# Solutions Form for Some Rational Systems of Difference Equations

Authors: H. El-Metwally
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

We deal with the form of the solutions for the following systems of rational difference equationsxn+1=(xn−1yn/(±xn−1±yn−2)), yn+1=(xnyn−1/(±yn−1±xn−2)), with nonzero real numbers initial conditions. Also we investigate some properties of the obtained solutions and present some numerical examples.


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## Body

## 1. Introduction

Our aim in this paper is to find the solutions form for the following systems of rational difference equations:(1)xn+1=xn-1yn±xn-1±yn-2,yn+1=xnyn-1±yn-1±xn-2,n=0,1,…,
with nonzero real numbers initial conditions and then investigate the obtained solutions.Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economy, physics, and so on. So, recently there has been an increasing interest in the study of qualitative analysis of scalar rational difference equations and systems of rational difference equations. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solutions. See [1–7] and the references cited therein.The periodicity of the positive solutions for the following system of rational difference equations(2)xn+1=myn,yn+1=pynxn-1yn-1
was studied by Cinar et al. [8].Özban [9] has studied the positive solutions for the following system:
(3)xn+1=ayn-3,yn+1=byn-3xn-qyn-q.
The behavior of the positive solutions for the following system
(4)xn+1=xn-11+xn-1yn,yn+1=yn-11+yn-1xn
has been studied by Kurbanlı et al. [10].Touafek and Elsayed [11] studied the periodicity and gave the form of the solutions for the following systems:
(5)xn+1=ynxn-1(±1±yn),yn+1=xnyn-1(±1±xn).
Yalcinkaya [12] investigated the sufficient condition for the global asymptotic stability for the following system of difference equations:
(6)zn+1=tnzn-1+atn+zn-1,tn+1=zntn-1+azn+tn-1.
Yang [13] investigated the positive solutions for the system
(7)xn=A+yn-1xn-pyn-q,yn=A+xn-1xn-ryn-s.
Clark et al. [14, 15] investigate the global asymptotic stability of the following difference equations:
(8)xn+1=xna+cyn,yn+1=ynb+dxn.
Camouzis and Papaschinopoulos [16] studied the global asymptotic behavior of the positive solutions of the system of rational difference equations as follows:
(9)xn+1=1+xnyn-m,yn+1=ynxn-m.


