# On the Dynamics of a Higher-Order Rational Difference Equation

Authors: A. M. Ahmed
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

The aim of this paper is to investigate the global asymptotic stability and the periodic character for the rational difference equationxn+1=αxn-1/(β+γΠi=lkxn-2ipi),n=0,1,2,…, where the parameters α,β,γ,pl,pl+1,…,pk are nonnegative real numbers, and l,k are nonnegative integers such that l≤k.


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## Body

## 1. Introduction

Difference equations have always played an important role in the construction and analysis of mathematical models of biology, ecology, physics, economic process, and so forth.The study of nonlinear rational difference equations of higher order is of paramount importance, since we still know so little about such equations.Amleh et al. [1] investigated the third-order rational difference equation (1.1)xn+1=a+bxn-1A+Bxn-2,n=0,1,2,…,
where a,b,A,B are nonnegative real numbers and the initial conditions are nonnegative real numbers.Ahmed [2] studied the global asymptotic behavior and the periodic character of solutions of the third-order rational difference equation (1.2)xn+1=bxn-1A+Bxnpxn-2q,n=0,1,2,…,
where the parameters b,A,B,p,q are nonnegative real numbers, and the initial conditions x-2,x-1,x0 are arbitrary nonnegative real numbers.For other related results, see [3] and also [4–15].In this paper, the global asymptotic behavior and the periodic character of solutions of the rational difference equation(1.3)xn+1=αxn-1β+γΠi=lkxn-2ipi,n=0,1,2,…,
where the parameters α,β,γ,pl,pl+1,…,pk are nonnegative real numbers, l,k are nonnegative integers such that l≤k, and the initial conditions x-2k,x-2k+1,…,x0 are arbitrary nonnegative real numbers such that (1.4)β+γ∏i=lkxn-2ipi>0,∀n≥0,
will be investigated.LetI be an interval of real numbers, and let f:I2k+1→I be a continuously differentiable function. Consider the difference equation (1.5)xn+1=f(xn,xn-1,…,xn-2k),n=0,1,2,…,
with x-2k,x-2k+1,…,x0∈I. Let x̅ be the equilibrium point of (1.5). The linearized equation of (1.5) about x̅ is (1.6)yn+1=c1yn+c2yn-1+⋯+c2k+1yn-2k,n=0,1,2,…,
where (1.7)c1=∂f∂xn(x̅,x̅,…,x̅),c2=∂f∂xn-1(x̅,x̅,…,x̅),…,c2k+1=∂f∂xn-2k(x̅,x̅,…,x̅).
The characteristic equation of (1.5) is (1.8)λ2k+1-c1λ2k-c2λ2k-1-⋯-c2k+1=0.Definition 1.1.
Letx̅ be an equilibrium point of (1.5).(i)
The equilibrium pointx̅ of (1.5) is called locally stable if, for every ϵ>0, there exists δ>0 such that, for all x-2k,x-2k+1,…,x0∈I with |x-2k-x̅|+|x-2k+1-x̅|+⋯+|x0-x̅|<δ, we have |xn-x̅|<ϵ for all n≥-2k.(ii)
The equilibrium pointx̅ of (1.5) is called locally asymptotically stable if it is locally stable, and if there exists γ>0 such that for all x-2k,x-2k+1,…,x0∈I with |x-2k-x̅|+|x-2k+1-x̅|+⋯+|x0-x̅|<γ, we have limn→∞xn=x̅.(iii)
The equilibrium pointx̅ of (1.5) is called global attractor if, for every x-2k,x-2k+1,…,x0∈I, we have limn→∞xn=x̅.(iv)
The equilibrium pointx̅ of (1.5) is called globally asymptotically stable if it is locally stable and global attractor.(v)
The equilibrium pointx̅ of (1.5) is called unstable if it is not stable.(vi)
The equilibrium pointx̅ of (1.5) is called source or repeller if there exists r>0 such that, for all x-2k,x-2k+1,…,x0∈I with 0<|x-2k-x̅|+|x-2k+1-x̅|+⋯+|x0-x̅|<r, there exists N≥1 such that |xN-x̅|≥r. Clearly, a repeller is an unstable equilibrium.Theorem A (linearized stability theorem).
The following statements are true.(1)
If all roots of (1.8) have modulus less than one, then the equilibrium point x̅ of (1.5) is locally asymptotically stable.(2)
If at least one of the roots of (1.8) has modulus greater than one, then the equilibrium point x̅ of (1.5) is unstable.
The equilibrium pointx̅ of (1.5) is called a “saddle point” if (1.8) has roots both inside and outside the unit disk.


