# Dynamical Behavior of a System of Third-Order Rational Difference Equation

Authors: Qianhong Zhang; Jingzhong Liu; Zhenguo Luo
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

This paper deals with the boundedness, persistence, and global asymptotic stability of positive solution for a system of third-order rational difference equationsxn+1=A+xn/yn-1yn-2, yn+1=A+yn/xn-1xn-2, n=0,1,…, where A∈(0,∞), x-i∈(0,∞); y-i∈(0,∞), i=0,1,2. Some examples are given to demonstrate the effectiveness of the results obtained.


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## Body

## 1. Introduction

It is known that difference equation appears naturally as discrete analogous and as numerical solutions of differential equation and delay differential equation having many applications in economics, biology, computer science, control engineering, and so forth. The study of discrete dynamical systems described by difference equations has now been paid great attention by many mathematical researchers. Particularly, the persistence, boundedness, local asymptotic stability, global character, and the existence of positive periodic solutions can be discussed in many papers.In 1998, DeVault et al. [1] proved every positive solution of the difference equation:(1)xn+1=A+xnxn-1,n=0,1,…,where A∈(0,∞) oscillates about the positive equilibrium c=1+A of (1). Moreover every positive solution of (1) is bounded away from zero and infinity. Also the positive equilibrium of (1) is globally asymptotically stable.In 2003, Abu-Saris and DeVault [2] studied the following recursive difference equation:(2)xn+1=A+xnxn-k,n=0,1,…,where A∈(1,+∞), x-k,x-k+1,…,x0 are positive real numbers. For similar results the reader can refer to [3–9].Difference equations or discrete dynamical systems are a diverse field which impact almost every branch of pure and applied mathematics. We refer to [10, 11] for basic theory of difference equations and rational difference equations. It is very interesting to investigate the qualitative behavior of the discrete dynamical systems of nonlinear difference equations. Recently there has been a lot of work concerning the global asymptotic stability, the periodicity, and the boundedness of nonlinear difference equations. Moreover similar results in [12–17] have been derived for systems of two nonlinear difference equations.Papaschinopoulos and Schinas [12] investigated the global behavior for a system of the following two nonlinear difference equations:(3)xn+1=A+ynxn-p,yn+1=A+xnyn-q,n=0,1,…,where A is a positive real number, p,q are positive integers, and x-p,…,x0,x-q,…,x0 are positive real numbers.In 2005, Yang [13] studied the global behavior of the following system:(4)xn=A+yn-1xn-pyn-q,yn=A+xn-1xn-ryn-s,n=1,2,…,where p≥2, q≥2, r≥2, s≥2, A is a positive constant, and initial values x1-max⁡{p,r},x2-max⁡{p,r},…,x0,y1-max⁡{q,s},y2-max⁡{q,s},…,y0 are positive real numbers.In 2012, Zhang et al. [14] investigated the global behavior for a system of the following third-order nonlinear difference equations:(5)xn+1=xn-2B+yn-2yn-1yn,yn+1=yn-2A+xn-2xn-1xn,where A,B∈(0,∞), and the initial values x-i,y-i∈(0,∞), i=0,1,2.Motivated by the above mentioned discussion, in this paper we study the behavior of solutions of the following system:(6)xn+1=A+xnyn-1yn-2,yn+1=A+ynxn-1xn-2,n=0,1,…,where A is a positive constant. And initial conditions x-i,y-i∈(0,∞), i=0,1,2.Clearly, ifA>0, system (6) has always positive equilibrium. Consider (7)c,c=A+A2+42,A+A2+42.If 1<A<2/3, system (6) has two additional positive equilibria: (8)a1,b1=2A-4-3A2,2A+4-3A2,a2,b2=2A+4-3A2,2A-4-3A2.Our results in this paper are listed below: (i)
IfA>1, every positive solution of system (6) is bounded.(ii)
IfA>2/3, (c,c) is locally asymptotically stable.(iii)
IfA>3, every positive solution of (6) approaches (c,c).(iv)
If1<A<2/3, (a1,b1) and (a2,b2) are locally asymptotically stable.