## 2. On the System:xn+1=xn-1yn/(xn-1+yn-2),  yn+1=xnyn-1/(yn-1+xn-2)

In this section, we study the existence of analytical forms of the solutions for the following system of difference equations:(10)xn+1=xn-1ynxn-1+yn-2,yn+1=xnyn-1yn-1+xn-2,n=0,1,…,
with nonzero real initials conditions x-2, x-1, x0, y-2, y-1, and y0.In the sequel we assume that∏i=0-1‍AiBi=1, for any real numbers Ai and Bi.Theorem 1.
Suppose that{xn,yn} is a solution for system (10), then for n=0,1,2,…, one obtains
(11)x2n-2=c(ae)n∏i=0n-1(ia+e)(ie+c),x2n-1=b(bd)n∏i=0n-1(id+b)((i+1)b+f),y2n-2=f(bd)n∏i=0n-1(id+b)(ib+f),y2n-1=e(ae)n∏i=0n-1(ia+e)((i+1)e+c),
where c=x-2, b=x-1, a=x0, f=y-2, e=y-1, and d=y0.Proof.
Forn=0 the result holds. Now suppose that n>0 and that our assumption holds for n-1. That is,
(12)x2n-5=b(bd)n-2∏i=0n-3(id+b)((i+1)b+f),x2n-4=c(ae)n-1∏i=0n-2(ia+e)(ie+c),x2n-3=b(bd)n-1∏i=0n-2(id+b)((i+1)b+f),y2n-5=e(ae)n-2∏i=0n-3(ia+e)((i+1)e+c),y2n-4=f(bd)n-1∏i=0n-2(id+b)(ib+f),y2n-3=e(ae)n-1∏i=0n-2(ia+e)((i+1)e+c).
Now, it follows from system (10) that
(13)x2n-2=x2n-4y2n-3x2n-4+y2n-5=(c(ae)n-1∏i=0n-2(ia+e)(ie+c))×(e(ae)n-1∏i=0n-2(ia+e)((i+1)e+c))×((e(ae)n-2∏i=0n-3(ia+e)((i+1)e+c))(c(ae)n-1∏i=0n-2(ia+e)(ie+c))+(e(ae)n-2∏i=0n-3(ia+e)((i+1)e+c)))-1=(c(ae)n∏i=0n-2(ia+e)(ie+c))×((ca∏i=0n-2(ia+e)((i+1)e+c)∏i=0n-2(ia+e)(ie+c))+(∏i=0n-2(ia+e)((i+1)e+c)∏i=0n-3(ia+e)((i+1)e+c)))-1=(c(ae)n∏i=0n-2(ia+e)(ie+c))×((a((n-1)e+c))+(((n-2)a+e)((n-1)e+c)))-1=(c(ae)n/∏i=0n-2(ia+e)(ie+c))((n-1)e+c)(a+((n-2)a+e))=(c(ae)n/∏i=0n-2(ia+e)(ie+c))((n-1)e+c)((n-1)a+e)=c(ae)n∏i=0n-1(ia+e)(ie+c),y2n-2=y2n-4x2n-3y2n-4+x2n-5=(f(bd)n-1∏i=0n-2(id+b)(ib+f))×(b(bd)n-1∏i=0n-2(id+b)((i+1)b+f))×((f(bd)n-1∏i=0n-2(id+b)(ib+f))+(b(bd)n-2∏i=0n-3(id+b)((i+1)b+f)))-1=(f(bd)n∏i=0n-2(id+b)(ib+f))×(d((n-1)b+f)+((n-2)d+b)((n-1)b+f))-1=(f(bd)n/∏i=0n-2(id+b)(ib+f))((n-1)b+f)[d+((n-2)d+b)]=f(bd)n∏i=0n-1(id+b)(ib+f).
Also, from system (10), we see that
