Skip to main content

On the solutions of some nonlinear systems of difference equations

Abstract

In this paper, we deal with the existence of solutions and the periodicity character of the following systems of rational difference equations with order three:

x n + 1 = x n y n 2 y n 1 ( ± 1 ± x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( ± 1 ± y n x n 2 ) ,

where the initial conditions x 2 , x 1 , x 0 , y 2 , y 1 and y 0 are nonzero real numbers.

MSC:39A10.

1 Introduction

Recently, rational difference equations have attracted the attention of many researchers for various reasons. On the one hand, they provide examples of nonlinear equations which are, in some cases, treatable but their dynamics present some new features with respect to the linear case. On the other hand, rational equations frequently appear in some biological models. Hence, their study is of interest also due to their applications. A good example of both facts is Ricatti difference equations because the richness of the dynamics of Ricatti equations is very well known (see, e.g., [1, 2]), and a particular case of these equations provides the classical Beverton-Holt model on the dynamics of exploited fish populations [3]. Obviously, higher-order rational difference equations and systems of rational equations have also been widely studied but still have many aspects to be investigated. The reader can find in the following books [46], and works cited therein, many results, applications, and open problems on higher-order equations and rational systems.

A preliminary study of planar rational systems can be found in the paper [7] by Camouzis et al. In the work, they give some results and provide some open questions for systems of equations of the following type:

x n + 1 = α 1 + β 1 x n + γ 1 y n A 1 + B 1 x n + C 1 y n , y n + 1 = α 2 + β 2 x n + γ 2 y n A 2 + B 2 x n + C 2 y n ,n=0,1,2,,

where the parameters are taken to be nonnegative. As shown in the cited paper, some of these systems can be reduced to some Ricatti equations or to some previously studied second-order rational equations. Furthermore, for some choices of the parameters, one obtains a system which is equivalent to the case with some other parameters. Camouzis et al. arrived at a list of 325 non-equivalent systems that should be focused on. They list such systems as pairs k, l, where k and l make reference to the number of the corresponding equation in their Tables 3 and 4. There are many papers that are related to the difference equations systems. For example, the periodicity of the positive solutions of the rational difference equations systems

x n + 1 = m y n , y n + 1 = p y n x n 1 y n 1

has been obtained by Cinar in [8].

In [9] Clark and Kulenovic investigated the global asymptotic stability of the system

x n + 1 = x n a + c y n , y n + 1 = y n b + d x n .

The behavior of the positive solutions of the following system:

x n + 1 = x n 1 1 + x n 1 y n , y n + 1 = y n 1 1 + y n 1 x n

has been studied by Kurbanli et al. [10].

Touafek et al. [11] investigated the periodic nature and got the form of the solutions of the following systems of rational difference equations:

x n + 1 = x n 3 ± 1 ± x n 3 y n 1 , y n + 1 = y n 3 ± 1 ± y n 3 x n 1 .

In [12] Yalçınkaya investigated the sufficient conditions for the global asymptotic stability of the following system of difference equations:

x n + 1 = x n + y n 1 x n y n 1 1 , y n + 1 = y n + x n 1 y n x n 1 1 .

Similarly, difference equations and nonlinear systems of the rational difference equations were investigated, see [139].

In this paper, we investigate the periodic nature and the form of the solutions of some nonlinear difference equations systems of order three

x n + 1 = x n y n 2 y n 1 ( ± 1 ± x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( ± 1 ± y n x n 2 ) ,

where the initial conditions x 2 , x 1 , x 0 , y 2 , y 1 and y 0 are nonzero real numbers.

Definition (Periodicity)

A sequence { x n } n = k is said to be periodic with period p if x n + p = x n for all nk.

2 On the solution of the system x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 + y n x n 2 )

In this section, we investigate the solutions of the two difference equations system

x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 + y n x n 2 ) ,
(1)

where n N 0 and the initial conditions are arbitrary nonzero real numbers.

Theorem 1 Assume that { x n , y n } are solutions of system (1). Then for n=0,1,2, , we see that all solutions of system (1) are given by the following formulas:

x 4 n 2 = a n f n c n 1 d n i = 0 n 1 ( 1 + ( 4 i ) c d ) ( 1 + ( 4 i + 2 ) a f ) , x 4 n 1 = b a n f n c n d n i = 0 n 1 ( 1 + ( 4 i + 1 ) c d ) ( 1 + ( 4 i + 3 ) a f ) , x 4 n = a n + 1 f n c n d n i = 0 n 1 ( 1 + ( 4 i + 2 ) c d ) ( 1 + ( 4 i + 4 ) a f ) , x 4 n + 1 = a n + 1 f n + 1 e c n d n ( 1 + a f ) i = 0 n 1 ( 1 + ( 4 i + 3 ) c d ) ( 1 + ( 4 i + 5 ) a f )

and

y 4 n 2 = c n d n a n f n 1 i = 0 n 1 ( 1 + ( 4 i ) a f ) ( 1 + ( 4 i + 2 ) c d ) , y 4 n 1 = e c n d n a n f n i = 0 n 1 ( 1 + ( 4 i + 1 ) a f ) ( 1 + ( 4 i + 3 ) c d ) , y 4 n = c n d n + 1 a n f n i = 0 n 1 ( 1 + ( 4 i + 2 ) a f ) ( 1 + ( 4 i + 4 ) c d ) , y 4 n + 1 = c n + 1 d n + 1 b a n f n ( 1 + c d ) i = 0 n 1 ( 1 + ( 4 i + 3 ) a f ) ( 1 + ( 4 i + 5 ) c d ) ,

where x 2 =c, x 1 =b, x 0 =a, y 2 =f, y 1 =e and y 0 =d.

