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

Abstract

We consider the second order rational difference equation 𝑥_𝑛+1=(𝛼+𝛽𝑥_𝑛+𝛾𝑥_𝑛−1)/(𝐴+𝐵𝑥_𝑛+𝐶𝑥_𝑛−1),  n = 0,1,2,…, where the parameters 𝛼,𝛽,𝛾,𝐴,𝐵,𝐶 are positive real numbers, and the initial conditions 𝑥₋₁,𝑥₀ 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, 𝑥_𝑛+1=𝛼+∑^𝑘_𝑖=0𝛼_𝑖𝑥_𝑛−𝑖𝛽+∑^ℓ_𝑗=0𝛽_𝑗𝑥_𝑛−𝑗,𝑛=0,1,2,3… (1.1) 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 𝑥_𝑛+1=𝛼+𝛽𝑥_𝑛+𝛾𝑥_𝑛−1𝐴+𝐵𝑥_𝑛+𝐶𝑥_𝑛−1,𝑛=0,1,2,…, (1.2) where the parameters 𝛼,𝛽,𝛾,𝐴,𝐵,𝐶 are positive real numbers, and the initial conditions 𝑥₋₁,𝑥₀ 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 11.4.3 in the Kulenović and Ladas monograph [26].

To this end and using transformations similar to the ones used by [22, 27], let 𝑥_𝑛=𝛾𝐶𝑦_𝑛 (1.3) then (1.2) reduces to 𝑦_𝑛+1=𝑟+𝑝𝑦_𝑛+𝑦_𝑛−1𝑧+𝑞𝑦_𝑛+𝑦_𝑛−1,𝑛=0,1,2,…, (1.4) where 𝑟=𝛼𝐶𝛾²𝛽,𝑝=𝛾𝐴,𝑧=𝛾𝐵,𝑞=𝐶 (1.5) are positive real numbers, and the initial conditions 𝑦₋₁,𝑦₀ 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 𝐼 be a nondegenerate interval of real numbers and let 𝑓∶𝐼×𝐼→𝐼 be a continuously differentiable function. Then for every set of initial conditions 𝑥₀,𝑥₋₁∈𝐼, the difference equation 𝑥_𝑛+1=𝑓𝑥_𝑛,𝑥_𝑛−1,𝑛=0,1,… (2.1) has a unique solution {𝑥_𝑛}^∞_𝑛=−1.

A constant sequence, 𝑥_𝑛=𝑥 for all 𝑛 where 𝑥∈𝐼, is called an equilibrium solution of (2.1) if 𝑥=𝑓(𝑥,𝑥). (2.2)

Definition 2.1. Let 𝑥 be an equilibrium solution of (2.1). (i)𝑥 is called locally stable if for every 𝜖>0, there exists 𝛿>0 such that for all 𝑥₀,𝑥₋₁∈𝐼, with |𝑥₀−𝑥|+|𝑥₋₁−𝑥|<𝛿, we have 𝑥_𝑛−𝑥<𝜖,∀𝑛≥−1. (2.3)(ii)𝑥 is called locally asymptotically stable if it is locally stable, and if there exists 𝛾>0, such that for all 𝑥₀,𝑥₋₁∈𝐼, with |𝑥₀−𝑥|+|𝑥₋₁−𝑥|<𝛾, we have lim_𝑛→∞𝑥_𝑛=𝑥. (2.4)(iii)𝑥 is called a global attractor if for every 𝑥₀,𝑥₋₁∈𝐼, we have lim_𝑛→∞𝑥_𝑛=𝑥. (2.5)(iv)𝑥 is called globally asymptotically stable if it is locally stable and a global attractor. (v)𝑥 is called unstable if it is not stable. (vi)𝑥 is called a source, or a repeller, if there exists 𝑟>0 such that for all 𝑥₀,𝑥₋₁∈𝐼, with 0<|𝑥₀−𝑥|+|𝑥₋₁−𝑥|<𝑟, there exists 𝑁≥1 such that 𝑥_𝑁−𝑥≥𝑟. (2.6) Clearly a source is an unstable equilibrium.

