,ifn=3+3kμ+k,α19α20μα22α23μ+1α25α26μx0,ifn=3+3kμ+k+1,α19α20μα22α23μ+1α25α26μz−k,ifn=3+3kμ+k+2,α19α20μα22α23μ+1α25α26μz−k+1,ifn=3+3kμ+k+3,⋮α19α20μα22α23μ+1α25α26μz−1,ifn=3+3kμ+2k+1,α19α20μα22α23μ+1α25α26μz0,ifn=3+3kμ+2k+2,α19α20α22α23α25α26μ+1y−k,ifn=3k+3μ+2k+3,α19α20α22α23α25α26μ+1y−k+1,ifn=3+3kμ+2k+3,⋮α19α20α22α23α25α26μ+1y−1,ifn=3+3kμ+3k+2,α19α20α22α23α25α26μ+1y0,ifn=3+3kμ+3k+3.
Finally,(52)0<zn≤α19α20μα22α23μα25α26μ+1y−k,ifn=3+3kμ+1,α19α20μα22α23μα25α26μ+1y−k+1,ifn=3+3kμ+2,⋮α19α20μα22α23μα25α26μ+1y−1,ifn=3+3kμ+k,α19α20μα22α23μα25α26μ+1y0,ifn=3+3kμ+k+1,α19α20μ+1α22α23μα25α26μ+1x−k,ifn=3+3kμ+k+2,α19α20μ+1α22α23μα25α26μ+1x−k+1,ifn=3+3kμ+k+3,⋮α19α20μ+1α22α23μα25α26μ+1x−1,ifn=3+3kμ+2k+1,α19α20μ+1α22α23μα25α26μ+1x0,ifn=3+3kμ+2k+2,α19α20α22α23α25α26μ+1z−k,ifn=3+3kμ+2k+3,α19α20α22α23α25α26μ+1z−k+1,ifn=3+3kμ+2k+3,⋮α19α20α22α23α25α26μ+1z−1,ifn=3+3kμ+3k+2,α19α20α22α23α25α26μ+1z0,ifn=3+3kμ+3k+3.
(ii) IfXn is the positive solution of (13), then for μ≥0, the following holds:
(53)0<xn≤α28α29μ+1α31α32μα34α35μy−k,ifn=3+3kμ+1,α28α29μ+1α31α32μα34α35μy−k+1,ifn=3+3kμ+2,⋮α28α29μ+1α31α32μα34α35μy−1,ifn=3+3kμ+k,α28α29μ+1α31α32μα34α35μy0,ifn=3+3kμ+k+1,α28α29μ+1α31α32μ+1α34α35μz−k,ifn=3+3kμ+k+2,α28α29μ+1α31α32μ+1α34α35μz−k+1,ifn=3+3kμ+k+3,⋮α28α29μ+1α31α32μ+1α34α35μz−1,ifn=3+3kμ+2k+1,α28α29μ+1α31α32μ+1α34α35μz0,ifn=3+3kμ+2k+2,α28α29α31α32α34α35μ+1x−k,ifn=3+3kμ+2k+3,α28α29α31α32α34α35μ+1x−k+1,ifn=3+3kμ+2k+4,⋮α28α29α31α32α34α35μ+1x−1,ifn=3+3kμ+3k+2,α28α29α31α32α34α35μ+1x0,ifn=3+3kμ+3k+3.
And(54)0<yn≤α28α29μα31α32μ+1α34α35μz−k,ifn=3+3kμ+1,α28α29μα31α32μ+1α34α35μz−k+1,ifn=3+3kμ+2⋮α28α29μα31α32μ+1α34α35μz−1,ifn=3+3kμ+k,α28α29μα31α32μ+1α34α35μz0,ifn=3+3kμ+k+1,α28α29μα31α32μ+1α34α35μx−k,ifn=3+3kμ+k+2,α28α29μα31α32μ+1α34α35μx−k+1,ifn=3+3kμ+k+3,⋮α28α29μα31α32μ+1α34α35μx−1,ifn=3+3kμ+2k+1,α28α29μα31α32μ+1α34α35μx0,ifn=3+3kμ+2k+2,α28α29α31α32α34α35μ+1y−k,ifn=3+3kμ+2k+3,α28α29α31α32α34α35μ+1y−k+1,ifn=3+3kμ+2k+4,⋮α28α29α31α32α34α35μ+1y−1,ifn=3+3kμ+3k+2,α28α29α31α32α34α35μ+1y0,ifn=3+3kμ+3k+3.
