Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 3825927, 19 pages
https://doi.org/10.1155/2019/3825927

Research Article

Global Dynamics of Higher-Order Exponential Systems of Difference Equations

A. Q. Khan,¹ M. S. M. Noorani,² and H. S. Alayachi²

¹Department of Mathematics, University of Azad Jammu & Kashmir, Muzaffarabad 13100, Pakistan
²School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor, Malaysia

Correspondence should be addressed to A. Q. Khan; abdulqadeerkhan1@gmail.com

Received 11 December 2018; Accepted 24 April 2019; Published 2 June 2019

Academic Editor: Zhengqiu Zhang

Copyright © 2019 A. Q. Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we study the global dynamics of three higher-order exponential systems of rational difference equations. This suggested work considerably extends and improves some existing results in the literature.

1. Introduction

Global dynamical properties of difference equations or systems of difference equations involving exponential term have been widely investigated in recent years. For instance, Ozturk et al. [1] have explored the dynamical properties of the following exponential difference equation:

where , and are positive real numbers. Cömert et al. [2] have explored the dynamical properties of the following higher-order exponential difference equation:

where , and are positive real numbers. Bozkurt [3] has explored the dynamical properties of the following exponential difference equation:

where , and are positive real numbers. In 2009, Ozturk et al. [4] have explored the dynamical properties of the following higher-order exponential difference equation:

where , and are positive real numbers. Papaschinopoulos et al. [5] have explored the dynamical properties of the following exponential systems of difference equations:

where and are positive real numbers. Khan and Sharif [6] have explored the dynamical properties of the following exponential systems of difference equations:

where and are positive real numbers. Khan and Qureshi [7] have explored the dynamical properties of the following exponential system of difference equations:

where and are positive real numbers. Psarros et al. [8] have explored the dynamical properties of the following six exponential systems of difference equations:

where and are positive real numbers. Mylona et al. [9] have explored the dynamical properties of the following two close-to-cyclic systems of exponential difference equations:

where and are positive real numbers. Mylona et al. [10] have explored the dynamical properties of the following systems of difference equations with exponential terms:

where and are positive real numbers, and for more other interesting results on difference equations as well as systems of difference equation, we refer the reader to [11–13] and the references cited therein. Motivated by the above systemic studies, in this paper we aim to explore the dynamical properties of the following higher-order exponential systems of difference equations, which are natural extension of the work studied by Ozturk et al. [4]:

where and are positive real numbers.

The rest of this paper is organized as follows: Section 2 is about the study of boundedness and persistence of systems (11), (12), and (13). This also includes construction of invariant rectangle of these systems. Section 3 is about the uniqueness and existence of fixed point of systems (11), (12), and (13). Section 4 deals with the study of local stability, whereas in Section 5 we study the global dynamics of the unique fixed point of these systems. Section 6 deals with the study of rate of convergence of systems (11), (12), and (13). Brief conclusion is given in Section 7.

2. Boundedness, Persistence, and Construction of Invariant Rectangle of Systems (11), (12), and (13)

Theorem 1. If , then every solution of system (11), system (12), and system (13) is bounded and persists.

Proof. If is a solution of (11), then

From (11) and (14) we get

Finally from (14) and (15) we get

If is a solution of (12), then

From (12) and (17) we have

Finally from (17) and (18) we get

If is a solution of (13), then

From (13) and (20) we get

Finally from (20) and (21) we get

Theorem 2. If , then following holds:

(i)If is a solution of (11), then is invariant rectangle for (11).(ii)If is a solution of (12), then is invariant rectangle for (12).(iii)If is a solution of (13), then is invariant rectangle for (13).

Proof. The proof follows from mathematical induction.

3. Existence and Uniqueness of Equilibrium of Systems (11), (12), and (13)

Theorem 3. Assume that , then the following statements hold:

(i)If and

 then (11) has a unique equilibrium .(ii)If

 then (12) has a unique equilibrium .(iii)If

 then (13) has a unique equilibrium where and are defined in (53) and (54).

Proof. Consider

From (26), we have

From (27), define

where

and . Now our claim is has a single solution . From (28) and (29) one gets

where

If is a solution of , then from (28) and (29) one gets

where

In view of (31), (32), and (33), (30) becomes

Now assume that (23) holds, then from (34) one gets .
. From (12) one has

From (35) one gets

From (36), define

where

and . We claim that has a single solution . From (37) and (38) one gets

where

If is a solution of , then from (37) and (38) one gets

where

In view of (40), (41), and (42), (39) becomes

Finally assume that (24) holds, then from (43) one gets .
From system (13)

From (44), define

where

and . We claim that has a single solution . From (45) and (46) one gets

where

If is a solution of , then from (45) and (46) one gets

where

In view of (48), (49), and (50), (47) becomes

where

and

Finally assume that (25) along with (53) and (54) holds, then from (51) one gets .

4. Local Asymptotic Stability about Equilibrium of Systems (11), (12), and (13)

Theorem 4. Assume that , then the following statements hold:

(i) If

 and

 then of (11) is a sink.(ii)If

 and

 then of (12) is a sink.(iii)If

 and

 then of (13) is a sink.

