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), thenFrom (11) and (14) we getFinally from (14) and (15) we get If is a solution of (12), thenFrom (12) and (17) we haveFinally from (17) and (18) we get If is a solution of (13), thenFrom (13) and (20) we getFinally 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. ConsiderFrom (26), we haveFrom (27), definewhereand . Now our claim is has a single solution . From (28) and (29) one getswhereIf is a solution of , then from (28) and (29) one getswhereIn view of (31), (32), and (33), (30) becomesNow assume that (23) holds, then from (34) one gets .
. From (12) one hasFrom (35) one getsFrom (36), definewhereand . We claim that has a single solution . From (37) and (38) one getswhereIf is a solution of , then from (37) and (38) one getswhereIn view of (40), (41), and (42), (39) becomesFinally assume that (24) holds, then from (43) one gets .
From system (13)From (44), definewhereand . We claim that has a single solution . From (45) and (46) one getswhereIf is a solution of , then from (45) and (46) one getswhereIn view of (48), (49), and (50), (47) becomeswhereand 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), thenMoreover, linearized equation of (11) about iswhereandLet eigenvalues of be . Let be a diagonal matrix with , , andSince is invertible. So,The following two inequalities,imply thatFurthermore,It is a well-known fact that and have the same eigenvalues. ThusHence of (11) is a sink.
. If is equilibrium of (12), thenand linearized equation of (12) about iswhere is defined in (63). Alsoand Using arrangements as in the proof of one getsand
Assume that (68) and (69) hold, then Finally,Hence equilibrium (12) is a sink.
If is a fixed point of (13), thenand linearized equation of (13) about iswhere is again the same as (63). Moreover whereAgain using similar arrangements as in the proof of one getsMoreover,
Assume that (68) and (69) hold true, thenFinally,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 formAssuming 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 getsAssuming 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 getsAssuming 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 getsAfter some tedious calculations, from (105) one getsSimilarly,From (106) and (107), we haveLetand then (108) becomeswhereMoreover,
Hence limiting system of error terms becomeswhere
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 vectorof every solution of (11) satisfieswhere are the characteristic roots of about .
6.2. Rate of Convergence of System (12)
If is any solution of (12) such that (104) holds, thenAfter calculations, from (118) one getsSimilarly,From (119) and (120), we haveLet (109) hold, then from (121) one gets
whereMoreover,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) satisfieswhere are the characteristic roots of about .
6.3. Rate of Convergence of System (13)
If is any solution of (13) such that (104) holds, thenAfter simplifying, one getsSimilarly,From (130) and (131), we have
Let (109) hold, then (132) can be represented aswhereMoreover, Hence limiting system of error terms becomes wherewhich 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) satisfieswhere 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.
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.