Abstract
In the proposed work, global dynamics of a system of rational difference equations has been studied in the interior of . It is proved that system has at least one and at most seven boundary equilibria and a unique equilibrium under certain parametric conditions. By utilizing method of Linearization, local dynamical properties about equilibria have been investigated. It is shown that every solution of the system is bounded, and equilibrium becomes a globally asymptotically stable if , . It is also shown that every solution of the system converges to . Finally theoretical results are verified numerically.
1. Introduction
The importance of difference equations cannot be overemphasized. These equations model discrete physical phenomena on one hand and on the other hand these integral parts of numerical schemes used to solve differential equations. This widens the applicability of such equations to many branches of scientific knowledge. Discrete dynamical systems are described by difference equations and have applications in many branches of modern science like population dynamics, ecology, psychology, economics, queuing problems, genetics in biology, statistical problems, electrical networks, number theory, neural networks, quanta in radiation, sociology, physics, engineering, economics, probability theory, combinatorial analysis, stochastic time series, geometry, and resource management [1, 2]. Though the study of the global dynamics of higher-order nonlinear difference equations or system of difference equations is a challenging task but it is rewarding. These results pave path towards the development of basic theory of difference equations of higher-order. In recent years several authors have explored the behavior of solution of such difference equations or system of difference equations by studying equilibrium point, local and global dynamics about equilibria, boundedness and persistence, periodicity nature, prime period 2-solution, semicycle analysis, forbidden set, and many more (see [3–19] and references cited therein). For instance, Gibbons et al. [20] have explored the global dynamical properties of following difference equation:where and are real numbers. Çinar [21] has explored the solution of following difference equation:where are real numbers. Amleh et al. [22, 23] have explored boundedness, periodicity nature, and global dynamical properties of following difference equation:where and are real numbers. Amleh et al. [22, 23] have pointed that difference equation depicted in (3) consists of numerous challenging and inspiring special cases having order , that can also arise from the following planer system when it is reduced to a difference equation (see [24]):where and are real numbers. Amleh et al. [22, 23] have also pointed that when one or more allowed to zero then (3) contains 49 special cases in which 19 are riccati, linear, trivial, reducible to riccati, or linear difference equations while remaining 30 are posed to conjectures and open problems. Shojaei et al. [25] have extended the work of Çinar [21] to explore the dynamical properties of following difference equation:where and are real numbers. Later Bajo and Liz [26] have explored the dynamical properties of following difference equation which is special case of (3):where and are real numbers. Zhang et al. [27] have extended the work studied by several authors [21, 25, 26], to explore the dynamical properties of following system of difference equations:where and are real numbers. Motivated from aforementioned studies, our aim is to extend the work studied by [21, 25–27], to explore the global dynamical properties of the following difference equations system:where and are real numbers.
The flow pattern of the remaining paper, which is our main contribution and meaningful addition towards discrete dynamical systems described by the system of difference equations, is as follows: Section 2 deals with the study of existence of equilibria in whereas linearized form of the under consideration system is given in Section 3. Section 4 deals with the study of local dynamical properties about equilibria. Boundeness of the solution of the system is discussed in Section 5 whereas global dynamics about of the system is studied in Section 6. In Section 7, it is proved that every solution of under consideration system converges to . Discussion and simulations are given in the last section.
2. Existence of Equilibria
This section deals with the study of existence of equilibria of the system (8) in . The result about the existence of equilibria can be summarized as follows.
Lemma 1. System (8) has at least one and at most eight equilibria in . More precisely (i)system (8) has a unique boundary equilibrium ;(ii)system (8) has a boundary equilibrium if ;(iii)system (8) has a boundary equilibrium if ;(iv)system (8) has a boundary equilibrium if ;(v)system (8) has a boundary equilibrium if ;(vi)system (8) has a boundary equilibrium if ;(vii)system (8) has a boundary equilibrium if ;(viii)system (8) has a equilibrium if . More specifically, is a unique equilibrium of the system (8) if .
3. Linearized Form of (8)
We have the following map in order to construct the corresponding linearized form of (8):where Moreover about under the map depicted in (9) iswhere .
4. Local Stability of System (8)
Hereafter we will study the local dynamical properties about equilibria of the system (8) by utilizing Theorem 1.5 of [1].
4.1. Local Stability about Boundary Equilibria
This subsection is purely dedicated to the study of local dynamical properties about boundary equilibria: of the system (8), respectively.
