Advances in Mathematical Physics

Volume 2014 (2014), Article ID 729273, 8 pages

http://dx.doi.org/10.1155/2014/729273

Research Article

On a System of Two High-Order Nonlinear Difference Equations

Qianhong Zhang and Wenzhuan Zhang

Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang, Guizhou 550004, China

Received 7 December 2013; Accepted 13 April 2014; Published 5 May 2014

Academic Editor: Pavel Kurasov

Copyright © 2014 Qianhong Zhang and Wenzhuan Zhang. 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.

This paper is concerned with dynamics of the solution to the system of two high-order nonlinear difference equations , , , , where , , and . Moreover the rate of convergence of a solution that converges to the equilibrium of the system is discussed. Finally, some numerical examples are considered to show the results obtained.

1. Introduction

Difference equations or discrete dynamical systems are diverse fields which impact almost every branch of pure and applied mathematics. Every dynamical system determines a difference equation and vice versa. Recently, there has been great interest in studying the system of difference equations. One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models describing real life situations in population biology, economic, probability theory, genetics psychology, and so forth. 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. Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering, and economics. It is very interesting to investigate the behavior of solutions of a system of nonlinear difference equations and to discuss the local asymptotic stability of their equilibrium points.

The study of properties of rational difference equations and systems of rational difference equations has been an area of interest in recent years. There are many papers in which systems of difference equations have been studied.

Çinar et al. [1] have obtained the positive solution of the difference equation system:

Çinar [2] has obtained the positive solution of the difference equation system:

Also, Çinar and Yalçinkaya [3] have obtained the positive solution of the difference equation system:

Özban [4] has investigated the positive solutions of the system of rational difference equations:

Papaschinopoulos and Schinas [5] investigated the global behavior for a system of the following two nonlinear difference equations:

where is a positive real number, are positive integers, and are positive real numbers.

Clark et al. [6, 7] investigated the system of rational difference equations:

where and the initial conditions and are arbitrary nonnegative numbers.

In 2012, Zhang et al. [8] investigated the global behavior for a system of the following third order nonlinear difference equations:

where , and the initial values .

Ibrahim [9] has obtained the positive solution of the difference equation system in the modeling competitive populations:

Din et al. [10] studied the global behavior of positive solution to the fourth-order rational difference equations:

where the parameters and the initial conditions are positive real numbers.

Although difference equations are sometimes very simple in their forms, they are extremely difficult to understand thoroughly the behavior of their solutions. In [11], Kocić and Ladas have studied global behavior of nonlinear difference equations of higher order. Similar nonlinear systems of rational difference equations were investigated (see [12, 13]). Other related results reader can refer to [14–22].

Our aim in this paper is to investigate the solutions, stability character, and asymptotic behavior of the system of difference equations:

where and initial conditions . This paper is natural extension of [8–10, 14].

2. Preliminaries

Let be some intervals of real number and let be continuously differentiable functions. Then for every initial conditions , the system of difference equations

has a unique solution . A point is called an equilibrium point of (11) if ; that is, for all .

Definition 1. Assume that is an equilibrium point of (11). Then one has the following

(i) is said to be stable relative to , if, for every , and any initial conditions , there exists such that , implies .(ii) is called an attractor relative to if for all , .(iii) is called asymptotically stable relative to if it is stable and an attractor.(iv) is unstable if it is not stable.

Definition 2. Let be an equilibrium point of a map , where and are continuously differentiable functions at . The linearized system of (11) about the equilibrium point is

where , and is Jacobian matrix of system (11) about the equilibrium point .

Theorem 3 (see [11]). Assume that , is a system of difference equations and is the equilibrium point of this system; that is, . If all eigenvalues of the Jacobian matrix , evaluated at , lie inside the open unit disk , then is locally asymptotically stable. If one of them has modulus greater than one, then is unstable.

Theorem 4 (see [12]). Assume that , is a system of difference equations and is the equilibrium point of this system, and the characteristic polynomial of this system about the equilibrium point is , with real coefficients and . Then all roots of the polynomial lie inside the open unit disk if and only if

where is the principal minor of order of the matrix:

3. Main Results

The equilibrium points of system (10) are and , for and . In addition, if , then every point on the -axis is an equilibrium point, and if , then every point on the -axis is an equilibrium point. Finally, if and , (0, 0) is the unique equilibrium point.

