Abstract
In the present work, we study the qualitative behavior of two systems of higherorder rational difference equations. More precisely, we study the local asymptotic stability, instability, global asymptotic stability of equilibrium points and rate of convergence of positive solutions of these systems. Our results considerably extend and improve some recent results in the literature. Some numerical examples are given to verify our theoretical results.
MSC: 39A10, 40A05.
Keywords:
system of difference equations; stability; global character; rate of convergence1 Introduction
Recently, studying the qualitative behavior of difference equations and systems is a topic of great interest. Applications of discrete dynamical systems and difference equations have appeared recently in many areas such as ecology, population dynamics, queuing problems, statistical problems, stochastic time series, combinatorial analysis, number theory, geometry, electrical networks, neural networks, quanta in radiation, genetics in biology, economics, psychology, sociology, physics, engineering, economics, probability theory and resource management. Unfortunately, these are only considered as the discrete analogs of differential equations. It is a wellknown fact that difference equations appeared much earlier than differential equations and were instrumental in paving the way for the development of the latter. It is only recently that difference equations have started receiving the attention they deserve. Perhaps this is largely due to the advent of computers where differential equations are solved by using their approximate difference equation formulations. The theory of discrete dynamical systems and difference equations developed greatly during the last twentyfive years of the twentieth century. 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. It is very interesting to investigate the behavior of solutions of a system of higherorder rational difference equations and to discuss the local asymptotic stability of their equilibrium points. Systems of rational difference equations have been studied by several authors. Especially there has been a great interest in the study of the attractivity of the solutions of such systems. For more results on the qualitative behavior of nonlinear difference equations, we refer the interested reader to [119].
Zhang et al.[20] studied the dynamics of a system of rational thirdorder difference equations
Din et al.[11] investigated the dynamics of a system of fourthorder rational difference equations
To be motivated by the above studies, our aim in this paper is to investigate the qualitative behavior of the following order systems of rational difference equations:
where the parameters α, β, γ, , , and initial conditions , are positive real numbers, and
where the parameters a, b, c, , , and initial conditions , are positive real numbers. This paper is a natural extension of [11,20,21].
Let us consider dimensional discrete dynamical system of the form
where and are continuously differentiable functions and I, J are some intervals of real numbers. Furthermore, a solution of system (3) is uniquely determined by initial conditions for . Along with system (3), we consider the corresponding vector map . An equilibrium point of (3) is a point that satisfies
The point is also called a fixed point of the vector map F.
Definition 1 Let be an equilibrium point of system (3).
(i) An equilibrium point is said to be stable if for every there exists such that for every initial condition , , implies for all , where is the usual Euclidian norm in .
(ii) An equilibrium point is said to be unstable if it is not stable.
(iii) An equilibrium point is said to be asymptotically stable if there exists such that and as .
(iv) An equilibrium point is called a global attractor if as .
(v) An equilibrium point is called an asymptotic global attractor if it is a global attractor and stable.
Definition 2 Let be an equilibrium point of the map
where f and g are continuously differentiable functions at . The linearized system of (3) about the equilibrium point is
where
and is the Jacobian matrix of system (3) about the equilibrium point .
Lemma 1[22]
Assume that, , is a system of difference equations andis the fixed point ofF. If all eigenvalues of the Jacobian matrixaboutlie inside an open unit disk, thenis locally asymptotically stable. If one of them has norm greater than one, thenis unstable.
Lemma 2[23]
Assume that, , is a system of difference equations andis the equilibrium point of this system. The characteristic polynomial of this system about the equilibrium pointis, with real coefficients and. Then all roots of the polynomiallie inside the open unit diskif and only iffor , whereis the principal minor of orderkof thematrix
Let us consider a system of difference equations
where is an mdimensional vector, is a constant matrix, and is a matrix function satisfying
as , where denotes any matrix norm which is associated with the vector norm
Proposition 1 (Perron’s theorem)[24]
Suppose that condition (6) holds. Ifis a solution of (5), then eitherfor all largenor
exists and is equal to the modulus of one of the eigenvalues of matrixA.
