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# Global dynamics of some systems of higher-order rational difference equations

Abdul Qadeer Khan, Muhammad Naeem Qureshi and Qamar Din*

Author Affiliations

Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan

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Advances in Difference Equations 2013, 2013:354  doi:10.1186/1687-1847-2013-354

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/354

 Received: 3 October 2013 Accepted: 7 November 2013 Published: 3 December 2013

© 2013 Khan et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In the present work, we study the qualitative behavior of two systems of higher-order rational difference equations. More precisely, we study the local asymptotic stability, instability, global asymptotic stability of equilibrium points and rate of convergence of positive solutions of these systems. Our results considerably extend and improve some recent results in the literature. Some numerical examples are given to verify our theoretical results.

MSC: 39A10, 40A05.

##### Keywords:
system of difference equations; stability; global character; rate of convergence

### 1 Introduction

Recently, studying the qualitative behavior of difference equations and systems is a topic of great interest. Applications of discrete dynamical systems and difference equations have appeared recently in many areas such as ecology, population dynamics, queuing problems, statistical problems, stochastic time series, combinatorial analysis, number theory, geometry, electrical networks, neural networks, quanta in radiation, genetics in biology, economics, psychology, sociology, physics, engineering, economics, probability theory and resource management. Unfortunately, these are only considered as the discrete analogs of differential equations. It is a well-known fact that difference equations appeared much earlier than differential equations and were instrumental in paving the way for the development of the latter. It is only recently that difference equations have started receiving the attention they deserve. Perhaps this is largely due to the advent of computers where differential equations are solved by using their approximate difference equation formulations. The theory of discrete dynamical systems and difference equations developed greatly during the last twenty-five years of the twentieth century. The theory of difference equations occupies a central position in applicable analysis. There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. Nonlinear difference equations of order greater than one are of paramount importance in applications. It is very interesting to investigate the behavior of solutions of a system of higher-order rational difference equations and to discuss the local asymptotic stability of their equilibrium points. Systems of rational difference equations have been studied by several authors. Especially there has been a great interest in the study of the attractivity of the solutions of such systems. For more results on the qualitative behavior of nonlinear difference equations, we refer the interested reader to [1-19].

Zhang et al.[20] studied the dynamics of a system of rational third-order difference equations

Din et al.[11] investigated the dynamics of a system of fourth-order rational difference equations

To be motivated by the above studies, our aim in this paper is to investigate the qualitative behavior of the following -order systems of rational difference equations:

(1)

where the parameters α, β, γ, , , and initial conditions , are positive real numbers, and

(2)

where the parameters a, b, c, , , and initial conditions , are positive real numbers. This paper is a natural extension of [11,20,21].

Let us consider -dimensional discrete dynamical system of the form

(3)

where and are continuously differentiable functions and I, J are some intervals of real numbers. Furthermore, a solution of system (3) is uniquely determined by initial conditions for . Along with system (3), we consider the corresponding vector map . An equilibrium point of (3) is a point that satisfies

The point is also called a fixed point of the vector map F.

Definition 1 Let be an equilibrium point of system (3).

(i) An equilibrium point is said to be stable if for every there exists such that for every initial condition , , implies for all , where is the usual Euclidian norm in .

(ii) An equilibrium point is said to be unstable if it is not stable.

(iii) An equilibrium point is said to be asymptotically stable if there exists such that and as .

(iv) An equilibrium point is called a global attractor if as .

(v) An equilibrium point is called an asymptotic global attractor if it is a global attractor and stable.

Definition 2 Let be an equilibrium point of the map

where f and g are continuously differentiable functions at . The linearized system of (3) about the equilibrium point is

where

and is the Jacobian matrix of system (3) about the equilibrium point .

Lemma 1[22]

Assume that,  , is a system of difference equations andis the fixed point ofF. If all eigenvalues of the Jacobian matrixaboutlie inside an open unit disk, thenis locally asymptotically stable. If one of them has norm greater than one, thenis unstable.