Definition 2.2. Let 𝑎=𝜕𝑓(𝜕𝑢𝑥,𝑥),𝑏=𝜕𝑓(𝜕𝑣𝑥,𝑥) (2.7) denote the partial derivatives of 𝑓(𝑢,𝑣) evaluated at the equilibrium 𝑥 of (2.1). Then the equation 𝑦_𝑛+1=𝑎𝑦_𝑛+𝑏𝑦_𝑛−1,𝑛=0,1,… (2.8) is called the linearized equation associated with (2.1) about the equilibrium solution 𝑥.

Definition 2.3. A solution {𝑥_𝑛} of (2.1) is said to be periodic with period 𝑃 if 𝑥_𝑛+𝑃=𝑥_𝑛,∀𝑛≥−1. (2.9)
A solution {𝑥_𝑛} of (2.1) is said to be periodic with prime period 𝑃, or a 𝑃-cycle if it is periodic with period 𝑃 and 𝑃 is the least positive integer for which (2.9) holds.

Theorem 2.4 (Linearized stability). (a) If both roots of the quadratic equation 𝜆²−𝑎𝜆−𝑏=0. (2.10) lie in the open unit disk |𝜆|<1, then the equilibrium 𝑥 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 𝑥 of (2.1) is unstable. (c) A necessary and sufficient condition for both roots of (2.10) to lie in the open unit disk |𝜆|<1 is |𝑎|<1−𝑏<2. (2.11) In this case, the locally asymptotically stable equilibrium 𝑥 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 |𝑏|>1,|𝑎|<|1−𝑏|. (2.12) In this case, 𝑥 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 𝑎²+4𝑏>0,|𝑎|>|1−𝑏|. (2.13) In this case, the unstable equilibrium 𝑥 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 |𝑎|=|1−𝑏| (2.14) or 𝑏=−1,|𝑎|≤2. (2.15) In this case, the equilibrium 𝑥 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 𝑦=1+𝑝−𝑧+1+𝑝−𝑧²+4𝑟𝑞+12.𝑞+1 (3.1) Furthermore, the linearized equation associated with (1.4) about the equilibrium solution is given by 𝑧_𝑛+1=𝑝−𝑞𝑦𝑧+𝑞+1𝑦𝑧_𝑛+1−𝑦𝑧+𝑞+1𝑦𝑧_𝑛−1. (3.2) Therefore, its characteristic equation is 𝜆²−𝑝−𝑞𝑦𝑧+𝑞+1𝑦𝜆−1−𝑦𝑧+𝑞+1𝑦=0. (3.3)

Lemma 3.1. (a) When 𝑝+𝑧≥1, (3.4) Equation (1.4) has no nonnegative prime period-two solution. (b) When 𝑝+𝑧<1, (3.5) Equation (1.4) has prime period-two solution, …,Φ,Ψ,Φ,Ψ,…; (3.6) if and only if 𝑞−𝑟<1−𝑝−𝑧1−𝑝−𝑧1+3𝑝−𝑧4, (3.7) where Φ and Ψ are the positive and distinct solutions of the quadratic equation 𝑡²−𝑝1−𝑧−𝑝𝑡+1−𝑧−𝑝+𝑟𝑞−1=0,𝑞>1. (3.8)

