Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 969813, 11 pages

doi:10.1155/2012/969813

Research Article

Local Stability of Period Two Cycles of Second Order Rational Difference Equation

S. Atawna,¹ R. Abu-Saris,² and I. Hashim¹

¹School of Mathematical Sciences, Universiti Kebangsaan Malaysia, Selangor, 43600 Bangi, Malaysia

²Department of Basic Sciences, King Saud bin Abdulaziz University for Health Sciences, P.O. Box 22490, Riyadh 11426, Saudi Arabia

Received 1 September 2012; Accepted 11 October 2012

Academic Editor: Mustafa Kulenovic

Copyright © 2012 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 second order rational difference equation   n = 0,1,2,…, 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. In particular, we solve Conjecture 5.201.2 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.3 in Kulenović and Ladas monograph (2002).

1. Introduction

Difference equations proved to be effective in modelling and analysing discrete dynamical systems that arise in signal processing, populations dynamics, health sciences, economics, and so forth. They also arise naturally in studying iterative numerical schemes. Furthermore, they appear when solving differential equations using series solution methods or studying them qualitatively using, for example, Poincaré maps. For an introduction to the general theory of difference equations, we refer the readers to Agarwal [1], Elaydi [2], and Kelley and Peterson [3].

Rational difference equations; particularly bilinear ones, that is,

attracted the attention of many researchers recently. For example, see the articles [4–24], monographs Kocić and Ladas [25], Kulenović and Ladas [26], and Camouzis and Ladas [27], and the references cited therein. We believe that behavior of solutions of rational difference equations provides prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one [26, page 1].

Our aim in this paper is to study the second order bilinear 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 the positive solutions of (1.2). Indeed, our interest in (1.2) was stimulated by the following conjecture proposed by Camouzis and Ladas in [27, Conjecture  5.201.2].

Conjecture 1.1. Show that the period-two solution of (1.2) is locally asymptotically stable.

It is worth mentioning that the aforementioned conjecture appeared previously in Conjecture in the Kulenović and Ladas monograph [26].

To this end and using transformations similar to the ones used by [22, 27], let

then (1.2) reduces to

where

are positive real numbers, and the initial conditions are nonnegative real numbers.

That being said, the remainder of this paper is organized as follows. In the next section, we present some definitions and results that are needed in the sections to follow. Next, using elementary mathematics and nontrivial combinations of ideas, we establish our main results in Section 3. Our main results provide positive confirmation of Conjecture 1.1. Finally, we conclude in Section 4 with suggestions for future research.

2. Preliminaries

For the sake of self-containment and convenience, we recall the following definitions and results from [26].

Let be a nondegenerate 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 constant sequence, for all where , is called an equilibrium solution of (2.1) if

Definition 2.1. Let be an equilibrium solution of (2.1).

(i) is called locally stable if for every , there exists such that for all , with , we have

(ii) is called locally asymptotically stable if it is locally stable, and if there exists , such that for all , with , we have

(iii) is called a global attractor if for every , we have

(iv) is called globally asymptotically stable if it is locally stable and a global attractor. (v) is called unstable if it is not stable. (vi) is called a source, or a repeller, if there exists such that for all , with , there exists such that

Clearly a source is an unstable equilibrium.

Definition 2.2. Let

denote the partial derivatives of evaluated at the equilibrium of (2.1). Then the equation

is called the linearized equation associated with (2.1) about the equilibrium solution .

Definition 2.3. A solution of (2.1) is said to be periodic with period if

A solution of (2.1) is said to be periodic with prime period , or a -cycle if it is periodic with period and is the least positive integer for which (2.9) holds.

Theorem 2.4 (Linearized stability). (a) If both roots of the quadratic equation

lie in the open unit disk , then the equilibrium of (2.1) is locally asymptotically stable.

(b) If at least one of the roots of (2.10) has absolute value greater than one, then the equilibrium of (2.1) is unstable. (c) A necessary and sufficient condition for both roots of (2.10) to lie in the open unit disk is

 In this case, the locally asymptotically stable equilibrium is also called a sink. (d) A necessary and sufficient condition for both roots of (2.10) to have absolute value greater than one is

 In this case, is a repeller. (e) A necessary and sufficient condition for one root of (2.10) to have absolute value greater than one and for the other to have absolute value less than one is

 In this case, the unstable equilibrium is called a saddle point. (f) A necessary and sufficient condition for a root of (2.10) to have absolute value equal to one is

 or

 In this case, the equilibrium is called a nonhyperbolic point.

3. Main Result

In this section, we give a necessary and sufficient condition for (1.4) to have a prime period-two solution. We show that the period-two solution of (1.4) is locally asymptotically stable.

Equation (1.4) has a unique positive equilibrium given by

Furthermore, the linearized equation associated with (1.4) about the equilibrium solution is given by

Therefore, its characteristic equation is

Lemma 3.1. (a) When

Equation (1.4) has no nonnegative prime period-two solution.

(b) When

 Equation (1.4) has prime period-two solution,

 if and only if

 where and are the positive and distinct solutions of the quadratic equation

Proof. (a) Suppose and assume, for the sake of contradiction, there exist distinct positive real numbers and such that

 is a prime period-two solution of (1.4). Then, and satisfy

 Furthermore,

 Subtracting (3.11) from (3.12), we have

 But, implies that which contradicts the hypothesis that and are distinct positive real numbers.(b) Now, suppose , and let and be two positive real numbers such that

 is a prime period-two solution of (1.4). Thus, and satisfy

 Moreover,

 Subtracting (3.17) from (3.18), we have

 Furthermore, adding (3.17) to (3.18), we have

 Hence, and satisfy the quadratic equation

 In other words, and are given by

Theorem 3.2. Suppose (1.4) has a prime period-two solution. Then, the period-two solution is locally asymptotically stable.

Proof. To start off, we first vectorize (1.4) by introducing the following change of variables:

and write (1.4) in the following equivalent form:

where

Now and generate a period-two solution of (1.4) if and only if

is a fixed point of , the second iterate of . Furthermore,

where

The prime period-two solution of (1.4) is asymptotically stable if the eigenvalues of the Jacobian matrix , evaluated at lie inside the unit disk. But,

and, hence, its characteristic equation is

where

By Theorem 2.4(c), the eigenvalues of

lie inside the unit disk if and only if

To this end, without loss of generality, assume that . Then, by (3.15),

Hence,

Similarly, we observe that

Furthermore, since , (3.19) implies the sum of and 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 the understanding that

we have

The second inequality follows immediately since and

However, the proof of the first inequality is a bit tricky.

By (3.19),

Therefore,

if and only if

which is true if and only if

The latter inequality holds true because and

This completes the proof.

4. Conclusion

Without dispute, the existence of a periodic solution or an equilibrium solution does not imply its local stability. As such, it is natural to ask about the class of difference equations which afford the ELAS property, that is, the existence of a periodic solution implies its local asymptotic stability. In particular, one may want to investigate the class of bilinear difference equations (1.1) and completely characterize those equations enjoying the ELAS property.

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