# On the Solutions of a General System of Difference Equations

Authors: E. M. Elsayed; M. M. El-Dessoky; Abdullah Alotaibi
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

We deal with the solutions of the systems of the difference equationsxn+1=1/xn-pyn-p, yn+1=xn-pyn-p/xn-qyn-q, and xn+1=1/xn-pyn-pzn-p, yn+1=xn-pyn-pzn-p/xn-qyn-qzn-q, zn+1=xn-qyn-qzn-q/xn-ryn-rzn-r, with a nonzero real numbers initial conditions. Also, the periodicity of the general system of k variables will be considered.


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## Body

## 1. Introduction

In this paper, we deal with the solutions of the systems of the difference equations(1.1)xn+1=1xn-pyn-p,yn+1=xn-pyn-pxn-qyn-q,xn+1=1xn-pyn-pzn-p,yn+1=xn-pyn-pzn-pxn-qyn-qzn-q,zn+1=xn-qyn-qzn-qxn-ryn-rzn-r,
with a nonzero real numbers initial conditions. Also, the periodicity of the general system of k variables will be considered.Recently, there has been a great interest in studying nonlinear difference equations and systems (cf. [1–14] and the references therein). One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models describing real-life situations in population biology, economy, probability theory, genetics, psychology, sociology, and so forth. Such equations also appear naturally as discrete analogues of differential equations which model various biological and economical systems.Cinar [3] has obtained the positive solution of the difference equation system:(1.2)xn+1=1yn,yn+1=ynxn-1yn-1.
Also, Çinar and Yalçinkaya [4] have obtained the positive solution of the difference equation system:(1.3)xn+1=1zn,yn+1=xnxn-1,zn+1=1xn-1.
Clark and Kulenović [5] have investigated the global stability properties and asymptotic behavior of solutions of the system(1.4)xn+1=xna+cyn,yn+1=ynb+dxn.
Elabbasy et al. [6] have obtained the solution of particular cases of the following general system of difference equations:(1.5)xn+1=a1+a2yna3zn+a4xn-1zn,yn+1=b1zn-1+b2znb3xnyn+b4xnyn-1,zn+1=c1zn-1+c2znc3xn-1yn-1+c4xn-1yn+c5xnyn.
Elsayed [10] has obtained the solution of systems of difference equations of rational form.Also, the behavior of the solutions of the following systems:(1.6)xn+1=xn-1±1+xn-1yn,yn+1=yn-1∓1+yn-1xn,
has been studied by Elsayed [15].Özban [16] has investigated the positive solutions of the system of rational difference equations:(1.7)xn+1=1yn-k,yn+1=ynxn-myn-m-k.
Özban [17] has investigated the solutions of the following system:(1.8)xn+1=ayn-3,yn+1=byn-3xn-qyn-q.
Yang et al. [18] have investigated the positive solutions of the systems:(1.9)xn=ayn-p,yn=byn-pxn-qyn-q.
Similar nonlinear systems of rational difference equations were investigated see [15–32].Definition 1.1 (Periodicity).
A sequence{xn}n=-k∞is said to be periodic with period p if xn+p=xnfor all n≥-k.