Lemma 2[23]

Assume that,  , is a system of difference equations andis the equilibrium point of this system. The characteristic polynomial of this system about the equilibrium pointis, with real coefficients and. Then all roots of the polynomiallie inside the open unit diskif and only iffor , whereis the principal minor of orderkof thematrix

(4)

Let us consider a system of difference equations

(5)

where is an m-dimensional vector, is a constant matrix, and is a matrix function satisfying

(6)

as , where denotes any matrix norm which is associated with the vector norm

Proposition 1 (Perron’s theorem)[24]

Suppose that condition (6) holds. Ifis a solution of (5), then eitherfor all largenor

(7)

exists and is equal to the modulus of one of the eigenvalues of matrixA.

Proposition 2[24]

Suppose that condition (6) holds. Ifis a solution of (5), then eitherfor all largenor

(8)

exists and is equal to the modulus of one of the eigenvalues of matrixA.

### 2 On the system ,

In this section, we shall investigate the qualitative behavior of system (1). Let be an equilibrium point of system (1), then for and , system (1) has two positive equilibrium points , , where and .

To construct the corresponding linearized form of system (1), we consider the following transformation:

(9)

where , , , … , and , , , … , . The Jacobian matrix about the fixed point under the transformation (9) is given by

where , , and .

Theorem 1Letand, then every solutionof system (1) is bounded.

Proof It is easy to verify that

and

Take and . Then and for all  . □

Theorem 2The equilibrium pointof system (1) is locally asymptotically stable.

Proof The linearized system of (1) about the equilibrium point is given by

where

and

Let denote the eigenvalues of matrix E. Let be a diagonal matrix, where , , , and

Clearly, D is invertible. Computing , we obtain

We obtain the following two inequalities:

which implies that

and

Furthermore,

and

It is a well-known fact that E has the same eigenvalues as . Hence, we obtain

Hence, the equilibrium point of system (1) is locally asymptotically stable. □

Theorem 3The positive equilibrium pointof system (1) is unstable.

Proof The linearized system of (1) about the equilibrium point is given by

where

and

where and . The characteristic polynomial of is given by

(10)

From (10), we have

(11)

It is clear that not all of . Therefore, by Lemma 1, the unique positive equilibrium point is unstable. □

Theorem 4Letand, and letbe a solution of system (1). Then, for, the following statements are true:

(i) If, then

(ii) If, then

Proof It follows from induction. □

Theorem 5Letand, then the equilibrium pointof system (1) is globally asymptotically stable.

Proof For and , from Theorem 2, is locally asymptotically stable. From Theorem 1, every positive solution is bounded, i.e., and for all  , where and . So, it is sufficient to prove that is decreasing. From system (1), one has

This implies that and . Hence, the subsequences

are decreasing, i.e., the sequence is decreasing. Also,

This implies that and . Hence, the subsequences

are decreasing, i.e., the sequence is decreasing. Hence, . □

Theorem 6Letand. Then, for a solutionof system (1), the following statements are true:

(i) If, then.

(ii) If, then.

#### 2.1 Rate of convergence

We investigate the rate of convergence of a solution that converges to the equilibrium point of system (1).

Assume that and . First we will find a system of limiting equations for the map F. The error terms are given as

Set and , one has

where for ,

Taking the limits, we obtain for , , for , for , for and . Hence, the limiting system of error terms at can be written as

(12)

where

which is similar to the linearized system of (1) about the equilibrium point . Using proposition (1), one has the following result.

Theorem 7Assume thatis a positive solution of system (1) such that, and, where. Then the error vectorof every solution of (1) satisfies both of the following asymptotic relations:

whereare the characteristic roots of the Jacobian matrixabout.

### 3 On the system ,

In this section, we shall investigate the qualitative behavior of system (2). Let be an equilibrium point of system (2), then system (2) has a unique equilibrium point . To construct the corresponding linearized form of system (2), we consider the following transformation:

(13)

, , , …, and , , , … , . The Jacobian matrix about the fixed point under the transformation (13) is given by

where , , and .

Theorem 8Letbe a positive solution of system (2), then for every, the following results hold.

Lemma 3Let, then every solutionof system (2) is bounded.

Proof Assume that

and

Then from Theorem 8 one can easily see that and for all  . □

Theorem 9The equilibrium pointof equation (2) is locally asymptotically stable.