(14)x2n-1=x2n-3y2n-2x2n-3+y2n-4=(b(bd)n-1∏i=0n-2(id+b)((i+1)b+f))×(f(bd)n∏i=0n-1(id+b)(ib+f))×((b(bd)n-1∏i=0n-2(id+b)((i+1)b+f))+(f(bd)n-1∏i=0n-2(id+b)(ib+f)))-1=(fbdb(bd)n-1∏i=0n-2(id+b)((i+1)b+f))×(bf((n-1)d+b)+f((n-1)d+b)((n-1)b+f))-1=(b(bd)n/∏i=0n-2(id+b)((i+1)b+f))((n-1)d+b)[b+(n-1)b+f]=b(bd)n∏i=0n-1(id+b)((i+1)b+f),y2n-1=y2n-3x2n-2y2n-3+x2n-4=(e(ae)n∏i=0n-2(ia+e)((i+1)e+c))×(e((n-1)a+e)+((n-1)a+e)((n-1)e+c))-1=e(ae)n∏i=0n-1(ia+e)((i+1)e+c).
The proof is complete.Lemma 2.
Every positive solution for system (10) is bounded, and limn→∞xn=limn→∞yn=0.Proof.
It follows from system (10) that
(15)xn+1=xn-1ynxn-1+yn-2<xn-1ynxn-1=yn,yn+1=xnyn-1yn-1+xn-2<xnyn-1yn-1=xn,
for n large, we see that
(16)xn+1<yn<xn-1,yn+1<xn<yn-1.
Then the subsequences{x2n-1}n=0∞, {x2n}n=0∞, {y2n-1}n=0∞, and {y2n}n=0∞ are decreasing and so are bounded from above by M, M, N, and N, respectively, where M=max{x-1,x0}N=max{y-1,y0}.The proofs of the following theorems are similar to that of Theorem1 and will be omitted.Theorem 3.
Assume that{xn,yn} is a solution for the system
(17)xn+1=xn-1ynxn-1+yn-2,yn+1=xnyn-1xn-2-yn-1.
Then for n=0,1,2,…,
(18)x2n-2=c(ae)n∏i=0n-1(e+ia)(c-ie),x2n-1=b(bd)n∏i=0n-1(b-id)(f+(i+1)b),y2n-2=f(bd)n∏i=0n-1(b-id)(f+ib),y2n-1=e(ae)n∏i=0n-1(e+ia)(c-(i+1)e).Theorem 4.
The solutions form for the following system:(19)xn+1=xn-1ynyn-2-xn-1,yn+1=xnyn-1yn-1+xn-2
are given by the following formulas:
(20)x2n-2=c(ae)n∏i=0n-1(e-ia)(c+ie),x2n-1=b(bd)n∏i=0n-1(b+id)(f-(i+1)b),y2n-2=f(bd)n∏i=0n-1(b+id)(f-ib),y2n-1=e(ae)n∏i=0n-1(e-ia)(c+(i+1)e).Theorem 5.
Let{xn,yn} be a solution for the following system of difference equations
(21)xn+1=xn-1ynyn-2-xn-1,yn+1=xnyn-1xn-2-yn-1.
Then for n=0,1,2,…,
(22)x2n-2=c(ae)n∏i=0n-1(e-ia)(c-ie),x2n-1=b(bd)n∏i=0n-1(b-id)(f-(i+1)b),y2n-2=f(bd)n∏i=0n-1(b-id)(f-ib),y2n-1=e(ae)n∏i=0n-1(e-ia)(c-(i+1)e).Example 6.
We consider an interesting numerical example for system (10) with the initial conditions x-2=0.18, x-1=-0.4, x0=0.2, y-2=0.03, y-1=0.5, and y0=0.26. See Figure 1.Figure 1