Proof. (a) Suppose 𝑝+𝑧≥1 and assume, for the sake of contradiction, there exist distinct positive real numbers Φ and Ψ such that …,Φ,Ψ,Φ,Ψ,… (3.9) is a prime period-two solution of (1.4). Then, Φ and Ψ satisfy Φ=𝑟+𝑝Ψ+Φ𝑧+𝑞Ψ+Φ,Ψ=𝑟+𝑝Φ+Ψ.𝑧+𝑞Φ+Ψ (3.10) Furthermore, ΦΨ𝑧+𝑞Ψ+Φ=𝑟+𝑝Ψ+Φ,𝑧+𝑞Φ+Ψ=𝑟+𝑝Φ+Ψ. (3.11)(3.12) Subtracting (3.11) from (3.12), we have (Φ+Ψ)=1−𝑧−𝑝. (3.13) But, 𝑝+𝑧≥1 implies that Φ+Ψ≤0 which contradicts the hypothesis that Φ and Ψ are distinct positive real numbers.(b) Now, suppose 𝑝+𝑧<1, and let Φ and Ψ be two positive real numbers such that …,Φ,Ψ,Φ,Ψ,… (3.14) is a prime period-two solution of (1.4). Thus, Φ and Ψ satisfy Φ=𝑟+𝑝Ψ+Φ,𝑧+𝑞Ψ+ΦΨ=𝑟+𝑝Φ+Ψ.𝑧+𝑞Φ+Ψ (3.15)(3.16) Moreover, ΦΨ𝑧+𝑞Ψ+Φ=𝑟+𝑝Ψ+Φ,𝑧+𝑞Φ+Ψ=𝑟+𝑝Φ+Ψ. (3.17)(3.18) Subtracting (3.17) from (3.18), we have (Φ+Ψ)=1−𝑧−𝑝. (3.19) Furthermore, adding (3.17) to (3.18), we have 𝑝ΦΨ=1−𝑧−𝑝+𝑟𝑞−1,𝑞>1. (3.20) Hence, Φ and Ψ>0 satisfy the quadratic equation 𝑡²−𝑝1−𝑧−𝑝𝑡+1−𝑧−𝑝+𝑟𝑞−1=0,𝑞>1. (3.21) In other words, Φ and Ψ are given by ±𝑡=1−𝑧−𝑝1−𝑧−𝑝²−4𝑝1−𝑧−𝑝+𝑟/𝑞−12,𝑞>1. (3.22)