## 2. Main Results

### 2.1. The First System

In this section, we deal with the solutions of the system of the difference equations(2.1)xn+1=1xn-pyn-p,yn+1=xn-pyn-pxn-qyn-q,
with a nonzero real numbers initial conditions and p≠q.Theorem 2.1.
Suppose that{xn,yn} are solutions of system (2.1). Also, assume that the initial conditionsare arbitrary nonzero real numbers. Then all solutions of equation system (2.1) are eventually periodic with period (2q+2).Proof.
From (2.1), we see that
(2.2)xn+1=1xn-pyn-p,yn+1=xn-pyn-pxn-qyn-q,xn+2=1xn-p+1yn-p+1,yn+2=xn-p+1yn-p+1xn-q+1yn-q+1,xn+3=1xn-p+2yn-p+2,yn+3=xn-p+2yn-p+2xn-q+2yn-q+2,⋮xn-p+q-2=1xn-2p+q-3yn-2p+q-3,yn-p+q-2=xn-2p+q-3yn-2p+q-3xn-p-3yn-p-3,xn-p+q-1=1xn-2p+q-2yn-2p+q-2,yn-p+q-1=xn-2p+q-2yn-2p+q-2xn-p-2yn-p-2,xn-p+q=1xn-2p+q-1yn-2p+q-1,yn-p+q=xn-2p+q-1yn-2p+q-1xn-p-1yn-p-1,xn-p+q+1=1xn-2p+qyn-2p+q,yn-p+q+1=xn-2p+qyn-2p+qxn-pyn-p,xn-p+q+2=1xn-2p+q+1yn-2p+q+1,yn-p+q+2=xn-2p+q+1yn-2p+q+1xn-p+1yn-p+1,⋮xn-p+2q-1=1xn-2p+2q-2yn-2p+2q-2,yn-p+2q-1=xn-2p+2q-2yn-2p+2q-2xn-p-3yn-p-3,xn-p+2q=1xn-2p+2q-1yn-2p+2q-1,yn-p+2q=xn-2p+2q-1yn-2p+2q-1xn-p-2yn-p-2,xn-p+2q+1=1xn-2p+qyn-2p+q,yn-p+2q+1=xn-2p+2qyn-2p+2qxn-p-1yn-p-1,xn-p+2q+2=1xn-2p+q+1yn-2p+q+1,yn-p+2q+2=xn-2p+2q+1yn-2p+2q+1xn-pyn-p,⋮xn+q-1=1xn-p+q-2yn-p+q-2,yn+q-1=xn-p+q-2yn-p+q-2xn-2yn-2,xn+q=1xn-p+q-1yn-p+q-1,yn+q=xn-p+q-1yn-p+q-1xn-1yn-1,xn+q+1=1xn-p+qyn-p+q,yn+q+1=xn-p+qyn-p+qxnyn,xn+q+2=1xn-p+q+1yn-p+q+1,yn+q+2=xn-p+q+1yn-p+q+1xn+1yn+1=xn-p+q+1yn-p+q+1xn-qyn-q,xn+q+3=1xn-p+q+2yn-p+q+2,yn+q+3=xn-p+q+2yn-p+q+2xn+2yn+2=xn-p+q+2yn-p+q+2xn-q+1yn-q+1,xn+q+4=1xn-p+q+3yn-p+q+3,yn+q+4=xn-p+q+3yn-p+q+3xn+3yn+3=xn-p+q+3yn-p+q+3xn-q+2yn-q+2,⋮xn+2q=1xn-p-3yn-p-3,yn+2q=xn-p-3yn-p-3xn-2yn-2,xn+2q+1=1xn-p-2yn-p-2,yn+2q+1=xn-p-2yn-p-2xn-1yn-1,xn+2q+2=1xn-p-1yn-p-1,yn+2q+2=xn-p-1yn-p-1xnyn,xn+2q+3=1xn-p+2q+2yn-p+2q+2=xn+1,yn+2q+3=xn-p+2q+2yn-p+2q+2xn+q+2yn+q+2=yn+1,xn+2q+4=1xn-p+2q+3yn-p+2q+3=xn+2,yn+2q+4=xn-p+2q+3yn-p+2q+3xn+q+3yn+q+3=yn+2.
Hence, the proof is completed.Numerical Example
In order to illustrate the results of this section and to support our theoretical discussions, we consider the following numerical example.Example 2.2.
Consider the difference system (2.1) with p=1, q=2, and the initial conditions x-2=0.9, x-1=0.7, x0=2, y-2=-0.8, y-1=4.2, and y0=6. (See Figure 1).Figure 1


