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Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 179401, 5 pages

http://dx.doi.org/10.1155/2013/179401

## Dynamics of a System of Rational Higher-Order Difference Equation

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12121, Thailand

Received 5 February 2013; Accepted 22 April 2013

Academic Editor: Cengiz Çinar

Copyright © 2013 Banyat Sroysang. 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

We focus on a system of a rational -order difference equation , , , where . We investigate the dynamical behavior of positive solution for the system.

#### 1. Introduction

In 2011, Kurbanli et al. [1] studied the behavior of positive solutions of the system of rational difference equations where the initial conditions are arbitrary nonnegative real numbers.

In the same year, Kurbanli [2] studied the behavior of solutions of the system of rational difference equations where the initial conditions are arbitrary real numbers. Moreover, Kurbanli [3] studied the behavior of the solutions of the difference equation system where such that , and .

In [4], Liu et al. gave more results of the solution of the system (2) including a new and simple expression of and the asymptotical behavior of the solution.

In [5], Stević showed that the system of difference equations can be solved.

In 2012, Gu and Ding [6] derived two canonical state space forms from multiple-input multiple-output systems described by difference equations.

The system of two nonlinear difference equations was studied by Papaschinopoulos and Schinas [7], where .

Moreover, the system of rational difference equations was studied by Clark et al. [8, 9], where and .

Liu et al. [10] studied the behavior of a system of rational difference equations where the initial conditions are nonzero real numbers.

In 2012, Zhang et al. [11] studied the solutions, stability character, and asymptotic behavior of the system of a rational third-order difference equation where .

In this paper, we studied the solutions, stability character, and asymptotic behavior of the system of a rational -order difference equation where .

#### 2. Preliminaries

Let and let and be continuously differentiable functions, where and are intervals in .

For any , the system of difference equations has a unique solution .

*Definition 1. *A point is called an equilibrium point of the system (10) if and .

*Definition 2. *The linearized system of the system (10) about the equilibrium is the system of linear difference equations

*Definition 3. *An equilibrium point of the system (10) is said to be stable relative to if for every , there exists such that for any , with
One has for all .

*Definition 4. *An equilibrium point of the system (10) is called an attractor relative to if for all , one has and .

*Definition 5. *An equilibrium point of the system (10) is said to be asymptotically stable relative to if it is stable, and it is also an attractor.

*Definition 6. *An equilibrium point of the system (10) is said to be unstable if it is not stable.

Theorem 7 (see [12]). *Let , , be a system of difference equations and let be the equilibrium point of the system. If all eigenvalues of the Jacobian matrix evaluated at lie inside the open unit disk, then is asymptotically stable. If one of them has a modulus greater than one, then is unstable. *

Theorem 8 (see [13]). *Let , , be a system of difference equations and let be the equilibrium point of the system. Assume that the characteristic polynomial of the system about is where for all and . Then all roots of the characteristic equation lie inside the open unit disk if and only if for all positive integer , where is the principal minor of order of the matrix
*

#### 3. Results

We note that(i)if and then the system (9) has equilibrium and ;(ii)if and then the system (9) has equilibrium and ;(iii)if and then the system (9) has equilibrium and ;(iv)if and then is the unique equilibrium point of the system (9).

Theorem 9. *Let be positive solution of the system (9). For all nonnegative integer , one has
*

*Proof. *Obviously, they are true for . Suppose that they are true for . Then

Thus, they are true for .

By the mathematical induction, this proof is completed.

Corollary 10. *Let be positive solution of the system (9). If and , then the sequence converges exponentially to the equilibrium point . *

Theorem 11. *Let and . Then the equilibrium point of the system (9) is asymptotically stable. *

*Proof. *The linearized system of the system (9) about the equilibrium is
where

The characteristic equation of the system (16) is
Thus, . By Theorem 7, the equilibrium point is asymptotically stable.

Theorem 12. *Let and . Then both the equilibrium points and of the system (9) are unstable. *

*Proof. *We note by the characteristic equation (18) that and then, by Theorem 7, the equilibrium point is unstable.