Finally,(55)0<zn≤α28α29μα31α32μα34α35μ+1x−k,ifn=3+3kμ+1,α28α29μα31α32μα34α35μ+1x−k+1,ifn=3+3kμ+2,⋮α28α29μα31α32μα34α35μ+1x−1,ifn=3+3kμ+k,α28α29μα31α32μα34α35μ+1x0,ifn=3+3kμ+k+1,α28α29μ+1α31α32μα34α35μ+1y−k,ifn=3+3kμ+k+2,α28α29μ+1α31α32μα34α35μ+1y−k+1,ifn=3+3kμ+k+3,⋮α28α29μ+1α31α32μα34α35μ+1y−1,ifn=3+3kμ+2k+1,α28α29μ+1α31α32μα34α35μ+1y0,ifn=3+3kμ+2k+2,α28α29α31α32α34α35μ+1z−k,ifn=3+3kμ+2k+3,α28α29α31α32α34α35μ+1z−k+1,ifn=3+3kμ+2k+4,⋮α28α29α31α32α34α35μ+1z−1,ifn=3+3kμ+3k+2,α28α29α31α32α34α35μ+1z0,ifn=3+3kμ+3k+3.
(iii) IfXn is the positive solution of (14), then for μ≥0, the following holds:
(56)0<xn≤α37α38μ+1α40α41μα43α44μz−k,ifn=3+3kμ+1,α37α38μ+1α40α41μα43α44μz−k+1,ifn=3+3kμ+2,⋮α37α38μ+1α40α41μα43α44μz−1,ifn=3+3kμ+k,α37α38μ+1α40α41μα43α44μz0,ifn=3+3kμ+k+1,α37α38μ+1α40α41μ+1α43α44μy−k,ifn=3+3kμ+k+2,α37α38μ+1α40α41μ+1α43α44μy−k+1,ifn=3+3kμ+k+3,⋮α37α38μ+1α40α41μ+1α43α44μy−1,ifn=3+3kμ+2k+1,α37α38μ+1α40α41μ+1α43α44μy0,ifn=3+3kμ+2k+2,α37α38α40α41α43α44μ+1x−k,ifn=3+3kμ+2k+3,α37α38α40α41α43α44μ+1x−k+1,ifn=3+3kμ+2k+4,⋮α37α38α40α41α43α44μ+1x−1,ifn=3+3kμ+3k+2,α37α38α40α41α43α44μ+1x0,ifn=3+3kμ+3k+3.
And(57)0<yn≤α37α38μα40α41μ+1α43α44μx−k,ifn=3+3kμ+1,α37α38μα40α41μ+1α43α44μx−k+1,ifn=3+3kμ+2,⋮α37α38μα40α41μ+1α43α44μx−1,ifn=3+3kμ+k,α37α38μα40α41μ+1α43α44μx0,ifn=3+3kμ+k+1,α37α38μα40α41μ+1α43α44μz−k,ifn=3+3kμ+k+2,α37α38μα40α41μ+1α43α44μz−k+1,ifn=3+3kμ+k+3,⋮α37α38μα40α41μ+1α43α44μz−1,ifn=3+3kμ+2k+1,α37α38μα40α41μ+1α43α44μz0,ifn=3+3kμ+2k+2,α37α38α40α41α43α44μ+1y−k,ifn=3+3kμ+2k+3,α37α38α40α41α43α44μ+1y−k+1,ifn=3+3kμ+2k+4,⋮α37α38α40α41α43α44μ+1y−1,ifn=3+3kμ+3k+2,α37α38α40α41α43α44m+1y0,ifn=3+3kμ+3k+3.