Proof The linearized system of (2) about the equilibrium point is given by

where

and

Let denote the eigenvalues of matrix E. Let be a diagonal matrix, where , , and

Clearly, D is invertible. Computing , we obtain

Next, we have the following two inequalities:

which implies that

and

Furthermore,

and

Now H has the same eigenvalues as , we obtain that

Hence, the equilibrium point of system (2) is locally asymptotically stable. □

Theorem 10Letand, then the equilibrium pointof system (2) is globally asymptotically stable.

Proof Assume that and . Then from Theorem 9 the equilibrium point of system (2) is locally asymptotically stable. Moreover, from Lemma 3 every positive solution is bounded, i.e., and for all  , where and . Now, it is sufficient to prove that is decreasing. From system (2) one has

This implies that and .

This implies that

Hence, and . Hence, the subsequences

and

are decreasing. Therefore the sequences and are decreasing. Hence, . □

#### 3.1 Rate of convergence

Assume that and . First we will find a system of limiting equations for system (2). The error terms are given as

Set and , then one has

where

for , , for ,

and . Taking the limits, we obtain for , for , , for , , for , So, the limiting system of error terms can be written as

where

and

which is similar to the linearized system of (2) about the equilibrium point . Using proposition (1), one has the following result.

Theorem 11Assume thatis a positive solution of system (2) such that, and, where. Then the error vectorof every solution of (2) satisfies both of the following asymptotic relations:

whereare the characteristic roots of the Jacobian matrixabout.

### 4 Examples

In order to verify our theoretical results, we consider some interesting numerical examples in this section. These examples show that the equilibrium point of both systems (1) and (2) is globally asymptotically stable.

Example 1 Consider system (1) with initial conditions , , , , , , , , , , , , , , , , , . Moreover, choose the parameters , , , , , . Then system (1) can be written as

(14)

with initial conditions , , , , , , , , , , , , , , , , , . Moreover, in Figure 1, the plot of is shown in Figure 1a, the plot of is shown in Figure 1b, and an attractor of system (14) is shown in Figure 1c.

Figure 1. Plots for system (14).

Example 2 Consider system (1) with initial conditions , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Moreover, choose the parameters , , , , , . Then system (1) can be written as

(15)

with initial conditions , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Moreover, in Figure 2, the plot of is shown in Figure 2a, the plot of is shown in Figure 2b, and an attractor of system (15) is shown in Figure 2c.

Figure 2. Plots for system (15).

Example 3 Consider system (2) with initial conditions , , , , , , , , , , , , , . Moreover, choose the parameters , , , , , . Then system (2) can be written as

(16)

with initial conditions , , , , , , , , , , , , , . Moreover, in Figure 3, the plot of is shown in Figure 3a, the plot of is shown in Figure 3b, and an attractor of system (16) is shown in Figure 3c.

Figure 3. Plots for system (16).

Example 4 Consider system (2) with initial conditions , , , , , , , , , , , , , , , , , , , , , , , , , . Moreover, choose the parameters , , , , , . Then system (2) can be written as

(17)

with initial conditions , , , , , , , , , , , , , , , , , , , , , , , , , . Moreover, in Figure 4, the plot of is shown in Figure 4a, the plot of is shown in Figure 4b, and an attractor of system (17) is shown in Figure 4c.

Figure 4. Plots for system (17).

### Conclusion

This work is a natural extension of [11,20,21]. In the paper, we have investigated the qualitative behavior of -dimensional discrete dynamical systems. Each system has only one equilibrium point which is stable under some restriction to parameters. The linearization method is used to show that equilibrium point is locally asymptotically stable. The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state. An approach to this problem consists of determining the possible global behaviors of the system and determining which initial conditions lead to these long-term behaviors. In case of higher-order dynamical systems, it is crucial to discuss global behavior of the system. Some powerful tools such as semiconjugacy and weak contraction cannot be used to analyze global behavior of systems (1) and (2). In the paper, we prove the global asymptotic stability of equilibrium point by using simple techniques. We have carried out a systematical local and global stability analysis of both systems. The most important finding here is that the unique equilibrium point can be a global asymptotic attractor for systems (1) and (2). Moreover, we have determined the rate of convergence of a solution that converges to the equilibrium point of systems (1) and (2). Some numerical examples are provided to support our theoretical results. These examples are experimental verifications of theoretical discussions.