### 2.2. The Second System

In this section, we deal with the solutions of the system of the difference equations(2.3)xn+1=1xn-pyn-pzn-p,yn+1=xn-pyn-pzn-pxn-qyn-qzn-q,zn+1=xn-qyn-qzn-qxn-ryn-rzn-r,
with a nonzero real numbers initial conditions and p≠q,q≠r.Theorem 2.3.
Suppose that{xn,yn,zn} are solutions of system (2.3). Also, assume that the initial conditionsare arbitrary nonzero real numbers. Then all solutions of equation system (2.3) are eventually periodic with period (2r+2).Proof.
From (2.3), we see that
(2.4)xn+1=1xn-pyn-pzn-p,yn+1=xn-pyn-pzn-pxn-qyn-qzn-q,zn+1=xn-qyn-qzn-qxn-ryn-rzn-r,xn+2=1xn-p+1yn-p+1zn-p+1,yn+2=xn-p+1yn-p+1zn-p+1xn-q+1yn-q+1zn-q+1,zn+2=xn-q+1yn-q+1zn-q+1xn-r+1yn-r+1zn-r+1,xn+3=1xn-p+2yn-p+2zn-p+2,yn+3=xn-p+2yn-p+2zn-p+2xn-q+2yn-q+2zn-q+2,zn+3=xn-q+2yn-q+2zn-q+2xn-r+2yn-r+2zn-r+2,⋮xn-p+r-1=1xn-2p+r-2yn-2p+r-2zn-2p+r-2,yn-p+r-1=xn-2p+r-2yn-2p+r-2zn-2p+r-2xn-q-p+r-2yn-q-p+r-2zn-q-p+r-2,zn-p+r-1=xn-q-p+r-2yn-q-p+r-2zn-q-p+r-2xn-p-2yn-p-2zn-p-2,xn-p+r=1xn-2p+r-1yn-2p+r-1zn-2p+r-1,yn-p+r=xn-2p+r-1yn-2p+r-1zn-2p+r-1xn-q-p+r-1yn-q-p+r-1zn-q-p+r-1,zn-p+r=xn-q-p+r-1yn-q-p+r-1zn-q-p+r-1xn-p-1yn-p-1zn-p-1,xn-p+r+1=1xn-2p+ryn-2p+rzn-2p+r,yn-p+r+1=xn-2p+ryn-2p+rzn-2p+rxn-q-p+ryn-q-p+rzn-q-p+r,zn-p+r+1=xn-q-p+ryn-q-p+rzn-q-p+rxn-pyn-pzn-p,xn-p+r+2=1xn-2p+r+1yn-2p+r+1zn-2p+r+1,yn-p+r+2=xn-p+r+1yn-p+r+1zn-p+r+1xn-q-p+r+1yn-q-p+r+1zn-q-p+r+1,zn-p+r+2=xn-q-p+r+1yn-q-p+r+1zn-q-p+r+1xn-p+1yn-p+1zn-p+1,xn-p+r+3=1xn-2p+r+2yn-2p+r+2zn-2p+r+2,yn-p+r+3=xn-p+r+2yn-p+r+2zn-p+r+2xn-q-p+r+2yn-q-p+r+2zn-q-p+r+2,zn-p+r+3=xn-q-p+r+2yn-q-p+r+2zn-q-p+r+2xn-p+2yn-p+2zn-p+2,⋮xn-p+2r=1xn-2p+2r-1yn-2p+2r-1zn-2p+2r-1,yn-p+2r=xn-2p+2r-1yn-2p+2r-1zn-2p+2r-1xn-q-p+2r-1yn-q-p+2r-1zn-q-p+2r-1,zn-p+2r=xn-q-p+2r-1yn-q-p+2r-1zn-q-p+2r-1xn-p+r-1yn-p+r-1zn-p+r-1=xn-q-p+2ryn-q-p+2rzn-q-p+2rxn-p-2yn-p-2zn-p-2,xn-p+2r+1=1xn-2p+2ryn-2p+2rzn-2p+2r,yn-p+2r+1=xn-2p+2ryn-2p+2rzn-2p+2rxn-q-p+2ryn-q-p+2rzn-q-p+2r,zn-p+2r+1=xn-q-p+2ryn-q-p+2rzn-q-p+2rxn-p+ryn-p+rzn-p+r=xn-q-p+2ryn-q-p+2rzn-q-p+2rxn-p-1yn-p-1zn-p-1,xn-p+2r+2=1xn-2p+2r+1yn-2p+2r+1zn-2p+2r+1,yn-p+2r+2=xn-p+2r+1yn-p+2r+1zn-p+2r+1xn-q-p+2r+1yn-q-p+2r+1zn-q-p+2r+1,zn-p+2r+2=xn-q-p+2r+1yn-q-p+2r+1zn-q-p+2r+1xn-p+r+1yn-p+r+1zn-p+r+1=xn-q-p+2r+1yn-q-p+2r+1zn-q-p+2r+1xn-pyn-pzn-p,xn-p+2r+3=1xn-2p+2r+2yn-2p+2r+2zn-2p+2r+2,yn-p+2r+3=xn-p+2r+2yn-p+2r+2zn-p+2r+2xn-q-p+2r+2yn-q-p+2r+2zn-q-p+2r+2,zn-p+2r+3=xn-q-p+2r+2yn-q-p+2r+2zn-q-p+2r+2xn-p+r+2yn-p+r+2zn-p+r+2=xn-q-p+2r+2yn-q-p+2r+2zn-q-p+2r+2xn-p+1yn-p+1zn-p+1,⋮xn-q+r-1=1xn-p-q+r-2yn-p-q+r-2zn-p-q+r-2,yn-q+r-1=xn-p-q+r-2yn-p-q+r-2zn-p-q+r-2xn-2q+r-2yn-2q+r-2zn-q-p+r-2,zn-q+r-1=xn-2q+r-2yn-2q+r-2zn-q-p+r-2xn-q-2yn-q-2zn-q-2,xn-q+r=1xn-