Finally,(58)0<zn≤α37α38μα40α41μα43α44μ+1y−k,ifn=3+3kμ+1,α37α38μα40α41μα43α44μ+1y−k+1,ifn=3+3kμ+2,⋮α37α38μα40α41μα43α44μ+1y−1,ifn=3+3kμ+k,α37α38μα40α41μα43α44μ+1y0,ifn=3+3kμ+k+1,α37α38μ+1α40α41μα43α44μ+1x−k,ifn=3+3kμ+k+2,α37α38μ+1α40α41μα43α44μ+1x−k+1,ifn=3+3kμ+k+3,⋮α37α38μ+1α40α41μα43α44μ+1x−1,ifn=3+3kμ+2k+1,α37α38μ+1α40α41μα43α44μ+1x0,ifn=3+3kμ+2k+2α37α38α40α41α43α44μ+1z−k,ifn=3+3kμ+2k+3,α37α38α40α41α43α44μ+1z1−k,ifn=3+3kμ+2k+4,⋮α37α38α40α41α43α44μ+1z−1,ifn=3+3kμ+3k+2,α37α38α40α41α43α44μ+1z0,ifn=3+3kμ+3k+3.Proof.
Same as the proof of Theorem3.Corollary 6.
The following statements are true for (12)–(14):
(i)
If (40) holds, then Xn of discrete system (12) is bounded(ii)
If (41) holds, then Xn of discrete system (13) is bounded(iii)
If (42) holds, then Xn of discrete system (14) is boundedProof.
Its consequence of Theorem5.


## 5. Global Dynamics aboutP0 of (11)–(14)

Theorem 7.
If (30) holds, then P0 of (11) is globally stable.Proof.
If (30) is true, then from discrete system (11), one has
(59)xn+1=α10yn−kα11+α12∏i=0kzn−i≤α10α11yn−k<yn−k,yn+1=α13zn−kα14+α15∏i=0kxn−i≤α13α14zn−k<zn−k,zn+1=α16xn−kα17+α18∏i=0kyn−i≤α16α17xn−1<xn−k.
From (59), one gets
(60)x3k+3n+1<y3k+3n−kandx3k+3n+3k+4<y3k+3n+2k+3,y3k+3n+1<z3k+3−kandy3k+3n+3k+4<z3k+3n+2k+3,z3k+3n+1<x3k+3n−kandz3k+3n+3k+4<x3k+3n+2k+3.
Moreover, from (60), one gets
(61)x3k+3n+3k+4<y3k+3n+2k+3<z3k+3n+k+2<x3k+3n+1,y3k+3n+3k+4<z3k+3n+2k+3<x3k+3n+k+2<y3k+3n+1,z3k+3n+3k+4<x3k+3n+2k+3<y3k+3n+k+2<z3k+3n+1.
From (61), one can conclude that x3k+3n+1,⋯,x3k+3n+3k+3; y3k+3n+1,⋯,y3k+3n+3k+3; and z3k+3n+1,⋯,z3k+3n+3k+3 are decreasing. Thus, xn,yn, and zn are decreasing, and hence, one can conclude that limn⟶∞xn,yn,zn=P0.Theorem 8.
If respective parametric conditions (40), (41), and (42) hold, then P0 of discrete systems (12)–(14) is globally stable.Proof.
Like as the proof of Theorem7.


## 6. Convergence Rate

Theorem 9.
If respective parametric conditions (30), (40), (41), and (42) hold, then error vector
(62)ϱn~=ϱn1⋮ϱn−k1ϱn2⋮ϱn−k2ϱn3⋮ϱn−k3,of positive solution: Xn of corresponding discrete systems (11)–(14) satisfies the following relations:
(63)limn⟶∞ϱnn=νJP0,limn⟶∞ϱn+1ϱn=νJP0,where νJP0 is equivalent to the modulus of one of the characteristic roots of JP0 calculated at trivial fixed point P0.Proof.