Proof For n=0 the result holds. Now suppose that n>0 and that our assumption holds for n1, that is,

x 4 n 6 = a n 1 f n 1 c n 2 d n 1 i = 0 n 2 ( 1 + ( 4 i ) c d ) ( 1 + ( 4 i + 2 ) a f ) , x 4 n 5 = b a n 1 f n 1 c n 1 d n 1 i = 0 n 2 ( 1 + ( 4 i + 1 ) c d ) ( 1 + ( 4 i + 3 ) a f ) , x 4 n 4 = a n f n 1 c n 1 d n 1 i = 0 n 2 ( 1 + ( 4 i + 2 ) c d ) ( 1 + ( 4 i + 4 ) a f ) , x 4 n 3 = a n f n e c n 1 d n 1 ( 1 + a f ) i = 0 n 2 ( 1 + ( 4 i + 3 ) c d ) ( 1 + ( 4 i + 5 ) a f ) , y 4 n 6 = c n 1 d n 1 a n 1 f n 2 i = 0 n 2 ( 1 + ( 4 i ) a f ) ( 1 + ( 4 i + 2 ) c d ) , y 4 n 5 = e c n 1 d n 1 a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 1 ) a f ) ( 1 + ( 4 i + 3 ) c d ) , y 4 n 4 = c n 1 d n a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 2 ) a f ) ( 1 + ( 4 i + 4 ) c d ) , y 4 n 3 = c n d n b a n 1 f n 1 ( 1 + c d ) i = 0 n 2 ( 1 + ( 4 i + 3 ) a f ) ( 1 + ( 4 i + 5 ) c d ) .

Now we find from Eq. (1) that

x 4 n 2 = x 4 n 3 y 4 n 5 y 4 n 4 ( 1 + x 4 n 3 y 4 n 5 ) x 4 n 2 = ( a n f n e c n 1 d n 1 ( 1 + a f ) i = 0 n 2 ( 1 + ( 4 i + 3 ) c d ) ( 1 + ( 4 i + 5 ) a f ) ) ( e c n 1 d n 1 a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 1 ) a f ) ( 1 + ( 4 i + 3 ) c d ) ) x 4 n 2 = / ( ( c n 1 d n a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 2 ) a f ) ( 1 + ( 4 i + 4 ) c d ) ) x 4 n 2 = × ( 1 + ( a n f n e c n 1 d n 1 ( 1 + a f ) i = 0 n 2 ( 1 + ( 4 i + 3 ) c d ) ( 1 + ( 4 i + 5 ) a f ) ) x 4 n 2 = × ( e c n 1 d n 1 a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 1 ) a f ) ( 1 + ( 4 i + 3 ) c d ) ) ) ) x 4 n 2 = ( a f ( 1 + a f ) i = 0 n 2 ( 1 + ( 4 i + 1 ) a f ) ( 1 + ( 4 i + 5 ) a f ) ) ( c n 1 d n a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 2 ) a f ) ( 1 + ( 4 i + 4 ) c d ) ) ( 1 + a f ( 1 + a f ) i = 0 n 2 ( 1 + ( 4 i + 1 ) a f ) ( 1 + ( 4 i + 5 ) a f ) ) x 4 n 2 = ( a f ( 1 + ( 4 n 3 ) a f ) ) ( c n 1 d n a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 2 ) a f ) ( 1 + ( 4 i + 4 ) c d ) ) ( 1 + a f ( 1 + ( 4 n 3 ) a f ) ) x 4 n 2 = a n f n c n 1 d n ( 1 + ( 4 n 2 ) a f ) i = 0 n 2 ( 1 + ( 4 i + 4 ) c d ) ( 1 + ( 4 i + 2 ) a f ) = a n f n c n 1 d n i = 0 n 1 ( 1 + ( 4 i ) c d ) ( 1 + ( 4 i + 2 ) a f ) , y 4 n 2 = y 4 n 3 x 4 n 5 x 4 n 4 ( 1 + y 4 n 3 x 4 n 5 ) y 4 n 2 = ( c n d n b a n 1 f n 1 ( 1 + c d ) i = 0 n 2 ( 1 + ( 4 i + 3 ) a f ) ( 1 + ( 4 i + 5 ) c d ) ) ( b a n 1 f n 1 c n 1 d n 1 i = 0 n 2 ( 1 + ( 4 i + 1 ) c d ) ( 1 + ( 4 i + 3 ) a f ) ) y 4 n 2 = / ( ( a n f n 1 c n 1 d n 1 i = 0 n 2 ( 1 + ( 4 i + 2 ) c d ) ( 1 + ( 4 i + 4 ) a f ) ) y 4 n 2 = × ( 1 + ( c n d n b a n 1 f n 1 ( 1 + c d ) i = 0 n 2 ( 1 + ( 4 i + 3 ) a f ) ( 1 + ( 4 i + 5 ) c d ) ) y 4 n 2 = × ( b a n 1 f n 1 c n 1 d n 1 i = 0 n 2 ( 1 + ( 4 i + 1 ) c d ) ( 1 + ( 4 i + 3 ) a f ) ) ) ) y 4 n 2 = c d ( 1 + c d ) i = 0 n 2 ( 1 + ( 4 i + 1 ) c d ) ( 1 + ( 4 i + 5 ) c d ) ( a n f n 1 c n 1 d n 1 i = 0 n 2 ( 1 + ( 4 i + 2 ) c d ) ( 1 + ( 4 i + 4 ) a f ) ) ( 1 + c d ( 1 + c d ) i = 0 n 2 ( 1 + ( 4 i + 1 ) c d ) ( 1 + ( 4 i + 5 ) c d ) ) y 4 n 2 = ( c d ( 1 + ( 4 n 3 ) c d ) ) ( a n f n 1 c n 1 d n 1 i = 0 n 2 ( 1 + ( 4 i + 2 ) c d ) ( 1 + ( 4 i + 4 ) a f ) ) ( 1 + c d ( 1 + ( 4 n 3 ) c d ) ) y 4 n 2 = c n d n a n f n 1 ( 1 + ( 4 n 2 ) c d ) i = 0 n 2 ( 1 + ( 4 i + 4 ) a f ) ( 1 + ( 4 i + 2 ) c d ) = c n d n a n f n 1 i = 0 n 1 ( 1 + ( 4 i ) a f ) ( 1 + ( 4 i + 2 ) c d ) .

