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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

Mathematics Subject Classification

39A10

Introduction

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

Equation (1)

where the initial values Equation 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

Equation(2)

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

Equation(3)

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

Equation

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

Equation

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

Equation

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

Equation(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 Equation be continuously differentiable functions. Then for all initial values Equation, k = −1,0, the system of difference equations [17].

Equation(5)

has a unique solution Equation.

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

Equation

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

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

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

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

Equation

The linear map Equation given by

Equation

is called the linearization of the map F at Equation.

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

Equation

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 Equation, and let Equation in I be an equilibrium point of the map F = ( f,g) [17].

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

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

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

Equation (6)

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

Equation(7)

4. An equilibrium point Equation 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

Equation (8)

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

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

Equation(9)

or

Equation (10)

Main Results

The first system

EquationEquation

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

Equation (11)

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

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

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

Equation (12)

system (12) implies

Equation (13)

Equation (14)

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

The map F associated to system (11) is

Equation (15)

The Jacobian matrix of F at the equilibrium point Equation is

Equation (16)

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

Equation (17)

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

Equation (18)

where

Equation (19)

Equation (20)

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

Equation

and

Equation

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

Remark 1: The following system

Equation

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

The second system

EquationEquation

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

Equation (21)

where the initial values Equationare nonzero real numbers such that Equation and Equation.

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

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Proof: From equation (21), we see that

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

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

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

It follows that

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Hence, the proof is completed.

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

Equation (22)

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

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

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:

Equation (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).

applied-computational-mathematics-system-local

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

Example 2

Consider the difference system:

Equation (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).

applied-computational-mathematics-the-solution

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

References

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