LetXn be a positive solution of (11) for which limn⟶∞xn,yn,zn=Λ. In order for error terms, we have
(64)xn+1−x=α10yn−kα11+α12∏i=0kzn−i−α10yα11+α12zk+1,=α10α11+α12∏i=0kzn−iyn−k−y−α10α12y∏i=1kzn−iα11+α12∏i=0kzn−iα11+α12zk+1zn−z−α10α12yz∏i=2kzn−iα11+α12∏i=0kzn−iα11+α12zk+1zn−1−z−⋯−α10α12yzk−1zn−kα11+α12∏i=0kzn−iα11+α12zk+1zn−k+1−z−α10α12yzkα11+α12∏i=0kzn−iα11+α12zk+1zn−k−z,(65)yn+1−y=α13zn−kα14+α15∏i=0kxn−i−α13zα14+α15xk+1,=−α13α15z∏i=1kxn−iα14+α15∏i=0kxn−iα14+α15xk+1xn−x−α13α15zx∏i=2kxn−iα14+α15∏i=0kxn−iα14+α15xk+1xn−1−x−⋯−α13α15zxk−1xn−kα14+α15∏i=0kxn−iα14+α15xk+1xn−k+1−x−α13α15zxkα14+α15∏i=0kxn−iα14+α15xk+1xn−k−x+α13α14+α15∏i=01xn−kzn−k−z,(66)zn+1−z=α16xn−kα17+α18∏i=0kyn−i−α16xα17+α18yk+1,=α16α17+α18∏i=0kyn−ixn−k−x−α16α18x∏i=1kyn−iα17+α18∏i=0kyn−iα17+α18yk+1yn−y−α16α18xy∏i=2kyn−iα17+α18∏i=0kyn−iα17+α18yk+1yn−1−y−⋯−α16α18xyk−1yn−kα17+α18∏i=0kyn−iα17+α18yk+1yn−k+1−yα16α18xykα17+α18∏i=0kyn−iα17+α18yk+1yn−k−y.
Now, set(67)ϱn1=xn−x,ϱn2=yn−y,ϱn3=zn−z.
Utilizing (67) in (64) and (66), one gets
(68)ϱn+11=Ann−kϱn−k2+Bnnϱn3+Bnn−1ϱn−13+⋯+Bnn−k+1ϱn−k+13+Bnn−kϱn−k3,ϱn+12=Cnnϱn1+Cnn−1ϱn−11+⋯+Cnn−k+1ϱn−k+11+Cnn−kϱn−k1+Dnn−kϱn−k3,ϱn+13=Enn−kϱn−k1+Fnnϱn2+Fnn−1ϱn−12+⋯+Fnn−k+1ϱn−k+12+Fnn−kϱn−k2,where
(69)Ann−k=α10α11+α12zk+1,Bnn=−α10α12y∏i=1kzn−iα11+α12∏i=0kzn−iα11+α12zk+1,Bnn−1=−α10α12yz∏i=2kzn−iα11+α12∏i=0kzn−iα11+α12zk+1,⋮Bnn−k+1=−α10α12yzk−1zn−kα11+α12∏i=0kzn−iα11+α12zk+1,Bnn−k=−α10α12yzkα11+α12∏i=0kzn−iα11+α12zk+1,Cnn=−α13α15z∏i=1kxn−iα14+α15∏i=0kxn−iα14+α15xk+1,Cnn−1=−α13α15zx∏i=2kxn−iα14+α15∏i=0kxn−iα14+α15xk+1,⋮Cnn−k+1=−α13α15zxk−1xn−kα14+α15∏i=0kxn−iα14+α15xk+1,Cnn−k=−α13α15zxkα14+α15∏i=0kxn−iα14+α15xk+1,(70)Dnn−k=α13α14+α15∏i=01xn−k,Enn−k=α16α17+α18∏i=0kyn−i,Fnn=−α16α18x∏i=1kyn−iα17+α18∏i=0kyn−iα17+α18yk+1,Fnn−1=−α16α18xy∏i=2kyn−iα17+α18∏i=0kyn−iα17+α18yk+1,⋮Fnn−k+1=−α16α18xyk−1yn−kα17+α18∏i=0kyn−iα17+α18yk+1,Fnn−k=α16α18xykα17+α18∏i=0kyn−iα17+α18yk+1.
