# Dynamics of a System of Rational Higher-Order Difference Equation

Authors: Banyat Sroysang
Journal: Discrete Dynamics in Nature and Society
 (2013)

---

## Abstract

We focus on a system of a rationalm-order difference equation xn+1=(xn-m+1)/(A+ynyn-1⋯yn-m+1),  yn+1=(yn-m+1)/(B+xnxn-1⋯xn-m+1), n=0,1,…, where A,B,x0,x-1,…,x-m+1,y0,y-1,…,y-m+1∈(0,∞). We investigate the dynamical behavior of positive solution for the system.


---

## Body

## 1. Introduction

In 2011, Kurbanli et al. [1] studied the behavior of positive solutions of the system of rational difference equations
(1)xn+1=xn-1ynxn-1+1,yn+1=yn-1xnyn-1+1,
where the initial conditions are arbitrary nonnegative real numbers.In the same year, Kurbanli [2] studied the behavior of solutions of the system of rational difference equations
(2)xn+1=xn-1ynxn-1-1,yn+1=yn-1xnyn-1-1,zn+1=zn-1ynzn-1-1,
where the initial conditions are arbitrary real numbers. Moreover, Kurbanli [3] studied the behavior of the solutions of the difference equation system
(3)xn+1=xn-1ynxn-1-1,yn+1=yn-1xnyn-1-1,zn+1=1ynzn,
where x0,x-1,y0,y-1,z0,z-1∈ℝ such that y0x-1≠1, x0y-1≠1 and y0z0≠1.In [4], Liu et al. gave more results of the solution of the system (2) including a new and simple expression of zn and the asymptotical behavior of the solution.In [5], Stević showed that the system of difference equations
(4)xn+1=axn-1bynxn-1+c,yn+1=αyn-1βxnyn-1+γ,aaaaaaaaaaaaaaaaaaaaaaaaan=0,1,…,
can be solved.In 2012, Gu and Ding [6] derived two canonical state space forms from multiple-input multiple-output systems described by difference equations.The system of two nonlinear difference equations(5)xn+1=A+ynxn-p,yn+1=A+xnyn-q,n=0,1,…,
was studied by Papaschinopoulos and Schinas [7], where p,q∈ℕ.Moreover, the system of rational difference equations(6)xn+1=xna+cyn,yn+1=ynb+dxn,n=0,1,…,
was studied by Clark et al. [8, 9], where a,b,c,d∈(0,∞) and x0,y0∈[0,∞).Liu et al. [10] studied the behavior of a system of rational difference equations
(7)xn+1=xn-1ynxn-1-1,yn+1=yn-1xnyn-1-1,zn+1=1xnzn-1,n=0,1,…,
where the initial conditions are nonzero real numbers.In 2012, Zhang et al. [11] studied the solutions, stability character, and asymptotic behavior of the system of a rational third-order difference equation
(8)xn+1=xn-2A+ynyn-1yn-2,yn+1=yn-2B+xnxn-1xn-2,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaan=0,1,…,
where A,B,x0,x-1,x-2,y0,y-1,y-2∈(0,∞).In this paper, we studied the solutions, stability character, and asymptotic behavior of the system of a rationalm-order difference equation
(9)xn+1=xn-m+1A+ynyn-1⋯yn-m+1,yn+1=yn-m+1B+xnxn-1⋯xn-m+1,n=0,1,…,
where A,B,x0,x-1,…,x-m+1,y0,y-1,…,y-m+1∈(0,∞).