p-q+r-1yn-p-q+r-1zn-p-q+r-1,yn-q+r=xn-p-q+r-1yn-p-q+r-1zn-p-q+r-1xn-2q+r-1yn-2q+r-1zn-2q+r-1,zn-q+r=xn-2q+r-1yn-2q+r-1zn-2q+r-1xn-q-1yn-q-1zn-q-1,xn-q+r+1=1xn-p-q+ryn-p-q+rzn-p-q+r,yn-q+r+1=xn-p-q+ryn-p-q+rzn-p-q+rxn-2q+ryn-2q+rzn-2q+r,zn-q+r+1=xn-2q+ryn-2q+rzn-2q+rxn-qyn-qzn-q,xn-q+r+2=1xn-p-q+r+1yn-p-q+r+1zn-p-q+r+1,yn-q+r+2=xn-p-q+r+1yn-p-q+r+1zn-p-q+r+1xn-2q+r+1yn-2q+r+1zn-2q+r+1,zn-q+r+2=xn-2q+r+1yn-2q+r+1zn-2q+r+1xn-q+1yn-q+1zn-q+1,xn-q+r+3=1xn-p-q+r+2yn-p-q+r+2zn-p-q+r+2,yn-q+r+3=xn-p-q+r+2yn-p-q+r+2zn-p-q+r+2xn-2q+r+2yn-2q+r+2zn-2q+r+2,zn-q+r+3=xn-2q+r+2yn-2q+r+2zn-2q+r+2xn-q+2yn-q+2zn-q+2,⋮xn-q+2r=1xn-p-q+2r-1yn-p-q+2r-1zn-p-q+2r-1,yn-q+2r=xn-p-q+2r-1yn-p-q+2r-1zn-p-q+2r-1xn-2q+2r-1yn-2q+2r-1zn-2q+2r-1,zn-q+2r=xn-2q+2r-1yn-2q+2r-1zn-2q+2r-1xn-q+r-1yn-q+r-1zn-q+r-1=xn-2q+2r-1yn-2q+2r-1zn-2q+2r-1xn-q-2yn-q-2zn-q-2,xn-q+2r+1=1xn-p-q+2ryn-p-q+2rzn-p-q+2r,yn-q+2r+1=xn-p-q+2ryn-p-q+2rzn-p-q+2rxn-2q+2ryn-2q+2rzn-2q+2r,zn-q+2r+1=xn-2q+2ryn-2q+2rzn-2q+2rxn-q+ryn-q+rzn-q+r,=xn-2q+2ryn-2q+2rzn-2q+2rxn-q-1yn-q-1zn-q-1,xn-q+2r+2=1xn-p-q+2r+1yn-p-q+2r+1zn-p-q+2r+1,yn-q+2r+2=xn-p-q+2r+1yn-p-q+2r+1zn-p-q+2r+1xn-2q+2r+1yn-2q+2r+1zn-2q+2r+1,zn-q+2r+2=xn-2q+2r+1yn-2q+2r+1zn-2q+2r+1xn-qyn-qzn-q,xn-q+2r+3=1xn-p-q+2r+2yn-p-q+2r+2zn-p-q+2r+2,yn-q+2r+3=xn-p-q+2r+2yn-p-q+2r+2zn-p-q+2r+2xn-2q+2r+2yn-2q+2r+2zn-2q+2r+2,zn-q+2r+3=xn-2q+2r+2yn-2q+2r+2zn-2q+2r+2xn-p+1yn-p+1zn-p+1,⋮xn+r=xn-p-2yn-p-2zn-p-2,yn+r=xn-q-2yn-q-2zn-q-2xn-p-2yn-p-2zn-p-2,zn+r=1xn-q-2yn-q-2zn-q-2xn-1yn-1zn-1,xn+r+1=xn-p-1yn-p-1zn-p-1,yn+r+1=xn-q-1yn-q-1zn-q-1xn-p-1yn-p-1zn-p-1,zn+r+1=1xn-q-1yn-q-1zn-q-1xnynzn,xn+r+2=xn-pyn-pzn-p,yn+r+2=xn-q+1yn-q+1zn-q+1xn-pyn-pzn-p,zn+r+2=xn-ryn-rzn-rxn-q+1yn-q+1zn-q+1,xn+r+3=xn-p+1yn-p+1zn-p+1,yn+r+3=xn-q+2yn-q+2zn-q+2xn-p+1yn-p+1zn-p+1,zn+r+3=xn-r+1yn-r+1zn-r+1xn-q+2yn-q+2zn-q+2,xn+r+4=xn-p+2yn-p+2zn-p+2,yn+r+4=xn-q+3yn-q+3zn-q+3xn-p+2yn-p+2zn-p+2,zn+r+4=xn-r+2yn-r+2zn-r+2xn-q+3yn-q+3zn-q+3,⋮xn+2r+1=1xn-p-2yn-p-2zn-p-2,yn+2r+1=xn-p-2yn-p-2zn-p-2xn-q-2yn-q-2zn-q-2,zn+2r+1=xn-q-2yn-q-2zn-q-2xn-1yn-1zn-1,xn+2r+2=1xn-p-1yn-p-1zn-p-1,yn+2r+2=xn-p-1yn-p-1zn-p-1xn-q-1yn-q-1zn-q-1,zn+2r+2=xn-q-1yn-q-1zn-q-1xnynzn,xn+2r+3=xn+1,yn+2r+3=yn+1,zn+2r+3=yn+1.
Hence, the proof is completed.Example 2.4.
Consider the difference system (2.3) with p=1, q=0, r=2, and the initial conditions x-2=0.3, x-1=5, x0=-0.7,y-2=8, y-1=0.1, y0=2, z-2=3, z-1=0.8, and z0=6(See Figure 2).Figure 2Proposition 2.5.
It is easy to see by induction that the following general system is periodic with period(2pk+2):
(2.5)xn+11=1∏i=1kxn-p1i,xn+12=∏i=1kxn-p1i∏i=1kxn-p2i,xn+13=∏i=1kxn-p2i∏i=1kxn-p3i,…,xn+1j=∏i=1kxn-pj-1i∏i=1kxn-pji,…,xn+1k=∏i=1kxn-pk-1i∏i=1kxn-pki.