From (69) and (70), we get
(71)limn⟶∞Ann−k=α10α11+α12zk+1,limn⟶∞Bnn=limn⟶∞Bnn−1=⋯=limn⟶∞Bnn−k+1=limn⟶∞Bnn−k=−α10α12yzkα11+α12zk+12,limn⟶∞Cnn=limn⟶∞Cnn−1=⋯=limn⟶∞Cnn−k+1=limn⟶∞Cnn−k=−α13α15zxkα14+α15xk+12,limn⟶∞Dnn−k=α13α14+α15xk+1,limn⟶∞Enn−k=α16α17+α18yk+1,limn⟶∞Fnn=limn⟶∞Fnn−1=⋯=limn⟶∞Fnn−k+1=limn⟶∞Fnn−k=−α16α18xykα17+α18yk+12.
So the limiting system [23]:
(72)ϱn+1~=K~ϱn~,where ϱn~ is depicted in (62) and K~ is same as J∧ about ∧. In particular about P0, it becomes
(73)ϱn+11⋮ϱn−k+11ϱn+12⋮ϱn−k+12ϱn+13⋮ϱn−k+13=00⋯0000⋯0α10α1100⋯0010⋯0000⋯0000⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯1000⋯0000⋯0000⋯0000⋯0000⋯0α13α1400⋯0010⋯0000⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯0000⋯1000⋯0000⋯0α16α1700⋯0000⋯0000⋯0000⋯0010⋯00⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮⋮00⋯0000⋯0000⋯10ϱn1⋮ϱn−k1ϱn2⋮ϱn−k2ϱn3⋮ϱn−k3,which is the same as the linearized system of (11) about P0. Adopting a similar procedure, one can prove the rest of the results for discrete systems (12)–(14).


## 7. Numerical Simulations

We give some simulations in this section to verify not only the main finding but also show different behavior of discrete systems (11)–(14). For the systems under consideration, these numerical simulations provide time plots for xn,yn,zn, as well as a global attractor. The following scenarios should be considered in this regard:Example 1.
Figure1 indicates the behavior of discrete system (11) about P0 if k=2, αℓℓ=1,2,⋯,9, respectively, are 4, 5, 3, 25, 26, 1, 32, 33, and 4 and initial values x−ℓ, ℓ−ℓ, z−ℓℓ=2,1,0, respectively, are 0.7, 0.9, 0.4, 1.1, 0.9, 0.4, 0.9, 0.9, and 0.8. Moreover, Figures 1(a)–1(c) show that P0 of discrete system (11) is a sink, and its attractor is shown in Figure 1(d).Example 2.
Figure2 indicates the behavior of discrete system (11) about P0 if k=2, αℓℓ=1,2,⋯,9, respectively, are 6, 5, 0.1, 25, 22, 0.3, 32, 30, and 0.4 and initial values x−ℓ, ℓ−ℓ, and z−ℓℓ=2,1,0, respectively, are 0.7, 0.9, 0.4, 1.1, 0.9, 0.4, 0.9, 0.9, and 0.8. Moreover, Figures 2(a)–2(c) show that P0 of discrete system (11) is unstable.Example 3.
Figure3 indicates the behavior of discrete system (12) about P0 if k=2, αℓℓ=1,2,⋯,9, respectively, are 14, 15, 3, 15, 16, 3, 3, 4, and 4 and x−ℓ, ℓ−ℓ, and z−ℓℓ=2,1,0, respectively, are 0.9, 0.9, 0.4, 1.1, 0.9, 0.4, 0.45, 0.23, and 0.2. Moreover, Figures 3(a)–3(c) show that P0 of discrete system (12) is a sink, and its attractor is shown in Figure 3(d).Example 4.
Figure4 indicates the behavior of discrete system (12) about P0 if k=2, αℓℓ=1,2,⋯,9, respectively, are 14, 13, 13, 17, 16, 13, 6, 4, and 14 and x−ℓ, ℓ−ℓ, and z−ℓℓ=2,1,0, respectively, are 0.9, 0.9, 0.4, 1.1, 0.9, 0.4, 0.45, 0.23, and 0.2. Moreover, Figures 4(a)–4(c) show that P0 of discrete system (12) is unstable.Example 5.