## 2. Preliminaries

Letm∈ℕ and let f:Ixm×Iym→Ix and g:Ixm×Iym→Iy be continuously differentiable functions, where Ix and Iy are intervals in ℝ.For any(x0,y0),(x-1,y-1),…,(x-m+1,y-m+1)∈Ix×Iy, the system of difference equations
(10)xn+1=f(xn,xn-1,…,xn-m+1,yn,yn-1,…,yn-m+1),yn+1=g(xn,xn-1,…,xn-m+1,yn,yn-1,…,yn-m+1),n=0,1,…,
has a unique solution {(xn,yn)}n=-m+1∞.Definition 1.
A point(x-,y-)∈Ix×Iy is called an equilibrium point of the system (10) if x-=f(x-,x-,…,x-,y-,y-,…,y-) and y-=g(x-,x-,x-,…,y-,y-,…,y-).Definition 2.
The linearized system of the system (10) about the equilibrium (x-,y-) is the system of linear difference equations
(11)xn+1=∑i=0m-1(∂f(x-,x-,…,x-,y-,y-,…,y-)∂xn-ixn-i‍+∂f(x-,x-,…,x-,y-,y-,…,y-)∂yn-iyn-i),yn+1=∑i=0m-1(∂g(x-,x-,…,x-,y-,y-,…,y-)∂xn-ixn-i‍+∂g(x-,x-,…,x-,y-,y-,…,y-)∂yn-iyn-i).Definition 3.
An equilibrium point(x-,y-) of the system (10) is said to be stable relative to Ix×Iy if for every ϵ>0, there exists δ>0 such that for any (x0,y0),(x-1,y-1),…,(x-m+1,y-m+1)∈Ix×Iy, with
(12)max{∑i=-m+10‍|xi-x-|,∑i=-m+10‍|yi-y-|}<δ.
One has max{|xn-x-|,|yn-y-|}<ϵ for all n≥-m+1.Definition 4.
An equilibrium point(x-,y-) of the system (10) is called an attractor relative to Ix×Iy if for all (x0,y0),(x-1,y-1),…,(x-m+1,y-m+1)∈Ix×Iy, one has limn→∞xn=x- and limn→∞yn=y-.Definition 5.
An equilibrium point(x-,y-) of the system (10) is said to be asymptotically stable relative to Ix×Iy if it is stable, and it is also an attractor.Definition 6.
An equilibrium point(x-,y-) of the system (10) is said to be unstable if it is not stable.Theorem 7 (see [12]).
LetX(n+1)=F(X(n)), n=0,1,…, be a system of difference equations and let X- be the equilibrium point of the system. If all eigenvalues of the Jacobian matrix evaluated at X- lie inside the open unit disk, then X- is asymptotically stable. If one of them has a modulus greater than one, then X- is unstable.Theorem 8 (see [13]).
LetX(n+1)=F(X(n)), n=0,1,…, be a system of difference equations and let X- be the equilibrium point of the system. Assume that the characteristic polynomial of the system about X- is a0λn+a1λn-1+⋯+an-1λ+an where ai∈ℝ for all i and a0>0. Then all roots of the characteristic equation lie inside the open unit disk if and only if Δk>0 for all positive integer k≤n, where Δk is the principal minor of order k of the n×n matrix
(13)Δn=(a1a3a5⋯0a0a2a4⋯00a1a3⋯0⋮⋮⋮⋱⋮000⋯an).