Also, we can prove the other relations. The proof is complete. □

The following theorem can be proved similarly.

Theorem 2 Assume that { x n , y n } are solutions of the following system:

x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 y n x n 2 ) .

Then for n=0,1,2, ,

x 4 n 2 = a n f n c n 1 d n i = 0 n 1 ( 1 ( 4 i ) c d ) ( 1 + ( 4 i + 2 ) a f ) , x 4 n 1 = b a n f n c n d n i = 0 n 1 ( 1 ( 4 i + 1 ) c d ) ( 1 + ( 4 i + 3 ) a f ) , x 4 n = a n + 1 f n c n d n i = 0 n 1 ( 1 ( 4 i + 2 ) c d ) ( 1 + ( 4 i + 4 ) a f ) , x 4 n + 1 = a n + 1 f n + 1 e c n d n ( 1 + a f ) i = 0 n 1 ( 1 ( 4 i + 3 ) c d ) ( 1 + ( 4 i + 5 ) a f ) , y 4 n 2 = c n d n a n f n 1 i = 0 n 1 ( 1 + ( 4 i ) a f ) ( 1 ( 4 i + 2 ) c d ) , y 4 n 1 = e c n d n a n f n i = 0 n 1 ( 1 + ( 4 i + 1 ) a f ) ( 1 ( 4 i + 3 ) c d ) , y 4 n = c n d n + 1 a n f n i = 0 n 1 ( 1 + ( 4 i + 2 ) a f ) ( 1 ( 4 i + 4 ) c d ) , y 4 n + 1 = c n + 1 d n + 1 b a n f n ( 1 c d ) i = 0 n 1 ( 1 + ( 4 i + 3 ) a f ) ( 1 ( 4 i + 5 ) c d ) .

Example 1 For confirming the results of this section, we consider numerical example for the difference system (1) with the initial conditions x 2 =3, x 1 =5, x 0 =1, y 2 =0.4, y 1 =0.16 and y 0 =7.See Figure 1.

Figure 1
figure1

This figure shows the solutions of the system x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 + y n x n 2 ) , with the initial conditions x 2 =3 , x 1 =5 , x 0 =1 , y 2 =0.4 , y 1 =0.16 and y 0 =7 .

Example 2 We assumethat the initial conditions for the difference system (1) are x 2 =0.2, x 1 =0.15, x 0 =0.51, y 2 =0.23, y 1 =0.16 and y 0 =0.7. See Figure 2.

Figure 2
figure2

This figure shows the dynamics of solutions of the difference system ( 1 ) when we put the initial conditions x 2 =0.2 , x 1 =0.15 , x 0 =0.51 , y 2 =0.23 , y 1 =0.16 and y 0 =0.7 .

3 On the solution of the system x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 + y n x n 2 )

In this section, we obtain the form of the solutions of the two difference equations system

x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 + y n x n 2 ) ,
(2)

where n N 0 and the initial conditions are arbitrary nonzero real numbers with x 2 y 0 1.

Theorem 3 Let { x n , y n } n = 2 + be solutions of system (2). Then { x n } n = 2 + and { y n } n = 2 + are given by the formula for n=0,1,2, ,

x 4 n 2 = a n f n c n 1 d n i = 0 n 1 1 ( 1 + ( 4 i + 2 ) a f ) , x 4 n 1 = b a n f n c n d n ( 1 + c d ) n i = 0 n 1 ( 1 + ( 4 i + 3 ) a f ) , x 4 n = a n + 1 f n c n d n i = 0 n 1 1 ( 1 + ( 4 i + 4 ) a f ) , x 4 n + 1 = a n + 1 f n + 1 e c n d n ( 1 + a f ) ( 1 + c d ) n i = 0 n 1 ( 1 + ( 4 i + 5 ) a f ) , y 4 n 2 = c n d n a n f n 1 i = 0 n 1 ( 1 + ( 4 i ) a f ) , y 4 n 1 = e c n d n a n f n i = 0 n 1 ( 1 + ( 4 i + 1 ) a f ) ( 1 + c d ) n , y 4 n = c n d n + 1 a n f n i = 0 n 1 ( 1 + ( 4 i + 2 ) a f ) , y 4 n + 1 = c n + 1 d n + 1 b a n f n i = 0 n 1 ( 1 + ( 4 i + 3 ) a f ) ( 1 + c d ) n + 1 .