Figure5 indicates the behavior of discrete system (13) about P0 if k=2, αℓℓ=1,2,⋯,9, respectively, are 24, 25, 3, 5, 6, 3, 22, 23, and 8 and x−ℓ, ℓ−ℓ, and z−ℓℓ=2,1,0, respectively, are 1.9, 1.9, 1.4, 1.1, 1.9, 1.4, 1.45, 1.23, and 1.2. Moreover, Figures 5(a)–5(c) show that P0 of discrete system (13) is a sink, and its attractor is shown in Figure 5(d).Example 6.
Figure6 indicates the behavior of discrete system (13) about P0 if k=2, αℓℓ=1,2,⋯,9, respectively, are 27, 25, 3, 8, 6, 3, 25, 23, and 8 and x−ℓ, ℓ−ℓ, and z−ℓℓ=2,1,0, respectively, are 1.9, 1.9, 1.4, 1.1, 1.9, 1.4, 1.45, 1.23, and 1.2. Moreover, Figures 6(a)–6(c) show that P0 of discrete system (13) is unstable.Example 7.
Figure7 indicates the behavior of discrete system (14) about P0 if k=2, αℓℓ=1,2,⋯,9, respectively, are 124, 125, 3, 5, 6, 3, 122, 123, and 8 and x−ℓ, ℓ−ℓ, and z−ℓℓ=2,1,0, respectively, are 1.9, 1.9, 1.4, 1.1, 1.9, 1.4, 1.45, 1.23, and 1.2. Moreover, Figures 7(a)–7(c) show that P0 of discrete system (14) is a sink, and its attractor is shown in Figure 7(d).Example 8.
Figure8 indicates the behavior of discrete system (14) about P0 if k=2, αℓℓ=1,2,⋯,9, respectively, are 124, 123, 3, 15, 14, 3, 122, 121, and 8 and x−ℓ, ℓ−ℓ, and z−ℓℓ=2,1,0, respectively, are 1.9, 1.9, 1.4, 1.1, 1.9, 1.4, 1.45, 1.23, and 1.2. Moreover, Figures 8(a)–8(c) show that P0 of discrete system (14) is unstable.Figure 1
Dynamical characteristics of (11).
(a)(b)(c)(d)Figure 2
Dynamical characteristics of (11).
(a)(b)(c)Figure 3
Dynamical characteristics of (12).
(a)(b)(c)(d)Figure 4
Dynamical characteristics of (12).
(a)(b)(c)Figure 5
Dynamical characteristics of (13).
(a)(b)(c)(d)Figure 6
Dynamical characteristics of (13).
(a)(b)(c)Figure 7
Dynamical characteristics of (14).
(a)(b)(c)(d)Figure 8
Dynamical characteristics of (14).
(a)(b)(c)


## 8. Conclusion and Future Work

In this paper, we explored global dynamical characteristics, boundedness, and convergence rate of certain higher-order discrete systems of difference equations which is a natural extension of [21, 22]. We have proved that trivial fixed point P0 of discrete systems (11)–(14) is globally stable if, respectively, parametric conditions (i) α10/α11<1,α13/α14<1,andα16/α17<1; (ii) α19/α20<1,α22/α23<1,andα25/α26<1; (iii) α28/α29<1,α31/α32<1,andα34/α35<1; and (iv) α37/α38<1,α40/α41<1,andα43/α44<1 hold. Further, we have proved that every positive solution of discrete systems (11)–(14) is bounded if (i) α10/α11<1,α13/α14<1,andα16/α17<1; (ii) α19/α20<1,α22/α23<1,andα25/α26<1; (iii) α28/α29<1,α31/α32<1,andα34/α35<1; and (iv) α37/α38<1,α40/α41<1,andα43/α44<1 hold. It is also examined that positive solution of discrete systems (11)–(14) converges to P0. Finally, numerical verification of theoretical results is performed. Closed-form solution and calculation of forbidden set for the discrete systems (11)–(14) are our next aim to study.



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