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: 𝑢_𝑛=𝑦_𝑛−1,𝑣_𝑛=𝑦_𝑛,for𝑛=0,1,2,…, (3.23) and write (1.4) in the following equivalent form: 𝑢_𝑛+1𝑣_𝑛+1𝑢=𝑇^𝑛𝑣_𝑛;𝑛=0,1,2,…, (3.24) where 𝑇𝑢𝑣=𝑣𝑟+𝑝𝑣+𝑢𝑧+𝑞𝑣+𝑢. (3.25)
Now Φ and Ψ generate a period-two solution of (1.4) if and only if ΦΨ (3.26) is a fixed point of 𝑇², the second iterate of 𝑇. Furthermore, 𝑇²𝑢𝑣=𝑔()ℎ()𝑢,𝑣𝑢,𝑣, (3.27) where 𝑔(𝑢,𝑣)=𝑟+𝑝𝑣+𝑢()=()𝑧+𝑞𝑣+𝑢,ℎ𝑢,𝑣𝑟+𝑝𝑔𝑢,𝑣+𝑣().𝑧+𝑞𝑔𝑢,𝑣+𝑣 (3.28)
The prime period-two solution of (1.4) is asymptotically stable if the eigenvalues of the Jacobian matrix 𝐽_𝑇², evaluated at ΦΨ lie inside the unit disk. But, 𝐽_𝑇²ΦΨ=1−Φ𝑧+𝑞Ψ+Φ𝑝−𝑞Φ)𝑧+𝑞Ψ+Φ𝑝−𝑞Ψ(1−Φ𝑧+𝑞Ψ+Φ𝑧+𝑞Φ+Ψ1−Ψ+𝑧+𝑞Φ+Ψ𝑝−𝑞Φ𝑝−𝑞Ψ𝑧+𝑞Φ+Ψ𝑧+𝑞Ψ+Φ (3.29) and, hence, its characteristic equation is 𝜆²−𝜂𝜆+𝜇=0, (3.30) where 𝜂=1−Φ+𝑧+𝑞Ψ+Φ1−Ψ+𝑧+𝑞Φ+Ψ𝑝−𝑞Φ𝑝−𝑞Ψ,()𝑧+𝑞Φ+Ψ𝑧+𝑞Ψ+Φ𝜇=1−Φ)(1−Ψ.𝑧+𝑞Ψ+Φ𝑧+𝑞Φ+Ψ (3.31) By Theorem 2.4(c), the eigenvalues of 𝐽_𝑇²ΦΨ (3.32) lie inside the unit disk if and only if 𝜂<1+𝜇<2or,equivalently,𝜂<1+𝜇,𝜇<1. (3.33)
To this end, without loss of generality, assume that 0<Φ<Ψ. Then, by (3.15), ()()1=𝑟/Φ+𝑝Ψ/Φ+1>1𝑧+𝑞Ψ+Φ.𝑧+𝑞Ψ+Φ (3.34) Hence, 𝑧+𝑞Ψ+Φ>1. (3.35) Similarly, we observe that 𝑧+𝑞Φ+Ψ>1. (3.36) Furthermore, since 𝑝+𝑧<1, (3.19) implies the sum of Φ and Ψ is less than 1 and, a fortiori, each is less than 1. Indeed, we have 10<Φ<minΨ,2<1. (3.37)
With that in mind, it is clear that ()𝜇=1−Φ)(1−Ψ𝑧+𝑞Ψ+Φ𝑧+𝑞Φ+Ψ<1. (3.38) In addition, with the understanding that 𝜂=1−Φ+𝑧+𝑞Ψ+Φ1−Ψ𝑧+𝑞Φ+Ψ_𝐿+𝑝−𝑞Φ𝑝−𝑞Ψ(𝑧+𝑞Ψ+Φ)(𝑧+𝑞Φ+Ψ)_𝑄,1+𝜇−𝐿=1−1−Φ𝑧+𝑞Ψ+Φ1−1−Ψ=𝑧+𝑞Φ+Ψ1−𝑧−2Φ−𝑞Ψ1−𝑧−2Ψ−𝑞Φ,𝑧+𝑞Ψ+Φ𝑧+𝑞Φ+Ψ−1−𝜇−𝐿=−1+1−Φ𝑧+𝑞Ψ+Φ1+1−Ψ𝑧+𝑞Φ+Ψ=−1+𝑧+𝑞Ψ1+𝑧+𝑞Φ,𝑧+𝑞Ψ+Φ𝑧+𝑞Φ+Ψ (3.39) we have 𝜂⟺<,−<.<1+𝜇⟺−1−𝜇−𝐿<𝑄<1+𝜇−𝐿𝑝−𝑞Φ𝑝−𝑞Ψ1−𝑧−2Φ−𝑞Ψ1−𝑧−2Ψ−𝑞Φ1+𝑧+𝑞Ψ1+𝑧+𝑞Φ𝑝−𝑞Φ𝑝−𝑞Ψ (3.40) The second inequality follows immediately since 𝑝+𝑧<1 and 𝑝−𝑞Φ𝑝−𝑞Ψ<𝑝+𝑞Φ𝑝+𝑞Ψ<1+𝑧+𝑞Φ1+𝑧+𝑞Ψ, (3.41) However, the proof of the first inequality is a bit tricky.
By (3.19), 𝑝−𝑞Φ𝑝−𝑞Ψ=1−𝑧−Φ−𝑞Ψ_𝑎−Ψ1−𝑧−Ψ−𝑞Φ_𝑏−Φ. (3.42) Therefore, 𝑝−𝑞Φ𝑝−𝑞Ψ<1−𝑧−2Φ−𝑞Ψ1−𝑧−2Ψ−𝑞Φ (3.43) if and only if (𝑎−Ψ)(𝑏−Φ)<(𝑎−Φ)(𝑏−Ψ), (3.44) which is true if and only if (𝑏−𝑎)Φ<(𝑏−𝑎)Ψ. (3.45) The latter inequality holds true because Φ<Ψ and 𝑏−𝑎=𝑞−1(Ψ−Φ)>0. (3.46) 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.