Research Article | Open Access

A. Q. Khan, M. S. M. Noorani, H. S. Alayachi, "Global Dynamics of Higher-Order Exponential Systems of Difference Equations", *Discrete Dynamics in Nature and Society*, vol. 2019, Article ID 3825927, 19 pages, 2019. https://doi.org/10.1155/2019/3825927

# Global Dynamics of Higher-Order Exponential Systems of Difference Equations

**Academic Editor:**Zhengqiu Zhang

#### 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. CÃ¶mert 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 [11â€“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*â€‰*then (11) has a unique equilibrium .*(ii)*If*â€‰*then (12) has a unique equilibrium .*(iii)*If*â€‰*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*â€‰*and*â€‰*then of (11) is a sink.*(ii)*If*â€‰*and*â€‰*then of (12) is a sink.*(iii)*If*â€‰*and*â€‰*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 gets