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Periodicty and Stability of Solutions of Rational Difference Systems | OMICS International
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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# Periodicty and Stability of Solutions of Rational Difference Systems

E. M. Elabbasy1* and S. M. Eleissawy2

1Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

2Department of Physics and Engineering Mathematics, Faculty of Engineering, Port-Said University, Egypt

*Corresponding Author:
E. M. Elabbasy
Department of Mathematics
Faculty of Science, Mansoura University
Mansoura, 35516, Egypt
E-mail: [email protected]

Received Date: May 25, 2012; Accepted Date: June 19, 2012; Published Date: June 22, 2012

Citation: Elabbasy EM, Eleissawy SM (2012) Periodicty and Stability of Solutions of Rational Difference Systems. J Appl Computat Math 1:114. doi: 10.4172/2168-9679.1000114

Copyright: © 2012 Elabbasy EM, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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#### Abstract

We study the stability character and periodic solutions of the following rational difference systems
1 1
1 1
1 1
= 1 , = 1 ,
1 1
n n
n n
n n n n
x x y y
y x x y
− −
+ +
− −
± ±
± ±
where the initial values 1 0 1 0 x , x , y , y − − are nonzero real numbers. Some numerical examples are given to illustrate our results.

#### Keywords

Difference equation; Rational systems; Periodic solutions; Stability

39A10

#### Introduction

In this paper we study stability character and periodic solutions for the following rational difference systems

(1)

where the initial values are nonzero real numbers . Difference equation is a hot topic in that it is widely used to investigate equations arising in mathematical models describing real life situations such as population biology, probability theory, genetics and so on. Recently, rational difference equations appeals great interests. In particular, it is popular to study the system of two rational difference equations [1-4].

In [5], Çinar has obtained the positive solution of the difference equation system

(2)

Also, Çinar et al. [6] has obtained the positive solution of the difference equation system

(3)

In [7], Kurbanli et al. investigated the periodicity of the solutions of the system of difference equations

In [8] Yalcinkaya et al. obtained a sufficient condition for the global asymptotic stability of the following system of difference equations

Elsayed [9] has obtained the form of solutions for the rational difference system

Yang et al. [10] investigated the system of rational difference equations

(4)

Other related results on system of rational difference equations can be found in references [11-16].

Let I be some interval of real numbers and let be continuously differentiable functions. Then for all initial values , k = −1,0, the system of difference equations [17].

(5)

has a unique solution .

A point is called an equilibrium point of the system (5) if

Let be an equilibrium point of the system (5) [17].

1. An equilibrium point is said to be stable if for any ε > 0 there exist δ > 0 such that for every initial points and for which , the iterates of and satisfies for all n > 0. An equilibrium point is said to be unstable if it is not stable. (By is said to be unstable if it is not stable. (By given by .

2. An equilibrium point is said to be asymptotically stable if there exists r > 0 such that as n→∞ for all and that satisfies

Let be an equilibrium point of a map F = ( f , g), where f and g are continuously differentiable functions at . The jacobian matrix of F at is the matrix [17].

The linear map given by

is called the linearization of the map F at .

A solution of (1) is periodic if there exist a positive integer ω such that

and ω is called a period.

Theorem 1 (Linearized Stability Theorem)

Let F = (f,g) be a continuously differentiable function defined on an open set I in , and let in I be an equilibrium point of the map F = ( f,g) [17].

1. If all the eigenvalues of the Jacobian matrix have modulus less than one, then the equilibrium point is asymptotically stable.

2. If at least one of the eigenvalues of the Jacobian matrix has modulus greater than one, then the equilibrium point is unstable.

3. An equilibrium point of the map F = ( f , g) is locally asymptotically stable if and only if every solution of the characteristic equation

(6)

lies inside the unit circle, that is, if and only if

(7)

4. An equilibrium point of the map F = ( f,g) is a saddle point if the characteristic equation (6) has one root that lies inside the unit circle and one root that lies outside the unit circle if and only if

(8)

5. An equilibrium point of the map F = ( f,g) is nonhyperbolic if at least one of the eigenvalues of the Jacobian matrix has modulus equal one.

6. The characteristic equation (6) has at least one root that lies on the unit circle if and only if

(9)

or

(10)

#### Main Results

The first system

In this subsection, we study the stability of solutions of the difference system

(11)

where the initial values are nonzero real numbers such that and .

Theorem 1: System (11) has the unique positive equilibrium point which is locally asymptotically stable.

Proof: The equilibrium point of the system (11) satisfies the following system of equations

(12)

system (12) implies

(13)

(14)

from equations (13) and (14), the unique positive equilibrium point is (1,1).

The map F associated to system (11) is

(15)

The Jacobian matrix of F at the equilibrium point is

(16)

The value of the Jacobian matrix of F at the equilibrium point is

(17)

Then the characteristic equation about (1,1) has the following form

(18)

where

(19)

(20)

The result follows from Theorem 1.1 (iii) and the following relations

and

Therefore, the equilibrium point (1,1) is locally asymptotically stable.

Remark 1: The following system

has the unique negative equilibrium point which is locally asymptotically stable, where the initial values are nonzero real numbers such that and .

The second system

In this subsection, we study the solutions of the following system

(21)

where the initial values are nonzero real numbers such that and .

Theorem 2: Let be nonzero real numbers such that and . Let be a solution of system (21). Then all solutions of system (21) are periodic with period ten and for n = 0,1,...

Proof: From equation (21), we see that

For n = 0 the result holds for the given solutions. Now suppose n > 0 that and our assumption holds for n −1. That is;

It follows that

Hence, the proof is completed.

Remark 2: Let be nonzero real numbers such that and . Let be a solution of system

(22)

Then all solutions of system (22) are periodic with period ten and for n = 0,1,...

#### Numerical Examples

In this section, we give some numerical simulations supporting our theoretical analysis via the software package Matlab 7.13. These examples represent the periodicity and stablity of solutions of two dimensional systems of rational difference equations (1).

Example 1

Consider the difference system:

(23)

with the initial conditions x-1=0.5, x0 =−2, 1 y-1 = 0.3, y0= −1.7 System (23) has local asymptotic stability of the equilibrium point (1,1) (Figure 1).

Figure 1: System (23) has local asymptotic stability of the equilibrium point.

Example 2

Consider the difference system:

(24)

with the initial conditions x-1 = 2, x0 = −5, y-1 = 3, y0 = 4. The solution of (24) is periodic with period 10. (Figure 2).

Figure 2: The solution of (24) is periodic with period 10.

#### References

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