Next, we consider the equilibrium point . The linearized system of the system (9) about the equilibrium is
where
in which

The characteristic polynomial of the system (19) is

We note the characteristic polynomial that . Thus, we obtain that not all of , . By Theorems 7 and 8, the equilibrium point is unstable.

Theorem 13. *Let and , . Assume that satisfies the system (9). Then*(i)*if , then ;*(ii)*if , then .*

*Proof. * (i) Assume that . Then, for any ,

Then .

Next, we suppose that where is a positive integer. Then

Then .

By the mathematical induction, .

(ii) This is similar to the proof of (i).

#### Acknowledgment

The author would like to thank the referees for their useful comments and suggestions.

#### References

- A. S. Kurbanli, C. Çinar, and I. Yalçinkaya, “On the behavior of positive solutions of the system of rational difference equations ${x}_{n+1}={x}_{n-1}/({y}_{n}{x}_{n-1}+1)$, ${y}_{n+1}={y}_{n-1}/({x}_{n}{y}_{n-1}+1)$,”
*Mathematical and Computer Modelling*, vol. 53, no. 5-6, pp. 1261–1267, 2011. View at Publisher · View at Google Scholar · View at Scopus - A. S. Kurbanli, “On the behavior of solutions of the system of rational difference equations: ${x}_{n+1}={x}_{n-1}/({y}_{n}{x}_{n-1}-1)$, ${y}_{n+1}={y}_{n-1}/({x}_{n}{y}_{n-1}-1)$, and ${z}_{n+1}={z}_{n-1}/({y}_{n}{z}_{n-1}-1)$,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 932362, 12 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus - A. S. Kurbanli, “On the behavior of solutions of the system of rational difference equations: ${x}_{n+1}={x}_{n-1}/({y}_{n}{x}_{n-1}-1)$, ${y}_{n+1}={y}_{n-1}/({x}_{n}{y}_{n-1}-1)$, and ${z}_{n+1}={z}_{n-1}/({y}_{n}{z}_{n-1}-1)$,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 932362, 12 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus - K. Liu, Z. Zhao, X. Li, and P. Li, “More on three-dimensional systems of rational difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 178483, 9 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - S. Stević, “On a system of difference equations,”
*Applied Mathematics and Computation*, vol. 218, no. 7, pp. 3372–3378, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Gu and R. Ding, “Observable state space realizations for multivariable systems,”
*Computers & Mathematics with Applications*, vol. 63, no. 9, pp. 1389–1399, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Papaschinopoulos and C. J. Schinas, “On a system of two nonlinear difference equations,”
*Journal of Mathematical Analysis and Applications*, vol. 219, no. 2, pp. 415–426, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Clark and M. R. S. Kulenović, “A coupled system of rational difference equations,”
*Computers & Mathematics with Applications*, vol. 43, no. 6-7, pp. 849–867, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Clark, M. R. S. Kulenović, and J. F. Selgrade, “Global asymptotic behavior of a two-dimensional difference equation modelling competition,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 52, no. 7, pp. 1765–1776, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Liu, Z. Wei, P. Li, and W. Zhong, “On the behavior of a system of rational difference equations ${x}_{n+1}={x}_{n-1}/({y}_{n}{x}_{n-1}-1),{y}_{n+1}={y}_{n-1}/({x}_{n}{y}_{n-1}-1),{z}_{n+1}=1/{x}_{n}{z}_{n-1}$,”
*Discrete Dynamics in Nature and Society*, vol. 2012, Article ID 105496, 9 pages, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. Zhang, L. Yang, and J. Liu, “Dynamics of a system of rational third-order difference equation,”
*Advances in Difference Equations*, vol. 2012, article 136, p. 6, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - H. Sedaghat,
*Nonlinear Difference Equations*, vol. 15 of*Mathematical Modelling: Theory and Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. View at MathSciNet - V. L. Kocić and G. Ladas,
*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications*, vol. 256 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. View at MathSciNet