## 3. On the System:xn+1=xn-1yn/(xn-1+yn-2),  yn+1=xnyn-1/(yn-1-xn-2)

In this section, we obtain the solutions form for the following system of two difference equations:(23)xn+1=xn-1ynxn-1+yn-2,yn+1=xnyn-1yn-1-xn-2,
with nonzero real numbers initial conditions x-2, x-1, x0, y-2, y-1, and y0 provided that x-2≠y-1 and x-1≠y0.Theorem 7.
Suppose that{xn,yn} is a solution for system (23). Then for n=0,1,2,…,
(24)x4n-2=(ae)2ncn-1(e-c)n∏i=0n-1(2ia+e)((2i+1)a+e),x4n-1=bn+1d2n(d-b)n∏i=0n-1((2i+1)b+f)((2i+2)b+f),x4n=a(ae)2ncn(e-c)n∏i=0n-1((2i+1)a+e)((2i+2)a+e),x4n+1=bn+1d2n+1(b+f)(d-b)n∏i=0n-1((2i+2)b+f)((2i+3)b+f),y4n-2=fbnd2n(d-b)n∏i=0n-1((2i)b+f)((2i+1)b+f),y4n-1=e(ae)2ncn(e-c)n∏i=0n-1((2i)a+e)((2i+1)a+e),y4n=bnd2n+1(d-b)n∏i=0n-1((2i+1)b+f)((2i+2)b+f),y4n+1=(ae)2n+1cn(e-c)n+1∏i=0n-1((2i+1)a+e)((2i+2)a+e).Proof.
Forn=0 the result holds. Now suppose that n>0 and that our assumption holds for n-1. That is,
(25)x4n-6=(ae)2n-2cn-2(e-c)n-1∏i=0n-2(2ia+e)((2i+1)a+e),x4n-5=bnd2n-2(d-b)n-1∏i=0n-2((2i+1)b+f)((2i+2)b+f),x4n-4=a(ae)2n-2cn-1(e-c)n-1∏i=0n-2((2i+1)a+e)((2i+2)a+e),x4n-3=bnd2n-1(b+f)(d-b)n-1∏i=0n-2((2i+2)b+f)((2i+3)b+f),y4n-6=fbn-1d2n-2(d-b)n-1∏i=0n-1((2i)b+f)((2i+1)b+f),y4n-5=e(ae)2n-2cn-1(e-c)n-1∏i=0n-2((2i)a+e)((2i+1)a+e),y4n-4=bn-1d2n-1(d-b)n-1∏i=0n-2((2i+1)b+f)((2i+2)b+f),y4n-3=(ae)2n-1cn-1(e-c)n∏i=0n-2((2i+1)a+e)((2i+2)a+e).
Now, it follows from system (23) that
(26)x4n-2=x4n-4y4n-3x4n-4+y4n-5=(a(ae)2n-2cn-1(e-c)n-1∏i=0n-2((2i+1)a+e)((2i+2)a+e))×((ae)2n-1cn-1(e-c)n∏i=0n-2((2i+1)a+e)((2i+2)a+e))×((a(ae)2n-2cn-1(e-c)n-1∏i=0n-2((2i+1)a+e)((2i+2)a+e))+(e(ae)2n-2cn-1(e-c)n-1∏i=0n-2((2i)a+e)((2i+1)a+e)))-1=(a(ae)2n-1cn-1(e-c)n∏i=0n-2((2i+1)a+e)((2i+2)a+e)∏i=0n-2((2i+1)a+e)((2i+2)a+e))×(a+(2n-2)a+e)-1=1((2n-1)a+e)×a(ae)2n-1cn-1(e-c)n∏i=0n-2((2i+1)a+e)((2i+2)a+e)=(ae)2ncn-1(e-c)n∏i=0n-1((2i)a+e)((2i+1)a+e),y4n-2=x4n-3y4n-4y4n-4-x4n-5=(bnd2n-1(∏i=0n-2((2i+2)b+f)((2i+3)b+f))-1(b+f∏i=0n-2)-1(b+f)(d-b)n-1=(bnd2n-1×∏i=0n-2((2i+2)b+f)((2i+3)b+f))-1(b+f∏i=0n-2)-1)×(bn-1d2n-1(d-b)n-1∏i=0n-2((2i+1)b+f)((2i+2)b+f))×((bn-1d2n-1(d-b)n-1∏i=0n-2((2i+1)b+f)((2i+2)b+f))-(bnd2n-2(d-b)n-1∏i=0n-2((2i+1)b+f)((2i+2)b+f)))-1=(bnd2n(∏i=0n-2((2i+2)b+f)((2i+3)b+f)(b+f)(d-b)n-1bnd2n×∏i=0n-2((2i+2)b+f)((2i+3)b+f))-1)×(d-b)-1=fbnd2n(d-b)n∏i=0n-1((2i)b+f)((2i+1)b+f).