Theorem 2. For , the following hold: (i) of the system (8) is a sink if(ii) of the system (8) is unstable if
Proof. (i) We have the following linearized system of (8) about :where Let the eigenvalues of are . Also letdenoting the diagonal matrix whereandBy computing one gets Moreover,Equation (20) then implies thatFrom (17) and (18) one getsSince eigenvalues of are same as , so Thus of (8) is a sink.
(ii) Similarly it is easy to prove that of (8) is unstable if or or .
Theorem 3. of (8) is locally unstable.
Proof. The corresponding linearized form of (8) about is where Again let the eigenvalues of be and the diagonal matrix which is depicted in (16) where (17) holds. On computing one gets In view of (20) and (21) one has which implies that . Also Hence of the system (8) is unstable.
Theorem 4. of (8) is locally unstable.
Proof. The corresponding linearized form of (8) about is where Using same arrangement as in the proof of Theorems 2–3 and compute ; one gets In view of (17), (20), and (21) one has And which implies that . Also Hence of (8) is locally unstable.
Theorem 5. of the system (8) is locally unstable.
Proof. Its proof is similar to Theorems 3–4.
Theorem 6. of (8) is locally unstable.
Proof. The corresponding linearized system is where Again using same arrangements as in the proof of Theorems 2–4 and computing , one gets Further, and Hence of the system (8) is locally unstable.
Theorem 7. and of the system (8) are locally unstable.
Proof. Similar to Theorem 6.
4.2. Local Stability about the Unique Positive Equilibrium
Hereafter local dynamical properties about of (8) are explored. The following result shows that of system (8) is locally unstable.
Theorem 8. of (8) is unstable.
Proof. The corresponding linearized form of (8) about is where Also Moreover and Hence of the system (8) is locally unstable.
5. Boundedness
Theorem 9. Any positive solution of (8) satisfies following inequalities for :AndFinally,
Proof. Obviously (47), (48), and (49) hold for . Now suppose that for inequalities (47), (48), and (49) are true, ,AndFinally,Now, for using (8) one hasAndFinally,
Lemma 10. If the conditions depicted in (12) hold then of (8) is bounded.
Proof. Direct consequence of Theorem 9.
6. Global Stability about the Equilibrium
Now we will explore the global dynamics of (8) about .
Theorem 11. If the conditions depicted in (12) hold then of (8) is globally asymptotically stable.
Proof. Since of (8) is a sink by Theorem 2 and hence it is enough to prove From (8), one gets By induction, one getsFinally if (12) holds then from (58) one gets .
7. Rate of Convergence
Theorem 12. If (12) holds then error vector of every solution of (8) satisfies both of the following asymptotic relations: where is equal to the modulus of one of the eigenvalues of the Jacobian matrix evaluated at .
Proof. Let be an arbitrary solution of (8), , and . To find error term one has from the system (8) and SoSetIn view of (64), (63) becomes where Moreover that is, where as . Thus we have the system [28]:where and Therefore limiting system of error terms takes the following form:Hence (73) is same as linearized system of (8) about . Particularly about it becomes which is same as linearized system of (8) about .
8. Discussion and Numerical Simulations
This proposed work is about the global dynamical properties of a system of difference equations, which is our key finding towards discrete dynamical system. We investigated that discrete-time system (8) has equilibria: and the unique equilibrium point: under restriction to the parameters . By method of linearization, we studied the local dynamical properties about each equilibria and conclusion are presented in Table 1. Further we proved that of (8) is bounded and is globally asymptotically stable if (12) holds. Finally, we proved that every solution of the system converges to . The above obtained theoretical results can also be verified from following numerical simulations. Thus in the remaining section, theoretical results are verified numerically.
Example 1. If then (8) with , respectively, are , the following form is taken:The graphs of of (75) are shown in Figures 1(a), 1(b), and 1(c) while its attractor is shown in Figure 1(d); , if , then all solutions are attracted to . This coincides with the proof of Theorem 11.
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Example 2. If then (8) with , respectively, are , the following form is taken:The graphs of of (76) are shown in Figures 2(a), 2(b), and 2(c) while its attractor is shown in Figure 2(d).
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Example 3. If then (8) with , respectively, are , the following form is taken:The graphs of of (75) are shown in Figures 3(a), 3(b), and 3(c). In this case of (77) is unstable.
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Data Availability
All the data utilized in this article has been included and the sources adopted were cited accordingly.
Disclosure
The authors declare that they got no funding on any part of this research.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was partially supported by the Higher Education Commission of Pakistan.