Proposition 2[24]
Suppose that condition (6) holds. Ifis a solution of (5), then eitherfor all largenor
exists and is equal to the modulus of one of the eigenvalues of matrixA.
2 On the system ,
In this section, we shall investigate the qualitative behavior of system (1). Let be an equilibrium point of system (1), then for and , system (1) has two positive equilibrium points , , where and .
To construct the corresponding linearized form of system (1), we consider the following transformation:
where , , , … , and , , , … , . The Jacobian matrix about the fixed point under the transformation (9) is given by
Theorem 1Letand, then every solutionof system (1) is bounded.
Proof It is easy to verify that
and
Take and . Then and for all . □
Theorem 2The equilibrium pointof system (1) is locally asymptotically stable.
Proof The linearized system of (1) about the equilibrium point is given by
where
and
Let denote the eigenvalues of matrix E. Let be a diagonal matrix, where , , , and
Clearly, D is invertible. Computing , we obtain
We obtain the following two inequalities:
which implies that
and
Furthermore,
and
It is a wellknown fact that E has the same eigenvalues as . Hence, we obtain
Hence, the equilibrium point of system (1) is locally asymptotically stable. □
Theorem 3The positive equilibrium pointof system (1) is unstable.
Proof The linearized system of (1) about the equilibrium point is given by
where
and
where and . The characteristic polynomial of is given by
From (10), we have
It is clear that not all of . Therefore, by Lemma 1, the unique positive equilibrium point is unstable. □
Theorem 4Letand, and letbe a solution of system (1). Then, for, the following statements are true:
Proof It follows from induction. □
Theorem 5Letand, then the equilibrium pointof system (1) is globally asymptotically stable.
Proof For and , from Theorem 2, is locally asymptotically stable. From Theorem 1, every positive solution is bounded, i.e., and for all , where and . So, it is sufficient to prove that is decreasing. From system (1), one has
This implies that and . Hence, the subsequences
are decreasing, i.e., the sequence is decreasing. Also,
This implies that and . Hence, the subsequences
are decreasing, i.e., the sequence is decreasing. Hence, . □
Theorem 6Letand. Then, for a solutionof system (1), the following statements are true:
2.1 Rate of convergence
We investigate the rate of convergence of a solution that converges to the equilibrium point of system (1).
Assume that and . First we will find a system of limiting equations for the map F. The error terms are given as
Taking the limits, we obtain for , , for , for , for and . Hence, the limiting system of error terms at can be written as
where
which is similar to the linearized system of (1) about the equilibrium point . Using proposition (1), one has the following result.
Theorem 7Assume thatis a positive solution of system (1) such that, and, where. Then the error vectorof every solution of (1) satisfies both of the following asymptotic relations:
whereare the characteristic roots of the Jacobian matrixabout.
3 On the system ,
In this section, we shall investigate the qualitative behavior of system (2). Let be an equilibrium point of system (2), then system (2) has a unique equilibrium point . To construct the corresponding linearized form of system (2), we consider the following transformation:
, , , …, and , , , … , . The Jacobian matrix about the fixed point under the transformation (13) is given by
Theorem 8Letbe a positive solution of system (2), then for every, the following results hold.
Lemma 3Let, then every solutionof system (2) is bounded.
Proof Assume that
and
Then from Theorem 8 one can easily see that and for all . □
Theorem 9The equilibrium pointof equation (2) is locally asymptotically stable.
Proof The linearized system of (2) about the equilibrium point is given by
where
and
Let denote the eigenvalues of matrix E. Let be a diagonal matrix, where , , and
Clearly, D is invertible. Computing , we obtain
Next, we have the following two inequalities:
which implies that
and
Furthermore,
and
Now H has the same eigenvalues as , we obtain that
Hence, the equilibrium point of system (2) is locally asymptotically stable. □
Theorem 10Letand, then the equilibrium pointof system (2) is globally asymptotically stable.