Proof For n=0 the result holds. Now suppose that n>0 and that our assumption holds for n1, that is,

x 4 n 5 = b a n 1 f n 1 c n 1 d n 1 ( 1 + c d ) n 1 i = 0 n 2 ( 1 + ( 4 i + 3 ) a f ) , x 4 n 4 = a n f n 1 c n 1 d n 1 i = 0 n 2 1 ( 1 + ( 4 i + 4 ) a f ) , x 4 n 3 = a n f n e c n 1 d n 1 ( 1 + a f ) ( 1 + c d ) n 1 i = 0 n 2 ( 1 + ( 4 i + 5 ) a f ) , y 4 n 5 = e c n 1 d n 1 a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 1 ) a f ) ( 1 + c d ) n 1 , y 4 n 4 = c n 1 d n a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 2 ) a f ) , y 4 n 3 = c n d n b a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 3 ) a f ) ( 1 + c d ) n .

Now, we obtain from Eq. (2) that

x 4 n 2 = x 4 n 3 y 4 n 5 y 4 n 4 ( 1 + x 4 n 3 y 4 n 5 ) = ( a n f n e c n 1 d n 1 ( 1 + a f ) ( 1 + c d ) n 1 i = 0 n 2 ( 1 + ( 4 i + 5 ) a f ) ) ( e c n 1 d n 1 a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 1 ) a f ) ( 1 + c d ) n 1 ) / ( ( c n 1 d n a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 2 ) a f ) ) × ( 1 + ( a n f n e c n 1 d n 1 ( 1 + a f ) ( 1 + c d ) n 1 i = 0 n 2 ( 1 + ( 4 i + 5 ) a f ) ) × ( e c n 1 d n 1 a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 1 ) a f ) ( 1 + c d ) n 1 ) ) ) = ( a f ( 1 + ( 4 n 3 ) a f ) ) ( c n 1 d n a n 1 f n 1 i = 0 n 2 ( 1 + ( 4 i + 2 ) a f ) ) ( 1 + a f ( 1 + ( 4 n 3 ) a f ) ) = a n 1 f n 1 a f ( c n 1 d n i = 0 n 2 ( 1 + ( 4 i + 2 ) a f ) ) ( 1 + ( 4 n 3 ) a f + a f ) = a n f n ( c n 1 d n i = 0 n 2 ( 1 + ( 4 i + 2 ) a f ) ) ( 1 + ( 4 n 2 ) a f ) = a n f n c n 1 d n i = 0 n 1 ( 1 + ( 4 i + 2 ) a f ) .

Also, we can prove the other relations. This completes the proof. □

We consider the following systems, and the proof of the theorems is similar to the above theorem, and so it is left to the reader,

x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 y n x n 2 ) ,
(3)
x n + 1 = x n y n 2 y n 1 ( 1 x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 + y n x n 2 ) ,
(4)
x n + 1 = x n y n 2 y n 1 ( 1 x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 y n x n 2 ) .
(5)

The following theorems are devoted to the expressions of the form of the solutions of systems (3), (4), (5) with x 2 =c, x 1 =b, x 0 =a, y 2 =f, y 1 =e and y 0 =d.

Theorem 4 Let { x n , y n } n = 2 + be solutions of system (3) and x 2 y 0 1. Then for n=0,1,2, ,

x 4 n 2 = a n f n c n 1 d n i = 0 n 1 1 ( 1 + ( 4 i + 2 ) a f ) , x 4 n 1 = b a n f n c n d n ( 1 c d ) n i = 0 n 1 ( 1 + ( 4 i + 3 ) a f ) , x 4 n = a n + 1 f n c n d n i = 0 n 1 1 ( 1 + ( 4 i + 4 ) a f ) , x 4 n + 1 = a n + 1 f n + 1 e c n d n ( 1 + a f ) ( 1 c d ) n i = 0 n 1 ( 1 + ( 4 i + 5 ) a f ) , y 4 n 2 = c n d n a n f n 1 i = 0 n 1 ( 1 + ( 4 i ) a f ) , y 4 n 1 = e c n d n a n f n i = 0 n 1 ( 1 + ( 4 i + 1 ) a f ) ( 1 c d ) n , y 4 n = c n d n + 1 a n f n i = 0 n 1 ( 1 + ( 4 i + 2 ) a f ) , y 4 n + 1 = c n + 1 d n + 1 b a n f n i = 0 n 1 ( 1 + ( 4 i + 3 ) a f ) ( 1 c d ) n + 1 .