## 2. Boundedness

Theorem 1.
Let{(xn,yn)} be a positive solution of (6). Then the following statements hold:(i)
xn≥A and yn≥A for all n≥1.(ii)
IfA>1, then, for k≥4, we have the following:(9)xk≤1A2k-3x3-A3A2-1+A3A2-1,yk≤1A2k-3y3-A3A2-1+A3A2-1.Proof.
Assertion (i) is obviously true. We now prove assertion (ii). From (6) and in view of (i), we have, for all k≥4, that(10)xk=A+xk-1yk-2yk-3≤A+1A2xk-1,yk=A+yk-1xk-2xk-3≤A+1A2yk-1.Let vk,wk be the solution of following system, respectively,(11)vk=A+1A2vk-1,wk=A+1A2wk-1,such that (12)x3=v3,y3=w3.
We prove by induction that(13)xk≤vk,yk≤wk,k≥4.Suppose that (13) is true for k=m≥4. Then from (10) we get the following:(14)xm+1≤A+1A2xm≤A+1A2vm=vm+1,ym+1≤A+1A2ym≤A+1A2wm=wm+1.Therefore (13) is true. From (11) we have the following:(15)vk=1A2k-3x3-A3A2-1+A3A2-1,wk=1A2k-3y3-A3A2-1+A3A2-1,kkkkkkkkkkkkkkkkkkkkkkkkkkik≥4.Then from (10), (13), and (14) the proof of the relation (9) follows immediately.


