Dynamics of a fourth-order system of rational difference equations
© Din et al.; licensee Springer 2012
Received: 3 October 2012
Accepted: 3 December 2012
Published: 17 December 2012
In this paper, we study the equilibrium points, local asymptotic stability of an equilibrium point, instability of equilibrium points, periodicity behavior of positive solutions, and global character of an equilibrium point of a fourth-order system of rational difference equations of the form
, where the parameters α, β, γ, , , and initial conditions , , , , , , , are positive real numbers. Some numerical examples are given to verify our theoretical results.
1 Introduction and preliminaries
The theory of discrete dynamical systems and difference equations developed greatly during the last twenty-five years of the twentieth century. Applications of difference equations also experienced enormous growth in many areas. Many applications of discrete dynamical systems and difference equations have appeared recently in the areas of biology, economics, physics, resource management, and others. The theory of difference equations occupies a central position in applicable analysis. There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. Nonlinear difference equations of order greater than one are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering and economics. It is very interesting to investigate the behavior of solutions of a system of higher-order rational difference equations and to discuss the local asymptotic stability of their equilibrium points.
where , , , are some sequences or .
for all values of real parameters a, b.
The point is also called a fixed point of the vector map F.
An equilibrium point is said to be stable if for every , there exists such that for every initial condition , if implies for all , where is the usual Euclidean norm in .
An equilibrium point is said to be unstable if it is not stable.
An equilibrium point is said to be asymptotically stable if there exists such that and as .
An equilibrium point is called a global attractor if as .
An equilibrium point is called an asymptotic global attractor if it is a global attractor and stable.
and is a Jacobian matrix of the system (1.2) about the equilibrium point .
where , , and .
Theorem 1.3 For the system , , of difference equations such that is a fixed point of F. If all eigenvalues of the Jacobian matrix about lie inside the open unit disk , then is locally asymptotically stable. If one of them has a modulus greater than one, then is unstable.
Theorem 1.4 (Routh-Hurwitz criterion)
A necessary and sufficient condition for all of the roots of (1.5) to have a negative real part is for .
A necessary and sufficient condition for the existence of a root of (1.5) with a positive real part is for some .
2 Main results
where and .
Let and , then the equilibrium point of the system (1.1) is locally asymptotically stable.
If or , then the equilibrium point of the system (1.1) is unstable.
It is easy to see that if or , then there exists at least one root λ of Equation (2.1) such that . Hence, by Theorem 1.3 if or , then is unstable. □
Theorem 2.3 If and , then a positive equilibrium point of Equation (1.1) is unstable.
Hence, by Theorem 1.3 if and , then is unstable. □
Theorem 2.4 If and , then the equilibrium points , , of Equation (1.1) are unstable.
Proof The proof is similar to Theorem 2.3, so it is omitted. □
The following theorem is similar to Theorem 3.4 of .
If , then .
If , then .
Theorem 2.6 The system (1.1) has no prime period-two solutions.
From (2.3) and (2.4), one has for . Which is a contradiction. Hence, the system (1.1) has no prime period-two solutions. □
Theorem 2.7 Let and , then the equilibrium point of Equation (1.1) is globally asymptotically stable.
This implies that and . Hence, the subsequences , , , are decreasing, i.e., the sequence is decreasing. Hence, . □
If , then .
If , then .
In order to verify our theoretical results and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to the system of nonlinear difference equations (1.1). All plots in this section are drawn with mathematica.
This work is a natural extension of [9, 10]. In the paper, we investigated some dynamics of an eight-dimensional discrete system. The system has five equilibrium points all of which except are unstable. The linearization method is used to show that the equilibrium point is locally asymptotically stable. We prove that the system has no prime period-two solutions. The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state. An approach to this problem consists of determining the possible global behaviors of the system and determining which initial conditions lead to these long-term behaviors. In case of higher-order dynamical systems, it is crucial to discuss global behavior of the system. Some powerful tools such as semiconjugacy and weak contraction cannot be used to analyze global behavior of the system (1.1). In the paper, we prove the global asymptotic stability of the equilibrium point by using simple techniques. Some numerical examples are provided to support our theoretical results. These examples are experimental verifications of theoretical discussions.
Authors would like to thank the referees for their comments and suggestions on the manuscript. This work was supported by the Higher Education Commission of Pakistan.
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