Theorem 5 Assume that { x n , y n } are solutions of system (4) with x 2 y 0 1. Then for n=0,1,2, ,

x 4 n 2 = a n f n c n 1 d n i = 0 n 1 1 ( 1 ( 4 i + 2 ) a f ) , x 4 n 1 = b a n f n c n d n ( 1 + c d ) n i = 0 n 1 ( 1 ( 4 i + 3 ) a f ) , x 4 n = a n + 1 f n c n d n i = 0 n 1 1 ( 1 ( 4 i + 4 ) a f ) , x 4 n + 1 = a n + 1 f n + 1 e c n d n ( 1 + a f ) ( 1 + c d ) n i = 0 n 1 ( 1 ( 4 i + 5 ) a f ) , y 4 n 2 = c n d n a n f n 1 i = 0 n 1 ( 1 ( 4 i ) a f ) , y 4 n 1 = e c n d n a n f n i = 0 n 1 ( 1 ( 4 i + 1 ) a f ) ( 1 + c d ) n , y 4 n = c n d n + 1 a n f n i = 0 n 1 ( 1 ( 4 i + 2 ) a f ) , y 4 n + 1 = c n + 1 d n + 1 b a n f n i = 0 n 1 ( 1 ( 4 i + 3 ) a f ) ( 1 + c d ) n + 1 .

Theorem 6 Suppose that { x n , y n } are solutions of system (5) where x 2 y 0 1. Then for n=0,1,2, ,

x 4 n 2 = a n f n c n 1 d n i = 0 n 1 1 ( 1 ( 4 i + 2 ) a f ) , x 4 n 1 = b a n f n c n d n ( 1 c d ) n i = 0 n 1 ( 1 ( 4 i + 3 ) a f ) , x 4 n = a n + 1 f n c n d n i = 0 n 1 1 ( 1 ( 4 i + 4 ) a f ) , x 4 n + 1 = a n + 1 f n + 1 e c n d n ( 1 a f ) ( 1 c d ) n i = 0 n 1 ( 1 ( 4 i + 5 ) a f ) , y 4 n 2 = c n d n a n f n 1 i = 0 n 1 ( 1 ( 4 i ) a f ) , y 4 n 1 = e c n d n a n f n i = 0 n 1 ( 1 ( 4 i + 1 ) a f ) ( 1 c d ) n , y 4 n = c n d n + 1 a n f n i = 0 n 1 ( 1 ( 4 i + 2 ) a f ) , y 4 n + 1 = c n + 1 d n + 1 b a n f n i = 0 n 1 ( 1 ( 4 i + 3 ) a f ) ( 1 c d ) n + 1 .

Example 3 We consider an interesting numerical example for the difference system (2) with the initial conditions x 2 =0.2, x 1 =0.15, x 0 =0.11, y 2 =0.23, y 1 =0.16 and y 0 =0.17. SeeFigure 3.

Figure 3
figure3

This figure shows the behavior of the difference system x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 + y n x n 2 ) , when we take the initial conditions x 2 =0.2 , x 1 =0.15 , x 0 =0.11 , y 2 =0.23 , y 1 =0.16 and y 0 =0.17 .

Example 4 See Figure 4, where we take system (3) with the initial conditions x 2 =0.12, x 1 =0.15, x 0 =0.11, y 2 =0.3, y 1 =0.6 and y 0 =0.17.

Figure 4
figure4

This figure shows the qualitative behavior of the solutions of system ( 3 ) with the initial conditions x 2 =0.12 , x 1 =0.15 , x 0 =0.11 , y 2 =0.3 , y 1 =0.6 and y 0 =0.17 .

4 On the solution of the system x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 + y n x n 2 )

In this section, we get the form of the solutions of the difference equations system

x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 + y n x n 2 ) ,
(6)

where n=0,1,2, and the initial conditions x 2 , x 1 , x 0 , y 2 , y 1 and y 0 are arbitrary nonzero real numbers with x 0 y 2 , x 2 y 0 1.

Theorem 7 If { x n , y n } are solutions of difference equation system (6), then for n=0,1,2, ,

x 4 n 2 = a n f n c n 1 d n , x 4 n 1 = b a n f n c n d n ( 1 + c d ) n ( 1 + a f ) n , x 4 n = a n + 1 f n c n d n , x 4 n + 1 = a n + 1 f n + 1 ( 1 + c d ) n e c n d n ( 1 + a f ) n + 1 , y 4 n 2 = c n d n a n f n 1 , y 4 n 1 = e c n d n a n f n ( 1 + a f ) n ( 1 + c d ) n , y 4 n = c n d n + 1 a n f n , y 4 n + 1 = c n + 1 d n + 1 ( 1 + a f ) n b a n f n ( 1 + c d ) n + 1 .

Proof For n=0 the result holds. Now, suppose that n>1 and that our assumption holds for n1, that is,

x 4 n 6 = a n 1 f n 1 c n 2 d n 1 , x 4 n 5 = b a n 1 f n 1 ( 1 + c d ) n 1 c n 1 d n 1 ( 1 + a f ) n 1 , x 4 n 4 = a n f n 1 c n 1 d n 1 , x 4 n 3 = a n f n ( 1 + c d ) n 1 e c n 1 d n 1 ( 1 + a f ) n , y 4 n 6 = c n 1 d n 1 a n 1 f n 2 , y 4 n 5 = e c n 1 d n 1 ( 1 + a f ) n 1 a n 1 f n 1 ( 1 + c d ) n 1 , y 4 n 4 = c n 1 d n a n 1 f n 1 , y 4 n 3 = c n d n ( 1 + a f ) n 1 b a n 1 f n 1 ( 1 + c d ) n .