Similarly one can prove the other relations. The proof is complete.Lemma 8.
Every positive solution of the equationxn+1=xn-1yn/(xn-1+yn-2) is bounded, and limn→∞xn=0.The following theorems deal with the solutions form for the following systems, and their proofs will be omitted:(27)xn+1=xn-1ynyn-2-xn-1,yn+1=xnyn-1-yn-1-xn-2,(28)xn+1=xn-1ynyn-2-xn-1,yn+1=xnyn-1yn-1-xn-2,(29)xn+1=xn-1ynxn-1+yn-2,yn+1=xnyn-1-yn-1-xn-2.Theorem 9.
Assume that{xn,yn} is a solution for system (27) with x-2≠-y-1 and x-1≠-y0. Then for n=0,1,2,…,
(30)x4n-2=(-1)n(ae)2ncn-1(e+c)n∏i=0n-1(e-2ia)(e-(2i+1)a),x4n-1=(-1)nbn+1d2n(d+b)n∏i=0n-1((2i+1)b-f)((2i+2)b-f),x4n=(-1)na(ae)2ncn(e+c)n∏i=0n-1(e-(2i+1)a)(e-(2i+2)a),x4n+1=(-1)n+1bn+1d2n+1(b-f)(d+b)n∏i=0n-1((2i+2)b-f)((2i+3)b-f),y4n-2=(-1)nfbnd2n(d+b)n∏i=0n-1((2i)b-f)((2i+1)b-f),y4n-1=(-1)ne(ae)2ncn(e+c)n∏i=0n-1(e-(2i)a)(e-(2i+1)a),y4n=(-1)nbnd2n+1(d+b)n∏i=0n-1((2i+1)b-f)((2i+2)b-f),y4n+1=(-1)n+1(ae)2n+1cn(e+c)n+1∏i=0n-1(e-(2i+1)a)(e-(2i+2)a).Theorem 10.
Assume that{xn,yn} is a solution for system (28) with x-2≠y-1 and x-1≠y0. Then for n=0,1,2,…,
(31)x4n-2=(-1)n(ae)2ncn-1(c-e)n∏i=0n-1(e-2ia)(e-(2i+1)a),x4n-1=(-1)nbn+1d2n(b-d)n∏i=0n-1(f-(2i+1)b)(f-(2i+2)b),x4n=(-1)na(ae)2ncn(c-e)n∏i=0n-1(e-(2i+1)a)(e-(2i+2)a),x4n+1=(-1)nbn+1d2n+1(f-b)(b-d)n∏i=0n-1(f-(2i+2)b)(f-(2i+3)b),y4n-2=(-1)nfbnd2n(b-d)n∏i=0n-1(f-(2i)b)(f-(2i+1)b),y4n-1=(-1)ne(ae)2ncn(c-e)n∏i=0n-1(e-(2i)a)(e-(2i+1)a),y4n=(-1)nbnd2n+1(b-d)n∏i=0n-1(f-(2i+1)b)(f-(2i+2)b),y4n+1=(-1)n+1(ae)2n+1cn(c-e)n+1∏i=0n-1(e-(2i+1)a)(e-(2i+2)a).Theorem 11.
The solution form for system (29) is given by
(32)x4n-2=(-1)n(ae)2ncn-1(c+e)n∏i=0n-1(e+2ia)(e+(2i+1)a),x4n-1=(-1)nbn+1d2n(b+d)n∏i=0n-1(f+(2i+1)b)(f+(2i+2)b),x4n=(-1)na(ae)2ncn(c+e)n∏i=0n-1(e+(2i+1)a)(e+(2i+2)a),x4n+1=(-1)nbn+1d2n+1(f+b)(b+d)n∏i=0n-1(f+(2i+2)b)(f+(2i+3)b),y4n-2=(-1)nfbnd2n(b+d)n∏i=0n-1(f+(2i)b)(f+(2i+1)b),y4n-1=(-1)ne(ae)2ncn(c+e)n∏i=0n-1(e+(2i)a)(e+(2i+1)a),y4n=(-1)nbnd2n+1(b+d)n∏i=0n-1(f+(2i+1)b)(f+(2i+2)b),y4n+1=(-1)n+1(ae)2n+1cn(c+e)n+1∏i=0n-1(e+(2i+1)a)(e+(2i+2)a),
where x-2≠-y-1 and x-1≠-y0.Example 12.
Consider system (23) with the initial conditions x-2=8, x-1=4, x0=5, y-2=3, y-1=9, and y0=6. See Figure 2.Figure 2


