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Licensed Unlicensed Requires Authentication Published by De Gruyter January 11, 2018

Bifurcation Analysis and Chaos Control in a Second-Order Rational Difference Equation

  • Qamar Din EMAIL logo , A. A. Elsadany and Samia Ibrahim

Abstract

This work is related to dynamics of a second-order rational difference equation. We investigate the parametric conditions for local asymptotic stability of equilibria. Center manifold theorem and bifurcation theory are implemented to discuss the parametric conditions for existence and direction of period-doubling bifurcation and pitchfork bifurcation at trivial equilibrium point. Moreover, the parametric conditions for existence and direction of Neimark–Sacker bifurcation at positive steady state are investigated with the help of bifurcation theory. The chaos control in the system is discussed through implementation of OGY feedback control method. In particular, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control. Finally, numerical simulations are provided to illustrate theoretical results. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.

MSC 2010: 39A28; 39A30; 39A33

Acknowledgements:

The authors thank the anonymous referees for their valuable comments and suggestions leading to improvement of this paper.

References

[1] M. R. S. Kulenović and G. Ladas, Dynamics of second order rational difference equations, Chapman and Hall/CRC, 2002.10.1201/9781420035384Search in Google Scholar

[2] E. Camouzis and G. Ladas, Dynamics of third-order rational difference equations with open problems and conjectures, Chapman and Hall/CRC Press, Boca Raton, 2007.10.1201/9781584887669Search in Google Scholar

[3] E. A. Grove and G. Ladas, Periodicities in nonlinear difference equations, Chapman and Hall/CRC Press, Boca Raton, 2004.10.1201/9781420037722Search in Google Scholar

[4] M. R. S. Kulenović, E. Pilav and E. Silić, Naimark–Sacker bifurcation of a certain second order quadratic fractional difference equation, J. Math. Comput. Sci. 4(6) (2014), 1025–1043.10.1186/1687-1847-2014-68Search in Google Scholar

[5] S. Kalabušić, M. R. S. Kulenović and M. Mehuljić, Global dynamics and bifurcations of two quadratic fractional second order difference equations, J. Comput. Anal. Appl. 21(1) (2016), 132–143.Search in Google Scholar

[6] S. J. Hrustić, M. R. S. Kulenović and M. Nurkanović, Global dynamics and bifurcations of certain second order rational difference equation with quadratic terms, Qual. Theory Dyn. Syst. 15(1) (2016), 283–307.10.1007/s12346-015-0148-xSearch in Google Scholar

[7] M. R. S. Kulenović, S. Moranjkić and Z. Nurkanović, Naimark–Sacker bifurcation of second order rational difference equation with quadratic terms, J. Nonlinear Sci. Appl. 10 (2017), 3477–3489.10.22436/jnsa.010.07.11Search in Google Scholar

[8] R. Zhang and X. Ding, The Neimark–Sacker bifurcation of $$x_{n+1}=\frac{\delta x_{n-2}+x_{n-3}}{A+x_{n-3}}$$, J. Differ. Equ. Appl. 15 (2009), 775–784.10.1080/10236190802357669Search in Google Scholar

[9] Z. He and J. Qiu, Neimark–Sacker bifurcation of a third-order rational difference equation, J. Differ. Equ. Appl. 19 (2013), 1513–1522.10.1080/10236198.2013.764998Search in Google Scholar

[10] S. Kalabušić, M. R. S. Kulenović and M. Mehuljić, Global period-doubling bifurcation of quadratic fractional second order difference equation, Discrete Dyn. Nat. Soc. (2014), Article ID 920410, 13.10.1155/2014/920410Search in Google Scholar

[11] A. M. Amleh, E. Camouzis and G. Ladas, On the dynamics of a rational difference equation, Part I, Int. J. Difference Equ. 3 (2008), 1–35.Search in Google Scholar

[12] A. M. Amleh, E. Camouzis and G. Ladas, On the dynamics of a rational difference equation, Part II, Int. J. Difference Equ. 3 (2008), 195–225.Search in Google Scholar

