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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 179423, 10 pages
http://dx.doi.org/10.1155/2013/179423
Research Article

On the Period-Two Cycles of

1School of Mathematical Sciences, The National of University of Malaysia, 43600 Bangi, Selangor, Malaysia
2Department of Basic Sciences, King Saud bin Abdulaziz University for Health Sciences, Riyadh 11426, Saudi Arabia

Received 2 January 2013; Revised 13 April 2013; Accepted 13 April 2013

Academic Editor: Douglas Anderson

Copyright © 2013 S. Atawna et al. 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 consider the higher order nonlinear rational difference equation , where the parameters are positive real numbers and the initial conditions are nonnegative real numbers, . We give a necessary and sufficient condition for the equation to have a prime period-two solution. We show that the period-two solution of the equation is locally asymptotically stable.

1. Introduction

Recently, dynamics of nonnegative solutions of higher order rational difference equation has been an area of intense interest. Related to this subject, researches are done by Dehghan et al. [14], Zayed [57], Huang and Knopf [8, 9], Karatas [10, 11], and others. For the general theory of difference equations, one can refer to the monographes of Kocic and Ladas [12], Elaydi [13], Agarwal [14], Kulenovic and Ladas [15], and Camouzis and Ladas [16]. Other related results can be found in [1724].

Our aim in this paper is to study the higher order nonlinear rational difference equation where the parameters are positive real numbers and the initial conditions are nonnegative real numbers, . Our concentration is on the periodic character of all positive solutions of (1).

The periodic character of positive solutions of (1) for has been investigated by the authors in [25]. They showed that the period-two solution of (1) for is locally asymptotically stable if it exists.

Motivated by the above results, our interest is now to study and generalize the previous results to the general case depicted in (1).

The change of variable reduces (1) to where are positive real numbers and the initial conditions are nonnegative real numbers.

This paper is organized besides this introduction in three sections. In Section 2, we present some preliminaries and some results which can be mainly deduced from the general situation studied in [1216, 26]. Our main results are presented in Section 3; we give a necessary and sufficient condition for the equation to have a prime period-two solution, in addition to providing a necessary and sufficient conditions for the prime period-two solution of the equation to be locally asymptotically stable. In order to illustrate the results of the previous section and to support our theoretical discussion, we consider several numerical examples in Section 4; we use MATLAB to see how the behaviors of (1) look like. Finally, we conclude in Section 5 with suggestions for future research.

2. Preliminaries

For the sake of self-containment and convenience, we recall the following definitions and results from [16].

Let be some interval of real numbers and let be a continuously differentiable function. Then for every set of initial conditions , the difference equation has a unique solution .

A solution of (6) that is constant for all is called an equilibrium solution of (6). If is an equilibrium solution of (6), then is called an equilibrium point or simply an equilibrium of (6).

Definition 1. (i) The equilibrium point of (6) is called locally stable if for every , there exists such that if , and , we have
(ii) The equilibrium point of (6) is called locally asymptotically stable if it is locally stable, and if there exists , if , and , we have
(iii) The equilibrium point of (6) is called a global attractor if for every , we have
(iv) The equilibrium point of (6) is called globally asymptotically stable if it is locally stable and a global attractor.
(v) The equilibrium point of (6) is called unstable if it is not stable.

Definition 2. (i) A solution of (6) is said to be periodic with period if
(ii) A solution of (6) is said to be periodic with prime period if is the least positive integer for which (11) holds.

Definition 3. Let denote the partial derivatives of evaluated at the equilibrium of (6). Then the equation is called the linearized equation associated with (6) about the equilibrium point . Its characteristic equation is

Theorem 4 (linearized stability). (a) If all roots of (14) lie in the open unit disk , then the equilibrium of (6) is locally asymptotically stable.
(b) If at least one of the roots of (14) has absolute value greater than one, then is unstable.

The following result from [26] will become handy in the sequel.

Lemma 5. If then satisfies the following recursive formula:

The aforementioned lemma leads to the following conclusion.

Corollary 6. If then

3. Main Result

In this section, we give a necessary and sufficient condition for (1) to have a prime period-two solution. We show that the period-two solution of (1) is locally asymptotically stable.

