Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2019, Article ID 9797242, 14 pages
https://doi.org/10.1155/2019/9797242
Research Article

Global Dynamics of a 3 × 6 System of Difference Equations

Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan

Correspondence should be addressed to A. Q. Khan; moc.liamg@1nahkreedaqludba

Received 4 April 2019; Revised 24 May 2019; Accepted 8 June 2019; Published 1 July 2019

Academic Editor: Douglas R. Anderson

Copyright © 2019 S. M. Qureshi and A. Q. Khan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [319] 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, 2527], 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 23 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 34.

Theorem 6. of (8) is locally unstable.

Proof. The corresponding linearized system is where Again using same arrangements as in the proof of Theorems 24 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.

Table 1: Number of Equilibria along their qualitative behavior of (8).

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.

Figure 1: Phase portrait of (75).

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).

Figure 2: Phase portrait of (76).

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.

Figure 3: Phase portrait of (77).

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.

References

  1. H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, vol. 15, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  2. V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Application, Kluwer Academic, Dordrecht, The Netherlands, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  3. E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. View at MathSciNet
  4. M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman & Hall, CRC, London, UK, 2002. View at MathSciNet
  5. E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations: With Open Problems and Conjectures, Chapman and Hall/HRC, Boca Raton, Fla, USA, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  6. A. E. Hamza and R. Khalaf-Allah, “Global behavior of a higher order difference equation,” Journal of Mathematics and Statistics, vol. 3, no. 1, pp. 17–20, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. R. Khalaf-Allah, “Asymptotic behavior and periodic nature of two difference equations,” Ukrainian Mathematical Journal, vol. 61, no. 6, pp. 834–838, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  8. V. L. Kocic and G. Ladas, “Global attractivity in a second-order nonlinear difference equation,” Journal of Mathematical Analysis and Applications, vol. 180, no. 1, pp. 144–150, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  9. M. R. Kulenović, G. Ladas, and N. R. Prokup, “A rational difference equation,” Computers & Mathematics with Applications, vol. 41, no. 5-6, pp. 671–678, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  10. Q. Zhang, W. Zhang, Y. Shao, and J. Liu, “On the system of higher-order rational difference equations,” International Scholarly Research Notices, vol. 2014, Article ID 760502, 5 pages, 2014. View at Google Scholar
  11. S. Kalabušić, M. R. S. Kulenović, and E. Pilav, “Global dynamics of a competitive system of rational difference equations in the plane,” Advances in Difference Equations, vol. 2009, Article ID 132802, 30 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  12. S. Kalabušić, M. R. S. Kulenović, and E. Pilav, “Dynamics of a two-dimensional system of rational difference equations of Leslie-Gower type,” Advances in Difference Equations, vol. 2011, article 29, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  13. S. Kalabušić, M. R. Kulenović, and E. Pilav, “Multiple attractors for a competitive system of rational difference equations in the plane,” Abstract and Applied Analysis, vol. 2011, Article ID 295308, 35 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  14. M. Garić-Demirović, M. R. S. Kulenović, and M. Nurkanović, “Global behavior of four competitive rational systems of difference equations in the plane,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 153058, 34 pages, 2009. View at Publisher · View at Google Scholar
  15. E. M. Elsayed, “Solutions of rational difference systems of order two,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 378–384, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. E. M. Elsayed, “Behavior and expression of the solutions of some rational difference equations,” Journal of Computational Analysis and Applications, vol. 15, no. 1, pp. 73–81, 2013. View at Google Scholar · View at MathSciNet · View at Scopus
  17. A. Q. Khan and M. N. Qureshi, “Global dynamics of a competitive system of rational difference equations,” Mathematical Methods in the Applied Sciences, vol. 38, no. 18, pp. 4786–4796, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. R. DeVault, G. Ladas, and S. W. Schultz, “On the recursive sequence