[13] M. Dehghan, C. M. Kent, R. Mazrooei-Sebdani, N. L. Ortiz and H. Sedaghat, Dynamics of rational difference equations containing quadratic terms, J. Differ. Equ. Appl. 14 (2008), 191–208.10.1080/10236190701565636Search in Google Scholar

[14] M. Dehghan, C. M. Kent, R. Mazrooei-Sebdani, N. L. Ortiz and H. Sedaghat, Monotone and oscillatory solutions of a rational difference equation containing quadratic terms, J. Differ. Equ. Appl. 14 (2008), 1045–1058.10.1080/10236190802332266Search in Google Scholar

[15] C. M. Kent and H. Sedaghat, Global attractivity in a quadratic–linear rational difference equation with delay, J. Differ. Equ. Appl. 15 (2009), 913–925.10.1080/10236190802040992Search in Google Scholar

[16] C. M. Kent and H. Sedaghat, Global attractivity in a rational delay difference equation with quadratic terms, J. Differ. Equ. Appl. 17 (2011), 457–466.10.1080/10236190903049009Search in Google Scholar

[17] H. Sedaghat, Global behaviours of rational difference equations of orders two and three with quadratic terms, J. Differ. Equ. Appl. 15 (2009), 215–224.10.1080/10236190802054126Search in Google Scholar

[18] H. L. Smith, Planar competitive and cooperative difference equations, J. Differ. Equ. Appl. 3 (1998), 335–357.10.1080/10236199708808108Search in Google Scholar

[19] E. Drymonis and G. Ladas, On the global character of the rational system $$x_{n+1}=\frac{\alpha_1}{A_1+B_1 x_n+y_n}$$ and $$y_{n+1}=\frac{\alpha_2+\beta_2 x_n}{A_2+B_2 x_n+C_2 y_n}$$, Sarajev. J. Math. 8 (21) (2012), 293–309.10.5644/SJM.08.2.10Search in Google Scholar

[20] M. Garić-Demirović, M. R. S. Kulenović and M. Nurkanović, Global dynamics of certain homogeneous second order quadratic fractional difference equation, Sci. World J. Math. Anal. (2013), Article ID 210846, 10.10.1155/2013/210846Search in Google Scholar PubMed PubMed Central

[21] M. Garić-Demirović and M. Nurkanović, Dynamics of an anti–competitive two dimensional rational system of difference equations, Sarajev. J. Math. 7 (19) (2011), 39–56.Search in Google Scholar

[22] S. Kalabušić, M. R. S. Kulenović and E. Pilav, Global dynamics of anti–competitive systems in the plane, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 20 (2013), 477–505.Search in Google Scholar

[23] Moranjkić S. and Nurkanović Z., Basins of attraction of certain rational anti–competitive system of difference equations in the plane, Adv. Differ. Equ. (2012), 153.10.1186/1687-1847-2012-153Search in Google Scholar

[24] M. R. S. Kulenović and O. Merino, Global bifurcation for discrete competitive systems in the plane, Discrete Contin. Dyn. Syst. Ser. B 12 (2009), 133–149.10.3934/dcdsb.2009.12.133Search in Google Scholar

[25] M. R. S. Kulenović and O. Merino, Invariant manifolds for competitive discrete systems in the plane, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), 2471–2486.10.1142/S0218127410027118Search in Google Scholar

[26] J. K. Hale and H. Kocak, Dynamics and bifurcations: text in applied mathematics, Springer-Verlag, New York, 1991.10.1007/978-1-4612-4426-4Search in Google Scholar

[27] Y. Kuznetsov, Elements of applied bifurcation theory, 2nd Ed., Springer, NewYork, 1998.Search in Google Scholar

[28] C. Robinson, Dynamical systems: stability, symbolic dynamics and chaos, 2nd Ed., Boca Raton, London, New York, 1999.Search in Google Scholar

[29] S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Springer-Verlag, New York, 2003.Search in Google Scholar

[30] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag, New York, 1983.10.1007/978-1-4612-1140-2Search in Google Scholar

[31] J. Carr, Application of center manifold theory, Springer-Verlag, New York, 1981.10.1007/978-1-4612-5929-9Search in Google Scholar