Now, we conclude from Eq. (6) that

x 4 n 2 = x 4 n 3 y 4 n 5 y 4 n 4 ( 1 + x 4 n 3 y 4 n 5 ) = ( a n f n ( 1 + c d ) n 1 e c n 1 d n 1 ( 1 + a f ) n ) ( e c n 1 d n 1 ( 1 + a f ) n 1 a n 1 f n 1 ( 1 + c d ) n 1 ) ( c n 1 d n a n 1 f n 1 ) ( 1 + ( a n f n ( 1 + c d ) n 1 e c n 1 d n 1 ( 1 + a f ) n ) ( e c n 1 d n 1 ( 1 + a f ) n 1 a n 1 f n 1 ( 1 + c d ) n 1 ) ) = ( a f ( 1 + a f ) ) ( c n 1 d n a n 1 f n 1 ) ( 1 + a f ( 1 + a f ) ) = a n f n c n 1 d n , y 4 n 2 = y 4 n 3 x 4 n 5 x 4 n 4 ( 1 + y 4 n 3 x 4 n 5 ) = ( c n d n ( 1 + a f ) n 1 b a n 1 f n 1 ( 1 + c d ) n ) ( b a n 1 f n 1 ( 1 + c d ) n 1 c n 1 d n 1 ( 1 + a f ) n 1 ) ( a n f n 1 c n 1 d n 1 ) ( 1 + ( c n d n ( 1 + a f ) n 1 b a n 1 f n 1 ( 1 + c d ) n ) ( b a n 1 f n 1 ( 1 + c d ) n 1 c n 1 d n 1 ( 1 + a f ) n 1 ) ) = ( c d ( 1 + c d ) ) ( a n f n 1 c n 1 d n 1 ) ( 1 + ( c d ( 1 + c d ) ) ) = c n d n a n f n 1 .

Also, we can prove the other relations. This completes the proof. □

We consider the following system, and the proof of the theorem is similar to the above mentioned theorem and so it is left to the reader,

x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 y n x n 2 ) .
(7)

Theorem 8 Let { x n , y n } n = 2 + be solutions of system (7) and x 0 y 2 1, x 2 y 0 1. Then for n=0,1,2, ,

x 4 n 2 = a n f n c n 1 d n , x 4 n 1 = b a n f n c n d n ( 1 c d ) n ( 1 + a f ) n , x 4 n = a n + 1 f n c n d n , x 4 n + 1 = a n + 1 f n + 1 ( 1 c d ) n e c n d n ( 1 + a f ) n + 1 , y 4 n 2 = c n d n a n f n 1 , y 4 n 1 = e c n d n a n f n ( 1 + a f ) n ( 1 c d ) n , y 4 n = c n d n + 1 a n f n , y 4 n + 1 = c n + 1 d n + 1 ( 1 + a f ) n b a n f n ( 1 c d ) n + 1 .

Lemma 1 The solution of system (6) is unbounded except in the following case.

Theorem 9 System (6) has a periodic solution of period four iff cd=af=2 and it will take the following form: { x n }={c,b,a, a f e ,c,b,a,}, { y n }={f,e,d, c d b ,f,e,d,}.

Proof First, suppose that a prime period four solution exists

{ x n }= { c , b , a , a f e , c , b , a , } ,{ y n }= { f , e , d , c d b , f , e , d , }

of system (6). We see from the form of the solution of system (6) that

x 4 n 2 = c = a n f n c n 1 d n , x 4 n 1 = b = b a n f n c n d n ( 1 + c d ) n ( 1 + a f ) n , x 4 n = a = a n + 1 f n c n d n , x 4 n + 1 = a f e = a n + 1 f n + 1 ( 1 + c d ) n e c n d n ( 1 + a f ) n + 1 , y 4 n 2 = f = c n d n a n f n 1 , y 4 n 1 = e = e c n d n a n f n ( 1 + a f ) n ( 1 + c d ) n , y 4 n = d = c n d n + 1 a n f n , y 4 n + 1 = c d b = c n + 1 d n + 1 ( 1 + a f ) n b a n f n ( 1 + c d ) n + 1 .

Then we get cd=af=2. Second, assume that cd=af=2. Then we see from the form of the solution of system (6) that

x 4 n 2 = a n f n c n 1 d n = c , x 4 n 1 = b a n f n c n d n ( 1 + c d ) n ( 1 + a f ) n = b , x 4 n = a n + 1 f n c n d n = a , x 4 n + 1 = a n + 1 f n + 1 ( 1 + c d ) n e c n d n ( 1 + a f ) n + 1 = a f e , y 4 n 2 = c n d n a n f n 1 = f , y 4 n 1 = e c n d n a n f n ( 1 + a f ) n ( 1 + c d ) n = e , y 4 n = c n d n + 1 a n f n = c , y 4 n + 1 = c n + 1 d n + 1 ( 1 + a f ) n b a n f n ( 1 + c d ) n + 1 = c d b .

Thus, we have a periodic solution of period four and the proof is complete. □

Lemma 2 The solution of system (7) is unbounded except in the following case.

Theorem 10 System (7) has a periodic solution of period eight iff cd=2, af=2 and it will take the following form: { x n }={c,b,a, a f e ,c,b,a, a f e ,c,b,a,}, { y n }={f,e,d, c d b ,f,e,d, c d b ,f,e,d,}.

Example 5 We consider a numerical example for the difference system (6) when we put the initial conditions x 2 =0.12, x 1 =0.5, x 0 =0.31, y 2 =0.23, y 1 =0.6 and y 0 =0.7.See Figure 5.