Equation (3) has a unique positive equilibrium given by The linearized equation associated with (3) about the equilibrium is given by and its characteristic equation is

Theorem 7. (a) If ?then (3) has no nonnegative prime period-two solution.
(b) If ?then (3) has prime period-two solution ?if and only if is odd and ?where the values of and are the positive and distinct solutions of the quadratic equation

Proof. Assume that there exist distinct nonnegative real numbers and , such that is a prime period-two solution of (3); there are two cases to be considered.
Case??1??(k is even). In this case and satisfy Furthermore, Subtracting (30) from (29), we have so This contradicts the hypothesis that and are distinct nonnegative real numbers. Also, contradicts the hypothesis that and are positive real numbers.
Case??2??(k is odd). (a) If then in this case and satisfy Furthermore, Subtracting (35) from (36), we have But, , this implies that which contradicts the hypothesis that are distinct positive real numbers.
(b) If then in this case and satisfy Moreover, Subtracting (41) from (42), we have Furthermore, one adds (41) to (42), makes use of (43), and then does some elementary algebraic manipulation; we have Equation (44) leads to the following conclusion: that follows from the facts that Notice that when then adding (41) to (42) gives , which is impossible.
Construct the quadratic equation So and are the positive and distinct solutions of the above quadratic equation, that is,

Theorem 8. Suppose (3) has a prime period-two solution. Then, the period-two solution is locally asymptotically stable.

Proof. To investigate the local stability of the two cycles we first vectorize (3) by introducing the following change of variables: and write (3) in the equivalent form: where Now and generate a period-two solution of (3) only if is a fixed point of , the second iterate of . Furthermore, where The prime period-two solution of (3) is asymptotically stable if the eigenvalues of the Jacobian matrix , evaluated at lie inside the unit disk.
We have
Now let be the characteristic polynomial of . Then, by the Laplace expansion in the row,
However; by Lemma 5, Corollary 6, and the fact that is odd, Therefore,
Hence, the characteristic polynomial is given by where
Assume that . Then, by (39), Hence, Similarly, we observe that Furthermore, since , (43) implies the sum of , is less than and, a fortiori, each is less than 1. Indeed, we have
With that in mind, it is clear that In addition, with understanding that and the fact that we have to establish First, we will establish inequality (69). To this end, observe that inequality (69) is equivalent to which is true if and only if which is true if and only if which is true if and only if which is true if and only if which is true if and only if which is clearly satisfied.
Next we will establish inequality (70). Observe that inequality (70) is equivalent to which is true if and only if which is true if and only if which is true if and only if which is true if and only if which is true if and only if Now observe that the righthand side of (82) is The lefthand side of (82) is Hence, inequality (70) is true if and only if or equivalently which is clearly satisfied (condition (25)).
Now, by applying Theorem 4 we shall show that the zeros of in (60) lie in the open unit disk . To do so, suppose to the contrary that has a zero such that . Then, by the triangle inequality, Thus,
However, by the Descartes’ Rule of Signs has either two or no positive zeros. Furthermore, and so, by the Intermediate Value Theorem, has two positive zeros in the open interval . Moreover, since , we conclude that for all which contradicts inequality (88).
The proof is complete.

Remark 9. The characteristic equation of the linearized equation at the equilibrium solution is given by Since the magnitude of the constant term is less than , the equation has at least one root inside the unit disk. As such, by the Stable Manifold Theorem, there is a manifold of solutions, of dimension bigger than or equal to 1, that converge to the equilibrium solution. Hence, the period-two solution cannot be globally asymptotically stable.

4. Numerical Examples

In order to illustrate the results of the previous section and to support our theoretical discussion, we consider several numerical examples generated by MATLAB.

Case??1??(k is even). For this case we consider the following example: The dynamics of (91) is shown in Figure 1, no prime period-two solution.

179423.fig.001
Figure 1: Dynamics of .

Case??2??(k is odd). There are two cases to be considered.

Subcase??2.1?? . For this case we consider the following example: The dynamics of (92) is shown in Figure 2, no prime period-two solution.

179423.fig.002
Figure 2: Dynamics of .

Subcase??2.2?? and . For this case we consider the following example: The dynamics of (93) is shown in Figure 3; it has prime period-two solution.

179423.fig.003
Figure 3: Dynamics of .

5. Conclusion

In this paper, we showed that the period-two solution of the higher order nonlinear rational difference equation where the parameters are positive real numbers and the initial conditions are nonnegative real numbers, , is locally asymptotically stable if it exists.

We consider the aforementioned result as a step forward in investigating bigger classes of difference equations which afford the ELAS property; that is, the existence of a periodic solution implies its local asymptotic stability.

Acknowledgment

The authors are very grateful to the anonymous referees for carefully reading the paper and for their comments and valuable suggestions that lead to an improvement in the paper.

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