Proof. If is equilibrium of (11), then

Moreover, linearized equation of (11) about is

where

and

Let eigenvalues of be . Let be a diagonal matrix with , , and

Since is invertible. So,

The following two inequalities,

imply that

Furthermore,

It is a well-known fact that and have the same eigenvalues. Thus

Hence of (11) is a sink.
. If is equilibrium of (12), then

and linearized equation of (12) about is

where is defined in (63). Also

and

Using arrangements as in the proof of one gets

and

Assume that (68) and (69) hold, then

Finally,

Hence equilibrium (12) is a sink.

If is a fixed point of (13), then

and linearized equation of (13) about is

where is again the same as (63). Moreover

where

Again using similar arrangements as in the proof of one gets

Moreover,

Assume that (68) and (69) hold true, then

Finally,

Hence equilibrium of (13) is a sink.

Hereafter we study the global dynamics about unique equilibrium of systems (11), (12), and (13). Since investigating the global stability of difference equations or systems of difference equation for higher-order is a challenging task in recent year, here we will investigate the global dynamics about unique equilibrium of systems (11), (12), and (13) by utilizing discrete-time Lyapunov function motivated by the work of [14–16].

5. Global Stability about Equilibrium of Systems (11), (12), and (13)

Theorem 5. Assume that , then the following statements hold:

(i) If

 and

 then of (11) is globally asymptotically stable.(ii)If

 and

 then of (12) is globally asymptotically stable.(iii) If

 and

 then of (13) is globally asymptotically stable.

Proof. Consider

where its positivity follows from the inequality below:

Moreover,

Now

In view of (99), (100) takes the form

Assuming that (91) and (92) hold, then from (101) one gets for all . Hence, . Therefore, of (11) is globally asymptotically stable.
. Using similar construction as in the proof of one gets

Assuming that (93) and (94) hold, then from (102) one gets for all . Hence, . Therefore, of (12) is globally asymptotically stable.
Again using similar arrangements, one gets

Assuming that (95) and (96) hold, then from (103) one gets for all . Hence, . Thus, of (13) is globally asymptotically stable.

Hereafter we will study rate of convergence of systems (11), (12), and (13) motivated by the work of [15, 17, 18].

6. Rate of Convergence of Systems (11), (12), and (13)

6.1. Rate of Convergence of System (11)

Let be any solution of (11) such that the following hold:

Now computing error terms one gets

After some tedious calculations, from (105) one gets

Similarly,

From (106) and (107), we have

Let

and then (108) becomes

where

Moreover,

Hence limiting system of error terms becomes

where

which is similar to linearized system of (11). Finally we have the following theorem by utilizing Perron’s Theorems (see [19]).

Theorem 6. Assume that is a positive solution of (11) such that (104) along with the following relation holds:

Then the error vector

of every solution of (11) satisfies

where are the characteristic roots of about .

6.2. Rate of Convergence of System (12)

If is any solution of (12) such that (104) holds, then

After calculations, from (118) one gets

Similarly,

From (119) and (120), we have

Let (109) hold, then from (121) one gets

where

Moreover,

Hence limiting system of error terms becomes

where

which is the same as linearized system of (12) about . Finally we have following theorem.

Theorem 7. Assume that is a positive solution of (12) such that (104) along with the following relation holds:

Then the error vector defined in (116) satisfies

where are the characteristic roots of about .

6.3. Rate of Convergence of System (13)

If is any solution of (13) such that (104) holds, then

After simplifying, one gets

Similarly,

From (130) and (131), we have

Let (109) hold, then (132) can be represented as

where

Moreover,

Hence limiting system of error terms becomes

where

which is the same as linearized system of (12) about . Finally we have following theorem.

Theorem 8. Assume that is a positive solution of (13) such that (104) along with following relation holds:

Then the error vector defined in (116) satisfies

where are the characteristic roots of about .

7. Conclusion

This work is about the global dynamics of three higher-order exponential systems of difference equations. We proved that every positive solution is bounded and persistent, and further , , and , respectively, are invariant rectangle for systems (11), (12), and (13). We studied existence and uniqueness of equilibrium and global stability, and conclusions are presented in Table 1. Finally rate of convergence for (11), (12), and (13) is also investigated.

Table 1: Systems with corresponding qualitative behavior.

Systems
Corresponding behavior

(11)
Unique positive equilibrium point if and

  ;

is locally asymptotically stable if

  ,

and

  ;

is globally asymptotically stable if

(12)
Unique positive equilibrium point if

;

is locally asymptotically stable if

  ,

and

  ;

is globally asymptotically stable if

(13)
Unique positive equilibrium point if

  ;

is locally asymptotically stable if

  ,

and

  ;

is globally asymptotically stable if

Data Availability

All the data utilized are included in this article and their sources are cited accordingly.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

A. Q. Khan research is partially supported by the Higher Education Commission of Pakistan, while the work of M. S. M. Noorani and H. S. Alayachi is financially supported by UKM Grant DIP-2017-011 and Ministry of Education Malaysia Grant FRGS/1/2017/STG06/UKM/01/1.

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