## 4. On the System:xn+1=xn-1yn/(xn-1-yn-2), yn+1=xnyn-1/(yn-1+xn-2)

In this section, we present the solutions form for the following system:(33)xn+1=xn-1ynxn-1-yn-2,yn+1=xnyn-1yn-1+xn-2,
with nonzero real numbers initial conditions where x-1≠y-2 and x0≠y-1.The following theorems can be proved similarly to those in Sections2 and 3.Theorem 13.
Suppose that{xn,yn} is a solution for system (33). Assume that x-2, x-1, x0, y-2, y-1, and y0 are arbitrary nonzero real numbers. Then
(34)x4n-2=ca2nen(a-e)n∏i=0n-1(2ie+c)((2i+1)e+c),x4n-1=b2n+1d2nfn(b-f)n∏i=0n-1((2i)d+b)((2i+1)d+b),x4n=a2n+1en(a-e)n∏i=0n-1((2i+1)e+c)((2i+2)e+c),x4n+1=(bd)2n+1fn(b-f)n∏i=0n-1((2i+1)d+b)((2i+2)d+b),y4n-2=(bd)2nfn-1(b-f)n∏i=0n-1((2i)d+b)((2i+1)d+b),y4n-1=a2nen+1(a-e)n∏i=0n-1((2i+1)e+c)((2i+2)e+c),y4n=b2nd2n+1fn(b-f)n∏i=0n-1((2i+1)d+b)((2i+2)d+b),y4n+1=a2n+1en+1(c+e)(a-e)n∏i=0n-1((2i+2)e+c)((2i+3)e+c).Lemma 14.
Every positive solution of the equationyn+1=xnyn-1/(yn-1+xn-2) is bounded and limn→∞yn=0.Theorem 15.
Let{xn,yn} be a solution for the system
(35)xn+1=xn-1yn-xn-1-yn-2,yn+1=xnyn-1xn-2-yn-1,
with x-1≠-y-2 and x0≠-y-1. Then for n=0,1,2,…,
(36)x4n-2=(-1)nca2nen(a+e)n∏i=0n-1(c-2ie)(c-(2i+1)e),x4n-1=(-1)nb2n+1d2nfn(b+f)n∏i=0n-1(b-(2i)d)(b-(2i+1)d),x4n=(-1)na2n+1en(a+e)n∏i=0n-1(c-(2i+1)e)(c-(2i+2)e),x4n+1=(-1)n+1(bd)2n+1fn(b+f)n∏i=0n-1(b-(2i+1)d)(b-(2i+2)d),y4n-2=(-1)n(bd)2nfn-1(b+f)n∏i=0n-1(b-(2i)d)(b-(2i+1)d),y4n-1=(-1)na2nen+1(a+e)n∏i=0n-1(c-(2i+1)e)(c-(2i+2)e),y4n=(-1)nb2nd2n+1fn(b+f)n∏i=0n-1(b-(2i+1)d)(b-(2i+2)d),y4n+1=(-1)na2n+1en+1(c-e)(a+e)n∏i=0n-1(c-(2i+2)e)(c-(2i+3)e).Theorem 16.
The solution form for the following system(37)xn+1=xn-1yn-xn-1-yn-2,yn+1=xnyn-1xn-2+yn-1,
with x-1≠-y-2 and x0≠-y-1 is given by
(38)x4n-2=(-1)nca2nen(a+e)n∏i=0n-1(c+2ie)(c+(2i+1)e),x4n-1=(-1)nb2n+1d2nfn(b+f)n∏i=0n-1(b+(2i)d)(b+(2i+1)d),x4n=(-1)na2n+1en(a+e)n∏i=0n-1(c+(2i+1)e)(c+(2i+2)e),x4n+1=(-1)n+1(bd)2n+1fn(b+f)n∏i=0n-1(b+(2i+1)d)(b+(2i+2)d),y4n-2=(-1)n(bd)2nfn-1(b+f)n∏i=0n-1(b+(2i)d)(b+(2i+1)d),y4n-1=(-1)na2nen+1(a+e)n∏i=0n-1(c+(2i+1)e)(c+(2i+2)e),y4n=(-1)nb2nd2n+1fn(b+f)n∏i=0n-1(b+(2i+1)d)(b+(2i+2)d),y4n+1=(-1)na2n+1en+1(c+e)(a+e)n∏i=0n-1(c+(2i+2)e)(c+(2i+3)e).Theorem 17.
The following system(39)xn+1=xn-1ynxn-1-yn-2,yn+1=xnyn-1xn-2-yn-1
has a solution form given by the following relations:
(40)x4n-2=ca2nen(a-e)n∏i=0n-1(c-2ie)(c-(2i+1)e),x4n-1=b2n+1d2nfn(b-f)n∏i=0n-1(b-(2i)d)(b-(2i+1)d),x4n=a2n+1en(a-e)n∏i=0n-1(c-(2i+1)e)(c-(2i+2)e),x4n+1=(bd)2n+1fn(b-f)n∏i=0n-1(b-(2i+1)d)(b-(2i+2)d),y4n-2=(bd)2nfn-1(b-f)n∏i=0n-1(b-(2i)d)(b-(2i+1)d),y4n-1=a2nen+1(a-e)n∏i=0n-1(c-(2i+1)e)(c-(2i+2)e),y4n=b2nd2n+1fn(b-f)n∏i=0n-1(b-(2i+1)d)(b-(2i+2)d),y4n+1=a2n+1en+1(c-e)(a-e)n∏i=0n-1(c-(2i+2)e)(c-(2i+3)e),
where x-1≠y-2 and x0≠y-1.Example 18.
Consider system (33) with the initial values x-2=0.8, x-1=4, x0=0.15, y-2=3, y-1=-9, and y0=-0.6. See Figure 3.Figure 3