## 2. The Special CasesαβγΣi=lkpi=0

In this section, we examine the character of solutions of (1.3) when one or more of the parameters in (1.3) are zero.There are four such equations; namely,(2.1)xn+1=0,n=0,1,2,…,(2.2)xn+1=αβxn-1,n=0,1,2,…,(2.3)xn+1=αβ+γxn-1,n=0,1,2,…,(2.4)xn+1=αxn-1γΠi=lkxn-2ipi,n=0,1,2,….Equation (2.1) is trivial, (2.2) and (2.3) are linear, and (2.4) is a non-linear difference equation; the change of variables xn=eyn reduces it to a linear difference equation.


## 3. A General Oscillation Result

The change of variablesxn=(β/γ)1/Σi=lkpiyn reduces (1.3) to the difference equation (3.1)yn+1=ryn-11+Πi=lkyn-2ipi,n=0,1,2,…,
where r=α/β>0.Note thaty̅1=0 is always an equilibrium point. When r>1, (3.1) also possesses the unique positive equilibrium y̅2=(r-1)1/Σi=lkpi.Theorem B (see [8]).
Assume thatF∈C([0,∞)2k+1→[0,∞)) is nonincreasing in the odd arguments, and nondecreasing in the even arguments. Let  x̅ be an equilibrium point of the difference equation
(3.2)xn+1=F(xn,xn-1,…,xn-2k),n=0,1,2,…,
and let {xn}n=-2k∞ be a solution of (3.2) such that either
(3.3)x-2k,x-2k+2,…,x0≥x̅,x-2k+1,x-2k+3,…,x-1<x̅,
or
(3.4)x-2k,x-2k+2,…,x0<x̅,x-2k+1,x-2k+3,…,x-1≥x̅.
Then {xn}n=-2k∞ oscillates about x̅ with semicycles of length one.Corollary 3.1.
Assume thatr>1; let  {yn}n=-2k∞ be a solution of (3.1) such that either
(3.5)y-2k,y-2k+2,…,y0≥y̅2=(r-1)1/Σi=lkpi,y-2k+1,y-2k+3,…,y-1<y̅2=(r-1)1/Σi=lkpi,
or
(3.6)y-2k,y-2k+2,…,y0<y̅2=(r-1)1/Σi=lkpiy-2k+1,y-2k+3,…,y-1≥y̅2=(r-1)1/Σi=lkpi.
Then {yn}n=-2k∞ oscillates about the positive equilibrium point y̅2=(r-1)1/Σi=lkpi with semicycles of length one.Proof.
The proof follows immediately from Theorem B.