## 2.1. The First System

In this section, we deal with the solutions of the system of the difference equations(2.1)xn+1=1xn-pyn-p,yn+1=xn-pyn-pxn-qyn-q,
with a nonzero real numbers initial conditions and p≠q.Theorem 2.1.
Suppose that{xn,yn} are solutions of system (2.1). Also, assume that the initial conditionsare arbitrary nonzero real numbers. Then all solutions of equation system (2.1) are eventually periodic with period (2q+2).Proof.
From (2.1), we see that
(2.2)xn+1=1xn-pyn-p,yn+1=xn-pyn-pxn-qyn-q,xn+2=1xn-p+1yn-p+1,yn+2=xn-p+1yn-p+1xn-q+1yn-q+1,xn+3=1xn-p+2yn-p+2,yn+3=xn-p+2yn-p+2xn-q+2yn-q+2,⋮xn-p+q-2=1xn-2p+q-3yn-2p+q-3,yn-p+q-2=xn-2p+q-3yn-2p+q-3xn-p-3yn-p-3,xn-p+q-1=1xn-2p+q-2yn-2p+q-2,yn-p+q-1=xn-2p+q-2yn-2p+q-2xn-p-2yn-p-2,xn-p+q=1xn-2p+q-1yn-2p+q-1,yn-p+q=xn-2p+q-1yn-2p+q-1xn-p-1yn-p-1,xn-p+q+1=1xn-2p+qyn-2p+q,yn-p+q+1=xn-2p+qyn-2p+qxn-pyn-p,xn-p+q+2=1xn-2p+q+1yn-2p+q+1,yn-p+q+2=xn-2p+q+1yn-2p+q+1xn-p+1yn-p+1,⋮xn-p+2q-1=1xn-2p+2q-2yn-2p+2q-2,yn-p+2q-1=xn-2p+2q-2yn-2p+2q-2xn-p-3yn-p-3,xn-p+2q=1xn-2p+2q-1yn-2p+2q-1,yn-p+2q=xn-2p+2q-1yn-2p+2q-1xn-p-2yn-p-2,xn-p+2q+1=1xn-2p+qyn-2p+q,yn-p+2q+1=xn-2p+2qyn-2p+2qxn-p-1yn-p-1,xn-p+2q+2=1xn-2p+q+1yn-2p+q+1,yn-p+2q+2=xn-2p+2q+1yn-2p+2q+1xn-pyn-p,⋮xn+q-1=1xn-p+q-2yn-p+q-2,yn+q-1=xn-p+q-2yn-p+q-2xn-2yn-2,xn+q=1xn-p+q-1yn-p+q-1,yn+q=xn-p+q-1yn-p+q-1xn-1yn-1,xn+q+1=1xn-p+qyn-p+q,yn+q+1=xn-p+qyn-p+qxnyn,xn+q+2=1xn-p+q+1yn-p+q+1,yn+q+2=xn-p+q+1yn-p+q+1xn+1yn+1=xn-p+q+1yn-p+q+1xn-qyn-q,xn+q+3=1xn-p+q+2yn-p+q+2,yn+q+3=xn-p+q+2yn-p+q+2xn+2yn+2=xn-p+q+2yn-p+q+2xn-q+1yn-q+1,xn+q+4=1xn-p+q+3yn-p+q+3,yn+q+4=xn-p+q+3yn-p+q+3xn+3yn+3=xn-p+q+3yn-p+q+3xn-q+2yn-q+2,⋮xn+2q=1xn-p-3yn-p-3,yn+2q=xn-p-3yn-p-3xn-2yn-2,xn+2q+1=1xn-p-2yn-p-2,yn+2q+1=xn-p-2yn-p-2xn-1yn-1,xn+2q+2=1xn-p-1yn-p-1,yn+2q+2=xn-p-1yn-p-1xnyn,xn+2q+3=1xn-p+2q+2yn-p+2q+2=xn+1,yn+2q+3=xn-p+2q+2yn-p+2q+2xn+q+2yn+q+2=yn+1,xn+2q+4=1xn-p+2q+3yn-p+2q+3=xn+2,yn+2q+4=xn-p+2q+3yn-p+2q+3xn+q+3yn+q+3=yn+2.
Hence, the proof is completed.Numerical Example
In order to illustrate the results of this section and to support our theoretical discussions, we consider the following numerical example.Example 2.2.
Consider the difference system (2.1) with p=1, q=2, and the initial conditions x-2=0.9, x-1=0.7, x0=2, y-2=-0.8, y-1=4.2, and y0=6. (See Figure 1).Figure 1