## 5. Other Systems

In this section, we give the solutions form for the following systems of difference equations:(41)xn+1=xn-1ynxn-1-yn-2,yn+1=xnyn-1yn-1-xn-2,(42)xn+1=xn-1yn-xn-1-yn-2,yn+1=xnyn-1yn-1-xn-2,(43)xn+1=xn-1ynxn-1-yn-2,yn+1=xnyn-1-yn-1-xn-2,(44)xn+1=xn-1yn-xn-1-yn-2,yn+1=xnyn-1-yn-1-xn-2,
with nonzero real numbers initial conditions.Theorem 19.
Let{xn,yn} be a solution for system (41) with x-2≠y-1≠x0 and y-2≠x-1≠y0. Then
(45)x4n-2=a2nencn-1[(a-e)(e-c)]n,x4n-1=bn+1d2n[f(b-d)(f-b)]n,x4n=a2n+1en[c(a-e)(e-c)]n,x4n+1=-d2n+1bn+1(f-b)[f(b-d)(f-b)]n,y4n-2=fbnd2n[f(b-d)(f-b)]n,y4n-1=a2nen+1[c(a-e)(e-c)]n,y4n=bnd2n+1[f(b-d)(f-b)]n,y4n+1=a2n+1en+1(e-c)[c(a-e)(e-c)]n.Theorem 20.
Suppose that{xn,yn} is a solution for system (42) with x-2≠y-1, y-1≠-x0, y-2≠-x-1, and x-1≠y0. Then
(46)x4n-2=a2nencn-1[(a+e)(c-e)]n,x4n-1=bn+1d2n[f(b-d)(f+b)]n,x4n=a2n+1en[c(a+e)(c-e)]n,x4n+1=-d2n+1bn+1(f+b)[f(b-d)(f+b)]n,y4n-2=fbnd2n[f(b-d)(f+b)]n,y4n-1=a2nen+1[c(a+e)(c-e)]n,y4n=bnd2n+1[f(b-d)(f+b)]n,y4n+1=-a2n+1en+1(c-e)[c(a+e)(c-e)]n.Theorem 21.
The solution for system (43) is given by the following formula; for n=0,1,2,…;
(47)x4n-2=a2nencn-1[(e-a)(c+e)]n,x4n-1=bn+1d2n[f(b+d)(f-b)]n,x4n=a2n+1en[c(e-a)(c+e)]n,x4n+1=-d2n+1bn+1(f-b)[f(b+d)(f-b)]n,y4n-2=fbnd2n[f(b+d)(f-b)]n,y4n-1=a2nen+1[c(e-a)(c+e)]n,y4n=bnd2n+1[f(b+d)(f-b)]n,y4n+1=-a2n+1en+1(c+e)[c(e-a)(c+e)]n,
where x-2≠-y-1, y-1≠x0, y-2≠x-1, and x-1≠-y0.Theorem 22.
If{xn,yn} is a solution for system (44) with x-2≠-y-1, y-1≠-x0, y-2≠-x-1, and x-1≠-y0, then
(48)x4n-2=a2nencn-1[(e+a)(c+e)]n,x4n-1=bn+1d2n[f(b+d)(f+b)]n,x4n=a2n+1en[c(e+a)(c+e)]n,x4n+1=-d2n+1bn+1(f+b)[f(b+d)(f+b)]n,y4n-2=fbnd2n[f(b+d)(f+b)]n,y4n-1=a2nen+1[c(e+a)(c+e)]n,y4n=bnd2n+1[f(b+d)(f+b)]n,y4n+1=-a2n+1en+1(c+e)[c(e+a)(c+e)]n.Example 23.
Figure4 shows the behavior of the solution for system (41) with the initial conditions x-2=0.18, x-1=-0.7, x0=-0.5, y-2=1.3, y-1=0.9, and y0=-0.26.Figure 4



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