## 3. Stability

Theorem 2.
Assume thatA>2/3. Then the equilibrium (c,c)=((A+A2+4)/2,(A+A2+4)/2) is locally asymptotically stable.Proof.
We can obtain easily that the linearized system of (6) about the positive equilibrium (c,c) is(16)Ψn+1=BΨn,where(17)Ψn=xnxn-1xn-2ynyn-1yn-2,B=c-2000-c-2-c-21000000100000-c-2-c-2c-200000100000010.Let λ1,λ2,…,λ6 denote the eigenvalues of matrix B, let D=diag(d1,d2,…,d6) be a diagonal matrix, where d1=d4=1, dk=1-kɛ(k=2,3,5,6), and (18)0<ɛ<min⁡16,c2-36c2.Clearly, D is invertible. Computing matrix DBD-1, we obtain that(19)DBD-1=c-2000-c-2d1d5-1-c-2d1d6-1d2d1-1000000d3d2-100000-c-2d4d2-1-c-2d4d3-1c-200000d5d4-1000000d6d5-10.From d1>d2>d3>0 and d4>d5>d6>0 imply that(20)d2d1-1<1,d3d2-1<1,d5d4-1<1,d6d5-1<1.Furthermore,(21)c-2+c-2d1d5-1+c-2d1d6-1=c-21+d1d5-1+d1d6-1=c-21+11-5ɛ+11-6ɛ<c-231-6ɛ<1,c-2+c-2d4d2-1+c-2d4d3-1=c-2(1+d4d2-1+d4d3-1)=c-21+11-2ɛ+11-3ɛ<c-231-3ɛ<c-231-6ɛ<1.It is well known that B has the same eigenvalues as DBD-1: we have that (22)max⁡1≤i≤6λi≤DBD-1∞=max⁡d2d1-1,d3d2-1,d5d4-1,d6d5-1,kkkkkkkc-21+d1d5-1+d1d6-1,kkkkkkkc-21+d4d2-1+d4d3-1<1.This implies that the equilibrium (c,c) of (6) is locally asymptotically stable.
To examine the global attractivity of(c,c), we need the following result.Lemma 3.
Assumeα>β>3. Then α3/(α2-1)>β3/(β2-1).Proof.
Consider the functionf(x)=x3/(x2-1). For x>3, we have (23)f′x=x2x+3x-3x2-12>0.So f(x) is strictly increasing for x>3. We derive that α3/(α2-1)>β3/(β2-1).Theorem 4.
Assume thatA>3. Then every positive solution of (6) converges to (c,c).Proof.
Let{xn,yn} be an arbitrary positive solution of (6). Let (24)Λ1=limn→∞sup⁡{xn,xn+1,…},λ1=limn→∞inf⁡xn,xn+1,…,Λ2=limn→∞sup⁡{yn,yn+1,…},λ1=limn→∞inf⁡{yn,yn+1,…}.From Theorem 1, we have 0<A≤λ1≤Λ1<∞, 0<A≤λ2≤Λ2<∞. The previous and (6) imply that (25)Λ1≤A+Λ1λ22,Λ2≤A+Λ2λ12,λ1≥A+λ1Λ22,λ2≥A+λ2Λ12,which can derive that (26)Λ1Λ22λ22-1≤λ1λ22Λ22-1,Λ2Λ12λ12-1≤λ2λ12Λ12-1.It follows that Λ13Λ23(λ12-1)(λ22-1)≤λ13λ23(Λ12-1)(Λ22-1); namely,(27)Λ13Λ23Λ12-1Λ22-1≤λ13λ23λ12-1λ22-1.If Λ1>λ1 and Λ2>λ2, it follows from Lemma 3 and the condition A>3 that Λ13Λ23/(Λ12-1)(Λ22-1)>λ13λ23/(λ12-1)(λ22-1). Then a contradiction occurs. Thus we have either Λ1=λ1 or Λ2=λ2.
We assumeΛ1=λ1 (the discussion for the case Λ2=λ2 is similar). Then limn→∞xn exists. Using (27) and in view of Λ1=λ1, it is obvious that Λ2≤λ2, so Λ2=λ2. Then limn→∞yn exists. From the uniqueness of the positive equilibrium (c,c) of (6), we conclude that limn→∞xn=c, limn→∞yn=c.
Combining Theorems2 and 4, we obtain the following theorem.Theorem 5.
Assume thatA>3. Then the positive equilibrium (c,c) of (6) is globally asymptotically stable for all positive solutions.Theorem 6.
If1<A<2/3, then the equilibria (a1,b1) and (a2,b2) are locally asymptotically stable.Proof.
The proof is similar to the proof of Theorem2. We can obtain easily that the linearized system of (6) about the positive equilibrium (a1,b1) is(28)Ψn+1=HΨn,where(29)Ψn=xnxn-1xn-2ynyn-1yn-2,H=a1-2000-a1b1-3-a1b1-31000000100000-b1a1-3-b1a1-3a1-200000100000010.Let λ1′,λ2′,…,λ6′ denote the eigenvalues of matrix H, let D=diag(d1,d2,…,d6) be a diagonal matrix, where d1=d4=1, dk=1-kɛ(k=2,3,5,6), and (30)0<ɛ′<min⁡16,a12-36a12,b13-3a16b12.Clearly, H is invertible. Computing matrix DHD-1, we obtain that(31)DHD-1=b1-2000-a1b1-3d1d5-1-a1b1-3d1d6-1d2d1-1000000d3d2-100000-b1a1-3d4d2-1-b1a1-3d4d3-1a1-200000d5d4-1000000d6d5-10.From d1>d2>d3>0 and d4>d5>d6>0 we imply that(32)d2d1-1<1,d3d2-1<1,d5d4-1<1,d6d5-1<1.Furthermore, (33)b1-2+a1b1-3d1d5-1+a1b1-3d1d6-1=b1-21+a1b1d1d5-1+a1b1d1d6-1<a1b131+11-5ɛ′+11-6ɛ′<a1b1331-6ɛ′<1,a1-2+b1a1-3d4d2-1+b1a1-3d4d3-1=a1-21+b1a1d4d2-1+b1a1d4d3-1<1a121+11-2ɛ′+11-3ɛ′<1a1231-6ɛ′<1.It is well known that H has the same eigenvalues as DHD-1; we have that (34)max⁡1≤i≤6λi′≤DHD-1∞=max⁡d2d1-1,d3d2-1,d5d4-1,d6d5-1,kkkkkkkb1-2+a1b1-3d1d5-1+a1b1-3d1d6-1,kkkkkkka1-2+b1a1-3d4d2-1+b1a1-3d4d3-1<1.This implies that the equilibrium (a1,b1) of (6) is locally asymptotically stable.
Similarly, we can prove that the equilibrium(a2,b2) is locally asymptotically stable.


## 4. Numerical Example

In order to illustrate the results of the previous sections and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to nonlinear difference equations and system of nonlinear difference equations.Example 1.
Consider the following system of difference equations:(35)xn+1=A+xnyn-1yn-2,yn+1=A+ynxn-1xn-2,with A=2.4>2/3 and the initial values x-i=1.2,y-i=0.5(i=0,1,2). Then the solution (xn,yn) of system (35) is bounded and persists. And the system has a unique equilibrium (x,y)=(2.7620,2.7620), which is globally asymptotically stable (see Figure 1).
LetA=1.1 and the initial values x-i=2, y-i=1.2(i=0,1,2). Then the solution (xn,yn) of system (35) is bounded and persists (see Figure 2).Figure 1
The dynamics of system (35) at A=2.4.Figure 2
The dynamics of system (35) at A=1.1.



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