## 2.2. The Second System

In this section, we deal with the solutions of the system of the difference equations(2.3)xn+1=1xn-pyn-pzn-p,yn+1=xn-pyn-pzn-pxn-qyn-qzn-q,zn+1=xn-qyn-qzn-qxn-ryn-rzn-r,
with a nonzero real numbers initial conditions and p≠q,q≠r.Theorem 2.3.
Suppose that{xn,yn,zn} are solutions of system (2.3). Also, assume that the initial conditionsare arbitrary nonzero real numbers. Then all solutions of equation system (2.3) are eventually periodic with period (2r+2).Proof.
From (2.3), we see that
(2.4)xn+1=1xn-pyn-pzn-p,yn+1=xn-pyn-pzn-pxn-qyn-qzn-q,zn+1=xn-qyn-qzn-qxn-ryn-rzn-r,xn+2=1xn-p+1yn-p+1zn-p+1,yn+2=xn-p+1yn-p+1zn-p+1xn-q+1yn-q+1zn-q+1,zn+2=xn-q+1yn-q+1zn-q+1xn-r+1yn-r+1zn-r+1,xn+3=1xn-p+2yn-p+2zn-p+2,yn+3=xn-p+2yn-p+2zn-p+2xn-q+2yn-q+2zn-q+2,zn+3=xn-q+2yn-q+2zn-q+2xn-r+2yn-r+2zn-r+2,⋮xn-p+r-1=1xn-2p+r-2yn-2p+r-2zn-2p+r-2,yn-p+r-1=xn-2p+r-2yn-2p+r-2zn-2p+r-2xn-q-p+r-2yn-q-p+r-2zn-q-p+r-2,zn-p+r-1=xn-q-p+r-2yn-q-p+r-2zn-q-p+r-2xn-p-2yn-p-2zn-p-2,xn-p+r=1xn-2p+r-1yn-2p+r-1zn-2p+r-1,yn-p+r=xn-2p+r-1yn-2p+r-1zn-2p+r-1xn-q-p+r-1yn-q-p+r-1zn-q-p+r-1,zn-p+r=xn-q-p+r-1yn-q-p+r-1zn-q-p+r-1xn-p-1yn-p-1zn-p-1,xn-p+r+1=1xn-2p+ryn-2p+rzn-2p+r,yn-p+r+1=xn-2p+ryn-2p+rzn-2p+rxn-q-p+ryn-q-p+rzn-q-p+r,zn-p+r+1=xn-q-p+ryn-q-p+rzn-q-p+rxn-pyn-pzn-p,xn-p+r+2=1xn-2p+r+1yn-2p+r+1zn-2p+r+1,yn-p+r+2=xn-p+r+1yn-p+r+1zn-p+r+1xn-q-p+r+1yn-q-p+r+1zn-q-p+r+1,zn-p+r+2=xn-q-p+r+1yn-q-p+r+1zn-q-p+r+1xn-p+1yn-p+1zn-p+1,xn-p+r+3=1xn-2p+r+2yn-2p+r+2zn-2p+r+2,yn-p+r+3=xn-p+r+2yn-p+r+2zn-p+r+2xn-q-p+r+2yn-q-p+r+2zn-q-p+r+2,zn-p+r+3=xn-q-p+r+2yn-q-p+r+2zn-q-p+r+2xn-p+2yn-p+2zn-p+2,⋮xn-p+2r=1xn-2p+2r-1yn-2p+2r-1zn-2p+2r-1,yn-p+2r=xn-2p+2r-1yn-2p+2r-1zn-2p+2r-1xn-q-p+2r-1yn-q-p+2r-1zn-q-p+2r-1,zn-p+2r=xn-q-p+2r-1yn-q-p+2r-1zn-q-p+2r-1xn-p+r-1yn-p+r-1zn-p+r-1=xn-q-p+2ryn-q-p+2rzn-q-p+2rxn-p-2yn-p-2zn-p-2,xn-p+2r+1=1xn-2p+2ryn-2p+2rzn-2p+2r,yn-p+2r+1=xn-2p+2ryn-2p+2rzn-2p+2rxn-q-p+2ryn-q-p+2rzn-q-p+2r,zn-p+2r+1=xn-q-p+2ryn-q-p+2rzn-q-p+2rxn-p+ryn-p+rzn-p+r=xn-q-p+2ryn-q-p+2rzn-q-p+2rxn-p-1yn-p-1zn-p-1,xn-p+2r+2=1xn-2p+2r+1yn-2p+2r+1zn-2p+2r+1,yn-p+2r+2=xn-p+2r+1yn-p+2r+1zn-p+2r+1xn-q-p+2r+1yn-q-p+2r+1zn-q-p+2r+1,zn-p+2r+2=xn-q-p+2r+1yn-q-p+2r+1zn-q-p+2r+1xn-p+r+1yn-p+r+1zn-p+r+1=xn-q-p+2r+1yn-q-p+2r+1zn-q-p+2r+1xn-pyn-pzn-p,xn-p+2r+3=1xn-2p+2r+2yn-2p+2r+2zn-2p+2r+2,yn-p+2r+3=xn-p+2r+2yn-p+2r+2zn-p+2r+2xn-q-p+2r+2yn-q-p+2r+2zn-q-p+2r+2,zn-p+2r+3=xn-q-p+2r+2yn-q-p+2r+2zn-q-p+2r+2xn-p+r+2yn-p+r+2zn-p+r+2=xn-q-p+2r+2yn-q-p+2r+2zn-q-p+2r+2xn-p+1yn-p+1zn-p+1,⋮xn-q+r-1=1xn-p-q+r-2yn-p-q+r-2zn-p-q+r-2,yn-q+r-1=xn-p-q+r-2yn-p-q+r-2zn-p-q+r-2xn-2q+r-2yn-2q+r-2zn-q-p+r-2,zn-q+r-1=xn-2q+r-2yn-2q+r-2zn-q-p+r-2xn-q-2yn-q-2zn-q-2,xn-q+r=1xn-p-q+r-1yn-p-q+r-1zn-p-q+r-1,yn-q+r=xn-p-q+r-1yn-p-q+r-1zn-p-q+r-1xn-2q+r-1yn-2q+r-1zn-2q+r-1,zn-q+r=xn-2q+r-1yn-2q+r-1zn-2q+r-1xn-q-1yn-q-1zn-q-1,xn-q+r+1=1xn-p-q+ryn-p-q+rzn-p-q+r,yn-q+r+1=xn-p-q+ryn-p-q+rzn-p-q+rxn-2q+ryn-2q+rzn-2q+r,zn-q+r+1=xn-2q+ryn-2q+rzn-2q+rxn-qyn-qzn-q,xn-q+r+2=1xn-p-q+r+1yn-p-q+r+1zn-p-q+r+1,yn-q+r+2=xn-p-q+r+1yn-p-q+r+1zn-p-q+r+1xn-2q+r+1yn-2q+r+1zn-2q+r+1,zn-q+r+2=xn-2q+r+1yn-2q+r+1zn-2q+r+1xn-q+1yn-q+1zn-q+1,xn-q+r+3=1xn-p-q+r+2yn-p-q+r+2zn-p-q+r+2,yn-q+r+3=xn-p-q+r+2yn-p-q+r+2zn-p-q+r+2xn-2q+r+2yn-2q+r+2zn-2q+r+2,zn-q+r+3=xn-2q+r+2yn-2q+r+2zn-2q+r+2xn-q+2yn-q+2zn-q+2,⋮xn-q+2r=1xn-p-q+2r-1yn-p-q+2r-1zn-p-q+2r-1,yn-q+2r=xn-p-q+2r-1yn-p-q+2r-1zn-p-q+2r-1xn-2q+2r-1yn-2q+2r-1zn-2q+2r-1,zn-q+2r=xn-2q+2r-1yn-2q+2r-1zn-2q+2r-1xn-q+r-1yn-q+r-1zn-q+r-1=xn-2q+2r-1yn-2q+2r-1zn-2q+2r-1xn-q-2yn-q-2zn-q-2,xn-q+2r+1=1xn-p-q+2ryn-p-q+2rzn-p-q+2r,yn-q+2r+1=xn-p-q+2ryn-p-q+2rzn-p-q+2rxn-2q+2ryn-2q+2rzn-2q+2r,zn-q+2r+