We summarize the local stability of the equilibria of (10) as follows.

Theorem 5. For the equilibrium point of system (10), the following results hold.

(i)If and , then the unique equilibrium point of system (10) is locally asymptotically stable.(ii)If or , then the equilibrium point of system (10) is unstable.

Proof. (i) The linearized equation of system (10) about is

where , and

The characteristic equation of (15) is

This shows that all the roots of characteristic equation lie inside unit disk. So the unique equilibrium is locally asymptotically stable.

(ii) It is easy to see that if or , then there exists at least one root of (17) such that . Hence by Theorem 3 if or , then is unstable. The proof is complete.

Theorem 6. If and , then the positive equilibrium point of (10) is unstable.

Proof. The linearized system of (10) about the equilibrium point is given by

where , and

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

Clearly, is invertible. In computing matrix , we obtain that

From , and it implies that

On the other hand

It is well known that has the same eigenvalues as , and we have that

This implies that the equilibrium of (10) is unstable.

The following theorem is similar to Theorem 3.4 of [8].

Theorem 7. Let and , is a solution of system (10), and then, for , the following statements are true.

(i)If , then .(ii)If , then .

Theorem 8. Let be positive solution of system (10), then for the following results hold:

Proof. It is true for . Suppose that results are true for , namely,

Now, for , by virtue of system (10), we have

and similarly,

and similarly,

Hence, for all , the results are true.

Theorem 9. If and , then the unique equilibrium point of system (10) is globally asymptotically stable.

Proof. From (i) of Theorem 5, we obtain that the unique equilibrium point of system (10) is locally asymptotically stable. By virtue of Theorem 7, it is clear that every positive solution is bounded. That is, and , where .

Now, it is sufficient to prove that is decreasing. From system (10) one has

This implies that and ; hence, the subsequences , are decreasing. So sequence is decreasing. Similarly, it is easy to prove that sequence is also decreasing. Hence . Therefore the equilibrium point is globally asymptotically stable.

4. Rate of Convergence

In this section we will determine the rate of convergence of a solution that converges to the equilibrium point of the system (10). The following result gives the rate of convergence of solution of a system of difference equations:

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

where denotes any matrix norm which is associated with the vector norm.

Theorem 10 (see [23]). Assume that condition (31) holds, if is a solution of (30), then either for all large or

or

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

Assume that , we will find a system of limiting equations for the system (10). The error terms are given as

Set ; therefore, it follows that

where

Now it is clear that

Hence, the limiting system of error terms at can be written as

where , and

Using Theorem 10, we have the following result.

Theorem 11. Assume that , and are a positive solution of the system (10). Then, the error vector of every solution of (10) satisfies both of the following asymptotic relations:

where is equal to the modulus of one the eigenvalues of the Jacobian matrix evaluated at the equilibrium .

5. Numerical Examples

In order to illustrate the results of the previous sections and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to system of nonlinear difference equations.

Example 1. Consider the system (10) with initial conditions , , , and , moreover, choosing the parameters and . Then system (10) can be written as

The plot of system (42) is shown in Figure 1.

Figure 1: The plot of system (42).

Example 2. Consider the system (10) with initial conditions , ,, , , , and , moreover, choosing the parameters and . Then system (10) can be written as

The plot of system (43) is shown in Figure 2.

Figure 2: The plot of system (43).

6. Conclusions and Future Work

In this paper, we discussed the dynamics of high-order discrete system which is extension of [8, 10, 14]. We conclude that (i) the equilibrium point is globally asymptotically stable if , (ii) the equilibrium and if or is unstable. Some numerical examples are provided to support our theoretical results. It is our future work to study the dynamical behavior of system (10) when or .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the Editor and the anonymous referees for their careful reading and constructive suggestions. This work was financially supported by the National Natural Science Foundation of China (Grant no. 11361012) and the Scientific Research Foundation of Guizhou Provincial Science and Technology Department ([2013]J2083, [2009]J2061).

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