[32] E. M. Elabbasy, A. A. Elsadany and Y. Zhang, Bifurcation analysis and chaos in a discrete reduced Lorenz system, Appl. Math. Comput. 228 (2014), 184–194.10.1016/j.amc.2013.11.088Search in Google Scholar

[33] E. M. Elabbasy, H. N. Agiza, H. El-Metwally and A. A. Elsadany, Bifurcation analysis, chaos and control in the Burgers mapping, Int. J. Nonlinear Sci. 4(3) (2007), 171–185.Search in Google Scholar

[34] Q. Din, Neimark–Sacker bifurcation and chaos control in Hassell–Varley model, J. Differ. Equations Appl. 23(4) (2017), 741–762.10.1080/10236198.2016.1277213Search in Google Scholar

[35] Q. Din, Ö.A. Gümüş and H. Khalil, Neimark–Sacker bifurcation and chaotic behaviour of a modified Host–Parasitoid model, Z. Naturforsch. A 72(1) (2017), 25–37.10.1515/zna-2016-0335Search in Google Scholar

[36] Q. Din, Complexity and chaos control in a discrete-time prey-predator model, Commun. Nonlinear Sci. Numer. Simul. 49 (2017), 113–134.10.1016/j.cnsns.2017.01.025Search in Google Scholar

[37] Q. Din and U. Saeed, Bifurcation analysis and chaos control in a host–parasitoid model, Math. Meth. Appl. Sci. (2017), doi: 10.1002/mma.4395.Search in Google Scholar

[38] Q. Din, A. A. Elsadany and H. Khalil, Neimark-Sacker bifurcation and chaos control in a fractional-order plant-herbivore model, Discrete Dyn. Nat. Soc. (2017), Article ID 6312964, 1–15.10.1155/2017/6312964Search in Google Scholar

[39] Q. Din, Qualitative analysis and chaos control in a density–dependent host–parasitoid system, Int. J. Dynam. Control (2017), doi: 10.1007/s40435-017-0341-7.Search in Google Scholar

[40] Q. Din, Controlling chaos in a discrete–time prey-predator model with Allee effects, Int. J. Dynam. Control (2017), doi: 10.1007/s40435-017-0347-1.Search in Google Scholar

[41] Q. Din, Global stability and Neimark–Sacker bifurcation of a host-parasitoid model, Int. J. Syst. Sci. 48(6) (2017), 1194–1202.10.1080/00207721.2016.1244308Search in Google Scholar

[42] Q. Din, Global stability of beddington model, Qual. Theor. Dyn. Syst., 16(2) (2017), 391–415.10.1007/s12346-016-0197-9Search in Google Scholar

[43] S. M. Salman, A. M. Yousef and A. A. Elsadany, Stability, bifurcation analysis and chaos control of a discrete predator-prey system with square root functional response, Chaos Solitons Fract. 93 (2016), 20–31.10.1016/j.chaos.2016.09.020Search in Google Scholar

[44] E. Ott, C. Grebogi and J. A. Yorke, Controlling chaos, Phys. Rev. Letters 64(11) (1990), 1196–1199.10.1103/PhysRevLett.64.1196Search in Google Scholar

[45] S. Lynch, Dynamical systems with applications using mathematica, Birkhäuser, Boston, 2007.Search in Google Scholar

[46] A. L. Fradkov and R. J. Evans, Control of chaos: methods and applications in engineering, Annu. Rev. Control. 29 (2005), 33–56.10.1016/j.arcontrol.2005.01.001Search in Google Scholar

[47] X. S. Luo, G. R. Chen, B. H. Wang, et al., Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems, Chaos Solitons Fract. 18 (2004), 775–783.10.1016/S0960-0779(03)00028-6Search in Google Scholar

[48] L-G. Yuan and Q-G. Yang, Bifurcation, invariant curve and hybrid control in a discrete-time predator-prey system, Appl. Math. Model. 39(8) (2015), 2345–2362.10.1016/j.apm.2014.10.040Search in Google Scholar

Received: 2017-3-31
Accepted: 2017-11-13
Published Online: 2018-1-11
Published in Print: 2018-2-23

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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