## 3. Results

We note that(i)
ifA<1 and B<1 then the system (9) has equilibrium (0,0) and (1-Bm,1-Am);(ii)
ifA=1 and B<1 then the system (9) has equilibrium (0,0) and (1-Bm,0);(iii)
ifA<1 and B=1 then the system (9) has equilibrium (0,0) and (0,1-Am);(iv)
ifA>1 and B>1 then (0,0) is the unique equilibrium point of the system (9).Theorem 9.
Let(xn,yn) be positive solution of the system (9). For all nonnegative integer k, one has
(14)0≤xn≤{x-m+1Ak+1,n=mk+1;x-m+2Ak+1,n=mk+2;⋮x0Ak+1,n=mk+m,0≤yn≤{y-m+1Bk+1,n=mk+1;y-m+2Bk+1,n=mk+2;⋮y0Bk+1,n=mk+m.Proof.
Obviously, they are true fork=0. Suppose that they are true for k=l. Then(15)xn={xm(l+1)+1≤xm(l+1)-m+1A=1Axml+1≤1A(x-m+1Al+1),n=m(l+1)+1;xm(m+1)+2≤xm(l+1)-m+2A=1Axml+2≤1A(x-m+2Al+1),n=m(l+1)+2;⋮xm(l+1)+m≤xm(l+1)A=1Axml+m≤1A(x0Al+1),n=m(l+1)+m,yn={ym(l+1)+1≤ym(l+1)-m+1B=1Byml+1≤1B(y-m+1Bl+1),n=m(l+1)+1;y4(l+1)+2≤ym(l+1)-m+2B=1Byml+2≤1B(y-m+2Bl+1),n=4(m+1)+2;⋮ym(l+1)+m≤ym(l+1)B=1Byml+m≤1B(y0Bl+1),n=m(l+1)+m.
Thus, they are true fork=l+1.
By the mathematical induction, this proof is completed.Corollary 10.
Let(xn,yn) be positive solution of the system (9). If A>1 and B>1, then the sequence {(xn,yn)} converges exponentially to the equilibrium point (0,0).Theorem 11.
LetA>1 and B>1. Then the equilibrium point (0,0) of the system (9) is asymptotically stable.Proof.
The linearized system of the system (9) about the equilibrium (0,0) is
(16)Φn+1=DΦn,
where
(17)Φn=(xnxn-1xn-2⋮xn-m+1ynyn-1yn-2⋮yn-m+1),D=(00⋯01A00⋯0010⋯0000⋯0001⋯0000⋯00⋮⋱⋱⋱⋱⋱⋱⋮⋮⋮00⋯100000000⋯0000001B00⋯001000000⋯0001000⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮00⋯0000010).
The characteristic equation of the system (16) is
(18)(λm-1A)(λm-1B)=0.
Thus, |λ|<1. By Theorem 7, the equilibrium point (0,0) is asymptotically stable.Theorem 12.
LetA<1 and B<1. Then both the equilibrium points (0,0) and (1-Bm,1-Am) of the system (9) are unstable.Proof.
We note by the characteristic equation (18) that |λ|>1 and then, by Theorem 7, the equilibrium point (0,0) is unstable.
Next, we consider the equilibrium point(1-Bm,1-Am). The linearized system of the system (9) about the equilibrium (1-Bm,1-Am) is
(19)Φn+1=GΦn,
where
(20)Φn=(xnxn-1xn-2⋮xn-m+1ynyn-1yn-2⋮yn-m+1),G=(00⋯01αα⋯αα10⋯0000⋯0001⋯0000⋯00⋮⋱⋱⋱⋱⋱⋱⋮⋮⋮00⋯1000000ββ⋯ββ0000100⋯001000000⋯0001000⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮00⋯0000010),
in which
(21)α=-(1-A)m-1(1-B)m,β=-(1-A)(1-B)m-1m.
The characteristic polynomial of the system (19) is
(22)1-αβ-∑i=1m‍(iαβλi-1)-2λm-∑i=1m-1‍(iαβλ2m-(i+1))+λ2m.
We note the characteristic polynomiala0λ2m+a1λ2m-1+⋯+a2m-1λ+a2m that a1=0. Thus, we obtain that not all of Δk>0, k=1,2,…,2m. By Theorems 7 and 8, the equilibrium point (1-Bm,1-Am) is unstable.Theorem 13.
LetA,B<1 and Ω1=(0,1-Bm)×(1-Am,∞), Ω2=(1-Bm,∞)×(0,1-Am). Assume that {(xn,yn)}n=-m+1∞ satisfies the system (9). Then(i)
if{(xn,yn)}n=-m+10⊆Ω1, then {(xn,yn)}n=-m+1∞⊆Ω1;(ii)
if{(xn,yn)}n=-m+10⊆Ω2, then {(xn,yn)}n=-m+1∞⊆Ω2.Proof.
(i) Assume that{(xn,yn)}n=-m+10⊆Ω1. Then, for any i∈{0,1,…,m-1},
(23)xi+1=xi-m+1A+yiyi-1⋯yi-m+1<xi-m+1A+(1-Am)m=xi-m+1,yi+1=yi-m+1B+xixi-1⋯xi-m+1>yi-m+1B+(1-Bm)m=yi-m+1.
Then(x1,y1),(x2,y2),…,(xm,ym)∈Ω1.
Next, we suppose that(xk,yk),(xk-1,yk-1),…,(xk-m+1,yk-m+1)∈Ω1 where k is a positive integer. Then
(24)xk+1=xk-m+1A+ykyk-1⋯yk-m+1<xk-m+1A+(1-Am)m=xk-m+1,yk+1=yk-m+1B+xkxk-1⋯xk-m+1>yk-m+1B+(1-Bm)m=yk-m+1.
Then(xk+1,yk+1)∈Ω1.
By the mathematical induction,{(xn,yn)}n=-m+1∞⊆Ω1.
(ii) This is similar to the proof of (i).



---

*Source: 179401-2013-05-20.xml*