### Competing interests

The authors have no competing interests.

### Authors’ contributions

All authors contributed equally in drafting this manuscript and giving the main proofs.

### Acknowledgements

This work was supported by the Higher Education Commission of Pakistan.

### References

1. Cinar, C: On the positive solutions of the difference equation system ; . Appl. Math. Comput.. 158, 303–305 (2004)

2. Stević, S: On some solvable systems of difference equations. Appl. Math. Comput.. 218, 5010–5018 (2012)

3. Kurbanli, AS: On the behavior of positive solutions of the system of rational difference equations , , . J. Differ. Equ.. 2011, (2011) Article ID 40

4. Stević, S: On a third-order system of difference equations. Appl. Math. Comput.. 218, 7649–7654 (2012)

5. Bajo, I, Liz, E: Global behaviour of a second-order nonlinear difference equation. J. Differ. Equ. Appl.. 17(10), 1471–1486 (2011)

6. Kalabuŝić, S, Kulenović, MRS, Pilav, E: Dynamics of a two-dimensional system of rational difference equations of Leslie-Gower type. Adv. Differ. Equ. (2011). BioMed Central Full Text

7. Kalabuŝić, S, Kulenović, MRS, Pilav, E: Global dynamics of a competitive system of rational difference equations in the plane. Adv. Differ. Equ.. 2009, (2009) Article ID 132802

8. Kurbanli, AS, Çinar, C, Yalçinkaya, I: On the behavior of positive solutions of the system of rational difference equations . . Math. Comput. Model.. 53, 1261–1267 (2011)

9. Din, Q: Dynamics of a discrete Lotka-Volterra model. Adv. Differ. Equ.. 1, 1–13 (2013)

10. Din, Q, Donchev, T: Global character of a host-parasite model. Chaos Solitons Fractals. 54, 1–7 (2013)

11. Din, Q, Qureshi, MN, Khan, AQ: Dynamics of a fourth-order system of rational difference equations. Adv. Differ. Equ.. 1, 1–15 (2012)

12. Din, Q, Khan, AQ, Qureshi, MN: Qualitative behavior of a host-pathogen model. Adv. Differ. Equ. (2013). BioMed Central Full Text

13. Din, Q: Global behavior of a rational difference equation. Acta Univ. Apulensis. 34, 35–49 (2013)

14. Din, Q, Ibrahim, TF: Global behavior of a neural networks system. Indian J. Comput. Appl. Math.. 1(1), 79–92 (2013)

15. Qureshi, MN, Khan, AQ, Din, Q: Global behavior of third order system of rational difference equations. Int. J. Eng. Res. Technol.. 2(5), 2182–2191 (2013)

16. El-Metwally, H, Elsayed, EM: Form of solutions and periodicity for systems of difference equations. J. Comput. Anal. Appl.. 15(5), 852–857 (2013)

17. Touafek, N, Elsayed, EM: On the solutions of systems of rational difference equations. Math. Comput. Model.. 55, 1987–1997 (2012)

18. Elsayed, EM, El-Metwally, HA: On the solutions of some nonlinear systems of difference equations. Adv. Differ. Equ. (2013). BioMed Central Full Text

19. Elsayed, EM: Solution and attractivity for a rational recursive sequence. Discrete Dyn. Nat. Soc.. 2011, (2011) Article ID 982309

20. Zhang, Q, Yang, L, Liu, J: Dynamics of a system of rational third order difference equation. J. Differ. Equ. (2012). BioMed Central Full Text

21. Shojaei, M, Saadati, R, Adibi, H: Stability and periodic character of a rational third order difference equation. Chaos Solitons Fractals. 39, 1203–1209 (2009)

22. Sedaghat, H: Nonlinear Difference Equations: Theory with Applications to Social Science Models, Kluwer Academic, Dordrecht (2003)

23. Kocic, VL, Ladas, G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic, Dordrecht (1993)

24. Pituk, M: More on Poincare’s and Perron’s theorems for difference equations. J. Differ. Equ. Appl.. 8, 201–216 (2002)