Figure 5
figure5

This figure shows the solution of the system of difference equations x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 + y n x n 2 ) , when we put the initial conditions x 2 =0.12 , x 1 =0.5 , x 0 =0.31 , y 2 =0.23 , y 1 =0.6 and y 0 =0.7 .

Example 6 Figure 6 shows the behavior of the solution of the difference system (6)with the initial conditions x 2 =4, x 1 =7, x 0 =0.5, y 2 =4, y 1 =6 and y 0 =0.5.

Figure 6
figure6

This figure shows the periodicity of the solutions of difference equations system ( 6 ) with initial conditions x 2 =4 , x 1 =7 , x 0 =0.5 , y 2 =4 , y 1 =6 and y 0 =0.5 .

Example 7 We consider a numerical example for the difference system(7) when we put the initial conditions x 2 =0.32, x 1 =0.25, x 0 =0.31, y 2 =0.23, y 1 =0.26 and y 0 =0.17. See Figure 7.

Figure 7
figure7

This figure shows the behavior of the solutions of x n + 1 = x n y n 2 y n 1 ( 1 + x n y n 2 ) , y n + 1 = y n x n 2 x n 1 ( 1 y n x n 2 ) , since we consider the initial conditions equals x 2 =0.32 , x 1 =0.25 , x 0 =0.31 , y 2 =0.23 , y 1 =0.26 and y 0 =0.17 .

Example 8 Figure 8 shows the periodicity of the solution of the difference system (7) with the initial conditions x 2 =9, x 1 =7, x 0 =0.4, y 2 =5, y 1 =8 and y 0 =2/9.

Figure 8
figure8

This figure shows the periodic nature of the solutions of system ( 7 ), where the initial conditions x 2 =9 , x 1 =7 , x 0 =0.4 , y 2 =5 , y 1 =8 and y 0 =2/9 .

References

  1. 1.

    Cull P, Flahive M, Robson R Undergraduate Texts in Mathematics. In Difference Equations: From Rabbits to Chaos. Springer, New York; 2005.

    Google Scholar 

  2. 2.

    Elaydi S Undergraduate Texts in Mathematics. In An Introduction to Difference Equations. 3rd edition. Springer, New York; 2005.

    Google Scholar 

  3. 3.

    Beverton RJH, Holt SJ Fishery Investigations Series II 19. In On the Dynamics of Exploited Fish Populations. Blackburn Press, Caldwell; 2004.

    Google Scholar 

  4. 4.

    Ahlbrandt CD, Peterson AC Kluwer Texts in the Mathematical Sciences 16. In Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations. Kluwer Academic, Dordrecht; 1996.

    Google Scholar 

  5. 5.

    Kocic VL, Ladas G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic, Dordrecht; 1993.

    Google Scholar 

  6. 6.

    Kulenovic MRS, Ladas G: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall/CRC Press, Boca Raton; 2001.

    Google Scholar 

  7. 7.

    Camouzis E, Kulenovic MRS, Ladas G, Merino O: Rational systems in the plane. J. Differ. Equ. Appl. 2009, 15(3):303-323. 10.1080/10236190802125264

    MathSciNet  Article  Google Scholar 

  8. 8.

    Cinar C, Yalçinkaya I, Karatas R:On the positive solutions of the difference equation system x n + 1 =m/ y n , y n + 1 =p y n / x n 1 y n 1 . J. Inst. Math. Comput. Sci. 2005, 18: 135-136.

    Google Scholar 

  9. 9.

    Clark D, Kulenovic MRS: A coupled system of rational difference equations. Comput. Math. Appl. 2002, 43: 849-867. 10.1016/S0898-1221(01)00326-1

    MathSciNet  Article  Google Scholar 

  10. 10.

    Kurbanli AS, Cinar C, Yalçınkaya I: On the behavior of positive solutions of the system of rational difference equations. Math. Comput. Model. 2011, 53: 1261-1267. 10.1016/j.mcm.2010.12.009

    Article  Google Scholar 

  11. 11.

    Touafek N, Elsayed EM: On the solutions of systems of rational difference equations. Math. Comput. Model. 2012, 55: 1987-1997. 10.1016/j.mcm.2011.11.058

    MathSciNet  Article  Google Scholar 

  12. 12.

    Yalcinkaya I, Cinar C, Atalay M: On the solutions of systems of difference equations. Adv. Differ. Equ. 2008., 2008: Article ID 143943

    Google Scholar 

  13. 13.

    Agarwal RP: Difference Equations and Inequalities. 1st edition. Dekker, New York; 1992. (2nd edn. (2000))

    Google Scholar 

  14. 14.

    Agarwal RP, Elsayed EM: On the solution of fourth-order rational recursive sequence. Adv. Stud. Contemp. Math. 2010, 20(4):525-545.

    MathSciNet  Google Scholar 

  15. 15.

    Battaloglu N, Cinar C, Yalçınkaya I: The dynamics of the difference equation. Ars Comb. 2010, 97: 281-288.

    Google Scholar 

  16. 16.

    Din Q, Qureshi MN, Qadeer Khan A: Dynamics of a fourth-order system of rational difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 215

    Google Scholar 

  17. 17.

    Elabbasy EM, El-Metwally H, Elsayed EM: On the solutions of a class of difference equations systems. Demonstr. Math. 2008, 41(1):109-122.

    MathSciNet  Google Scholar 

  18. 18.

    Elabbasy EM, El-Metwally H, Elsayed EM: Global behavior of the solutions of difference equation. Adv. Differ. Equ. 2011., 2011: Article ID 28

    Google Scholar 

  19. 19.