## 4. The Dynamics of (3.1)

In this section, we investigate the dynamics of (3.1) with nonnegative initial conditions.Theorem 4.1.
For (3.1), we have the following results. (i)
Assume thatr<1, then the zero equilibrium point is locally asymptotically stable.(ii)
Assume thatr>1, then the zero equilibrium point is saddle point.(iii)
The positive equilibrium pointy̅2 is unstable.Proof.
The linearized equation associated with (3.1) about y̅1=0 has the form
(4.1)zn+1-rzn-1=0,n=0,1,2,…,
so, the characteristic equation of (3.1) about y̅1=0 is
(4.2)λ2k+1-rλ2k-1=0,
then the proof of (i) and (ii) follows immediately from Theorem A.
The linearized equation of (3.1) about y̅2=(r-1)1/Σi=lkpi is
(4.3)zn+1-zn-1+∑i=lkpi(1-1r)zn-2i=0,n=0,1,2,…,
so, the characteristic equation of (3.1) about y̅2=(r-1)1/Σi=lkpi is
(4.4)λ2k+1-λ2k-1+∑i=lkpi(1-1r)λ2k-2i=0.
Set
(4.5)f(λ)=λ2k+1-λ2k-1+∑i=lkpi(1-1r)λ2k-2i,
then f(-1)=((Σi=lkpi)(r-1))/r>0, and limλ→-∞f(λ)=-∞ so f(λ) has at least a root in (-∞,-1). Then the proof of (iii) follows.Theorem 4.2.
Assumer<1. Then the zero equilibrium point of (3.1) is globally asymptotically stable.Proof.
We know by Theorem4.1 that the equilibrium point y̅1=0 is locally asymptotically stable of (3.1), and so it suffices to show that y̅1=0 is a global atractor of (3.1) as follows:
(4.6)0≤yn+1=ryn-11+Πi=lkyn-2ipi≤ryn-1,
since r<1, then
(4.7)lim⁡n→∞yn=0.The next theorem shows that (3.1) has a prime-period two solutions when r=1.Theorem 4.3.
For (3.1), we have the following results.
(a) Equation (3.1) possesses the prime-period two solutions
(4.8)…,ϕ,0,ϕ,0,ϕ,…
with ϕ>0, when r=1.
(b) Assume thatr=1, then every solution of (3.1) converges to a period (not necessarily prime) two solutions (4.8) with ϕ≥0.Proof.
(a) Let(4.9)…,ϕ,ψ,ϕ,ψ,…
be period two solutions of (3.1). Then
(4.10)ϕ=rϕ1+ψΣi=lkpi,ψ=rψ1+ϕΣi=lkpi.
If ϕ≠0 and ψ≠0, then ϕ=ψ=(r-1)1/Σi=lkpi, which is impossible. Hence, ψ=0 which implies that (r-1)ϕ=0, so r=1.
(b) Assume thatr=1, and let {yn}n=-2k∞ be a solution of (3.1), then
(4.11)yn+1-yn-1=-yn-1Πi=lkyn-2ipi1+Πi=lkyn-2ipi≤0.
So the even terms of this solution decrease to a limit (say Φ≥0), and the odd terms decrease to a limit (say Ψ≥0). Thus,
(4.12)Φ=Φ1+ΨΣi=lkpi,Ψ=Ψ1+ΦΣi=lkpi,
which implies that
(4.13)ΦΨΣi=lkpi=0,ΨΦΣi=lkpi=0.
This completes the proof.The next theorem shows that whenr>1, (3.1) possesses unbounded solutions.Theorem 4.4.
Assumer>1. Then (3.1) possesses unbounded solutions. In particular, every solution of (3.1) which oscillates about the equilibrium y̅2=(r-1)1/Σi=lkpi with semicycles of length one is unbounded.Proof.
we will prove that every solution{yn}n=-2k∞ of (3.1) which oscillates with semicycles of length one is unbounded (see corollary 3.1).
Assume that{yn}n=-2k∞ is a solution of (3.1) such that
(4.14)y2n+1<y̅2=(r-1)1/Σi=lkpi,y2n>y̅2=(r-1)1/Σi=lkpi,n≥-k.
Then
(4.15)y2n+2=ry2n1+Πi=lky2n+1-2ipi>ry2n1+y̅2Σi=lkpi=y2n,y2n+3=ry2n+11+Πi=lky2n+2-2ipi<ry2n+11+y̅2Σi=lkpi=y2n+1.
From which it follows that
(4.16)lim⁡n→∞y2n=∞,lim⁡n→∞y2n+1=0,
which completes the proof.



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*Source: 419789-2011-10-16.xml*