1=xn-2q+2ryn-2q+2rzn-2q+2rxn-q+ryn-q+rzn-q+r,=xn-2q+2ryn-2q+2rzn-2q+2rxn-q-1yn-q-1zn-q-1,xn-q+2r+2=1xn-p-q+2r+1yn-p-q+2r+1zn-p-q+2r+1,yn-q+2r+2=xn-p-q+2r+1yn-p-q+2r+1zn-p-q+2r+1xn-2q+2r+1yn-2q+2r+1zn-2q+2r+1,zn-q+2r+2=xn-2q+2r+1yn-2q+2r+1zn-2q+2r+1xn-qyn-qzn-q,xn-q+2r+3=1xn-p-q+2r+2yn-p-q+2r+2zn-p-q+2r+2,yn-q+2r+3=xn-p-q+2r+2yn-p-q+2r+2zn-p-q+2r+2xn-2q+2r+2yn-2q+2r+2zn-2q+2r+2,zn-q+2r+3=xn-2q+2r+2yn-2q+2r+2zn-2q+2r+2xn-p+1yn-p+1zn-p+1,⋮xn+r=xn-p-2yn-p-2zn-p-2,yn+r=xn-q-2yn-q-2zn-q-2xn-p-2yn-p-2zn-p-2,zn+r=1xn-q-2yn-q-2zn-q-2xn-1yn-1zn-1,xn+r+1=xn-p-1yn-p-1zn-p-1,yn+r+1=xn-q-1yn-q-1zn-q-1xn-p-1yn-p-1zn-p-1,zn+r+1=1xn-q-1yn-q-1zn-q-1xnynzn,xn+r+2=xn-pyn-pzn-p,yn+r+2=xn-q+1yn-q+1zn-q+1xn-pyn-pzn-p,zn+r+2=xn-ryn-rzn-rxn-q+1yn-q+1zn-q+1,xn+r+3=xn-p+1yn-p+1zn-p+1,yn+r+3=xn-q+2yn-q+2zn-q+2xn-p+1yn-p+1zn-p+1,zn+r+3=xn-r+1yn-r+1zn-r+1xn-q+2yn-q+2zn-q+2,xn+r+4=xn-p+2yn-p+2zn-p+2,yn+r+4=xn-q+3yn-q+3zn-q+3xn-p+2yn-p+2zn-p+2,zn+r+4=xn-r+2yn-r+2zn-r+2xn-q+3yn-q+3zn-q+3,⋮xn+2r+1=1xn-p-2yn-p-2zn-p-2,yn+2r+1=xn-p-2yn-p-2zn-p-2xn-q-2yn-q-2zn-q-2,zn+2r+1=xn-q-2yn-q-2zn-q-2xn-1yn-1zn-1,xn+2r+2=1xn-p-1yn-p-1zn-p-1,yn+2r+2=xn-p-1yn-p-1zn-p-1xn-q-1yn-q-1zn-q-1,zn+2r+2=xn-q-1yn-q-1zn-q-1xnynzn,xn+2r+3=xn+1,yn+2r+3=yn+1,zn+2r+3=yn+1.
Hence, the proof is completed.Example 2.4.
Consider the difference system (2.3) with p=1, q=0, r=2, and the initial conditions x-2=0.3, x-1=5, x0=-0.7,y-2=8, y-1=0.1, y0=2, z-2=3, z-1=0.8, and z0=6(See Figure 2).Figure 2Proposition 2.5.
It is easy to see by induction that the following general system is periodic with period(2pk+2):
(2.5)xn+11=1∏i=1kxn-p1i,xn+12=∏i=1kxn-p1i∏i=1kxn-p2i,xn+13=∏i=1kxn-p2i∏i=1kxn-p3i,…,xn+1j=∏i=1kxn-pj-1i∏i=1kxn-pji,…,xn+1k=∏i=1kxn-pk-1i∏i=1kxn-pki.



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*Source: 892571-2012-04-17.xml*