    Elsayed EM: Solution and attractivity for a rational recursive sequence. Discrete Dyn. Nat. Soc. 2011., 2011: Article ID 982309

    Google Scholar 

  20. 20.

    Elsayed EM: On the solution of some difference equations. Eur. J. Pure Appl. Math. 2011, 4(3):287-303.

    MathSciNet  Google Scholar 

  21. 21.

    Elsayed EM: Behavior and expression of the solutions of some rational difference equations. J. Comput. Anal. Appl. 2013, 15(1):73-81.

    MathSciNet  Google Scholar 

  22. 22.

    Elsayed EM: Solutions of rational difference system of order two. Math. Comput. Model. 2012, 55: 378-384. 10.1016/j.mcm.2011.08.012

    MathSciNet  Article  Google Scholar 

  23. 23.

    Elsayed EM, El-Dessoky MM: Dynamics and behavior of a higher order rational recursive sequence. Adv. Differ. Equ. 2012., 2012: Article ID 69

    Google Scholar 

  24. 24.

    Elsayed EM, El-Dessoky M, Alotaibi A: On the solutions of a general system of difference equations. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 892571

    Google Scholar 

  25. 25.

    Keying L, Zhongjian Z, Xiaorui L, Peng L: More on three-dimensional systems of rational difference equations. Discrete Dyn. Nat. Soc. 2011., 2011: Article ID 178483

    Google Scholar 

  26. 26.

    Kurbanli AS: On the behavior of solutions of the system of rational difference equations. Adv. Differ. Equ. 2011., 2011: Article ID 40

    Google Scholar 

  27. 27.

    Mansour M, El-Dessoky MM, Elsayed EM: The form of the solutions and periodicity of some systems of difference equations. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 406821

    Google Scholar 

  28. 28.

    Papaschinopoulos G, Radin MA, Schinas CJ:On the system of two difference equations of exponential form: x n + 1 =a+b x n 1 e y n , y n + 1 =c+d y n 1 e x n . Math. Comput. Model. 2011, 54(11-12):2969-2977. 10.1016/j.mcm.2011.07.019

    MathSciNet  Article  Google Scholar 

  29. 29.

    Papaschinopoulos G, Schinas CJ: On the dynamics of two exponential type systems of difference equations. Comput. Math. Appl. 2012, 64(7):2326-2334. 10.1016/j.camwa.2012.04.002

    MathSciNet  Article  Google Scholar 

  30. 30.

    Touafek N, Elsayed EM: On the periodicity of some systems of nonlinear difference equations. Bull. Math. Soc. Sci. Math. Roum. 2012, 55(103)(2):217-224.

    MathSciNet  Google Scholar 

  31. 31.

    Yalçınkaya I: On the global asymptotic behavior of a system of two nonlinear difference equations. Ars Comb. 2010, 95: 151-159.

    Google Scholar 

  32. 32.

    Yang X, Liu Y, Bai S:On the system of high order rational difference equations x n = a y n p , y n = b y n p x n q y n q . Appl. Math. Comput. 2005, 171(2):853-856. 10.1016/j.amc.2005.01.092

    MathSciNet  Article  Google Scholar 

  33. 33.

    Wang C, Wang S, Wang W: Global asymptotic stability of equilibrium point for a family of rational difference equations. Appl. Math. Lett. 2011, 24(5):714-718. 10.1016/j.aml.2010.12.013

    MathSciNet  Article  Google Scholar 

  34. 34.

    Wang C, Shi Q, Wang S: Asymptotic behavior of equilibrium point for a family of rational difference equation. Adv. Differ. Equ. 2010., 2010: Article ID 505906

    Google Scholar 

  35. 35.

    Wang C, Gong F, Wang S, Li L, Shi Q: Asymptotic behavior of equilibrium point for a class of nonlinear difference equation. Adv. Differ. Equ. 2009., 2009: Article ID 214309

    Google Scholar 

  36. 36.

    Wang C, Wang S, Wang Z, Gong F, Wang R: Asymptotic stability for a class of nonlinear difference equation. Discrete Dyn. Nat. Soc. 2010., 2010: Article ID 791610

    Google Scholar 

  37. 37.

    Zayed EME, El-Moneam MA:On the rational recursive sequence x n + 1 =A x n +(β x n +γ x n k )/(C x n +D x n k ). Commun. Appl. Nonlinear Anal. 2009, 16: 91-106.

    MathSciNet  Google Scholar 

  38. 38.

    Zhang Q, Yang L, Liu J: Dynamics of a system of rational third order difference equation. Adv. Differ. Equ. 2012., 2012: Article ID 136

    Google Scholar 

  39. 39.

    Zkan O, Kurbanli AS: On a system of difference equation. Discrete Dyn. Nat. Soc. 2013., 2013: Article ID 970316

    Google Scholar 

Download references

Acknowledgements

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (130-028-D1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Elsayed M Elsayed.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. Both authors read and approved the final manuscript.

Authors’ original submitted files for images

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Elsayed, E.M., El-Metwally, H. On the solutions of some nonlinear systems of difference equations. Adv Differ Equ 2013, 161 (2013). https://doi.org/10.1186/1687-1847-2013-161

Download citation

Keywords

  • difference equations
  • periodic solution
  • solution of difference equation
  • system of difference equations