Proof Assume that and . Then from Theorem 9 the equilibrium point of system (2) is locally asymptotically stable. Moreover, from Lemma 3 every positive solution is bounded, i.e., and for all , where and . Now, it is sufficient to prove that is decreasing. From system (2) one has
This implies that
Hence, and . Hence, the subsequences
and
are decreasing. Therefore the sequences and are decreasing. Hence, . □
3.1 Rate of convergence
Assume that and . First we will find a system of limiting equations for system (2). The error terms are given as
where
and . Taking the limits, we obtain for , for , , for , , for , So, the limiting system of error terms can be written as
where
and
which is similar to the linearized system of (2) about the equilibrium point . Using proposition (1), one has the following result.
Theorem 11Assume thatis a positive solution of system (2) such that, and, where. Then the error vectorof every solution of (2) satisfies both of the following asymptotic relations:
whereare the characteristic roots of the Jacobian matrixabout.
4 Examples
In order to verify our theoretical results, we consider some interesting numerical examples in this section. These examples show that the equilibrium point of both systems (1) and (2) is globally asymptotically stable.
Example 1 Consider system (1) with initial conditions , , , , , , , , , , , , , , , , , . Moreover, choose the parameters , , , , , . Then system (1) can be written as
with initial conditions , , , , , , , , , , , , , , , , , . Moreover, in Figure 1, the plot of is shown in Figure 1a, the plot of is shown in Figure 1b, and an attractor of system (14) is shown in Figure 1c.
Figure 1. Plots for system (14).
Example 2 Consider system (1) with initial conditions , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Moreover, choose the parameters , , , , , . Then system (1) can be written as
with initial conditions , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Moreover, in Figure 2, the plot of is shown in Figure 2a, the plot of is shown in Figure 2b, and an attractor of system (15) is shown in Figure 2c.
Figure 2. Plots for system (15).
Example 3 Consider system (2) with initial conditions , , , , , , , , , , , , , . Moreover, choose the parameters , , , , , . Then system (2) can be written as
with initial conditions , , , , , , , , , , , , , . Moreover, in Figure 3, the plot of is shown in Figure 3a, the plot of is shown in Figure 3b, and an attractor of system (16) is shown in Figure 3c.
Figure 3. Plots for system (16).
Example 4 Consider system (2) with initial conditions , , , , , , , , , , , , , , , , , , , , , , , , , . Moreover, choose the parameters , , , , , . Then system (2) can be written as
with initial conditions , , , , , , , , , , , , , , , , , , , , , , , , , . Moreover, in Figure 4, the plot of is shown in Figure 4a, the plot of is shown in Figure 4b, and an attractor of system (17) is shown in Figure 4c.
Figure 4. Plots for system (17).
Conclusion
This work is a natural extension of [11,20,21]. In the paper, we have investigated the qualitative behavior of dimensional discrete dynamical systems. Each system has only one equilibrium point which is stable under some restriction to parameters. The linearization method is used to show that equilibrium point is locally asymptotically stable. 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 longterm behaviors. In case of higherorder 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 systems (1) and (2). In the paper, we prove the global asymptotic stability of equilibrium point by using simple techniques. We have carried out a systematical local and global stability analysis of both systems. The most important finding here is that the unique equilibrium point can be a global asymptotic attractor for systems (1) and (2). Moreover, we have determined the rate of convergence of a solution that converges to the equilibrium point of systems (1) and (2). Some numerical examples are provided to support our theoretical results. These examples are experimental verifications of theoretical discussions.
Competing interests
The authors have no competing interests.
Authors’ contributions
All authors contributed equally in drafting this manuscript and giving the main proofs.
Acknowledgements
This work was supported by the Higher Education Commission of Pakistan.
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