Stability and Hopf bifurcation analysis of a chaotic system using time-delayed feedback control method

Document Type : Research Paper

Authors

1 Department of Mathematics, Baghlan University, Pol-e-Khomri, Baghlan - Afghanistan

2 Department of Applied Mathematics, Shahrekord University, Shahrekord, Iran

Abstract

In this paper, we study the effect of delayed feedback on the dynamics of a three-dimensional chaotic dynamical system and stabilize its chaotic behavior and control the respective unstable steady state. We derive an explicit formula in which a Hopf bifurcation occurs under some analytical conditions. Then the existence and stability of the Hopf bifurcation are analyzed by considering the time delay $ \tau $ as a bifurcation parameter. Furthermore, by numerical calculation and appropriate ascertaining of both the feedback strength $ K $ and time delay $ \tau $, we find certain threshold values of time delay at which an unstable equilibrium of the considered system is successfully controlled. Finally, we use numerical simulations to examine the derived analytical results and reveal more dynamical behaviors of the system.

Keywords

References

[1] A. Elsonbaty, A.A. Elsadany, Bifurcation analysis of chaotic geomagnetic  eld model, Chaos, Solitons and Fractals 103, (2017) 325{335.
[2] A. Gjurchinovski, T. Sandev, V. Urumov, Delayed feedback control of fractional-order chaotic systems, arXiv:1005.2899v2 [physics.gen-ph] (2011) 1-17.
[3] B. Naderi, H. Kheiri, Exponential synchronization of chaotic system and application in secure communication, Optik 127, (2016) 2407{2412.
[4] B. Li, X. Zhou, Y. Wang, Combination synchronization of three di erent fractionalorder delayed chaotic systems, Complexity (2019) 1{9.
[5] C. Li, H. Li, Y. Tong, Analysis of a novel three-dimensional chaotic system, Optik 124, (2013) 1516{1522.
[6] E. Ott, C. Grebogi, J.A. Yorke, Controlling chaos, Phys Rev Lett 64, (11) (1990) 1196{1199.
[7] F. Mohabati, M. R. Molaei, T. Waezizadeh, A dynamical model and bifurcation analysis for glucagon and glucose regulatory system, Journal of Information and Optimization Sciences (2019) 1{29.
[8] F. Khellat, Delayed feedback control of Bao Chaotic System based on Hopf bifurcation analysis, Journal of Engineering Science and Technology Review 8, (2) (2015) 7{11.
[9] G. M. Mahmoud , A. A. Arafa, T. M. Abed-Elhameed, E. E. Mahmoud, Chaos control of integer and fractional orders of chaotic Burke{Shaw system using time delayed feedback control, Chaos, Solitons and Fractals 104, (2017) 680{692.
[10] H. Zhao, Y. Lin, Y. Dai, Bifurcation analysis and control of chaos for a hybrid ratiodependent three species food chain, Applied Mathematics and Computation 218, (2011) 1533{1546.
[11] H. Zhao, Y. Sun, Z. Wang, Control of Hopf bifurcation and chaos in a delayed Lotka-Volterra predator-prey system with time-delayed feedbacks, Abstract and Applied Analysis (2014) 1{11.
[12] J. Yang, E. Zhang, M. Liu, Bifurcation analysis and chaos control in a modi ed  nance system with delayed feedback, International Journal of Bifurcation and Chaos 26, (6) (2016) 1{14.
[13] K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys Lett A 170, (6) (1992) 421{428.
[14] K. Pyragas, V. Pyragas, H. Benner, Delayed feedback control of dynamical systems at a subcritical Hopf bifurcation, PHYSICAL REVIEW E 70, (2004) 1{4.
[15] M. Ababneh, A new four-dimensional chaotic attractor, Ain Shams Engineering Journal 9, (2018) 1849{1854.
[16] M. Xiao, J. Cao, Bifurcation analysis and chaos control for lu system with delayed feedback, International Journal of Bifurcation and Chaos 17, (12) (2007) 4309{4322.
[17] P. P. Singh, J. P. Singh, M. Borah, B. K. Roy, On the construction of a new chaotic system, IFAC-PapersOnLine 49, (1) (2016) 522{525.
[18] R. Rakkiyappan, K. Udhayakumar, G. Velmurugan, J. Cao, A. Alsaedi, Stability and Hopf bifurcation analysis of fractional-order complex-valued neural networks with time delays, Advances in Di erence Equations 225, (2017) 1-25.
[19] S. Wang, S. He, A. Yousefpour , H. Jahanshahi, R. Repnik M. Perc, Chaos and complexity in a fractional-order  nancial system with time delays, Chaos, Solitons and Fractals, 131, (2020) 109521.
[20] S. B. Bhalekar, V. D. Gejji, A new chaotic dynamical system and its synchronization, In Proceedings of the International Conference on Mathematical Sciences in honor of Prof. A. M. Mathai (2011) 3{5.
[21] S.G. Ruan, J.J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, J. Math. Appl. Med. Biol. 18, (1) (2001) 41{52.
[22] S.G. Ruan, J.J. Wei, On the zero of some transcendential functions with applications to stability of delay di erential equations with two delays, Dyn. Cont. Discrete Impulsive Syst. Ser. A 10, (6) (2003) 863{874.
[23] U. E. Kocamaz, A. Goksu, H. Taskn, Y. Uyaroglu, Control of chaotic two-predator one-prey model with single state control signals, Journal of Intelligent Manufacturing (2020) 1{10.
[24] X. Guan, G. Feng, C. Chen, G. Chen, A full delayed feedback controller design method for time-delay chaotic systems, Physica D 227, (2007) 36{42.
[25] Y. Song, J. Wei, Bifurcation analysis for Chen's system with delayed feedback and its application to control of chaos, Chaos, Solitons and Fractals 22, (2004) 75{91.
[26] Y. Feng, Z. Wei, Delayed feedback control and bifurcation analysis of the generalized Sprott B system with hidden attractors, Eur. Phys. J. Special Topics 224, (2015) 1619{1636.
[27] Y. Ding, W. Jiang, H.Wang, Delayed feedback control and bifurcation analysis of Rossler chaotic system, Nonlinear Dyn. 61, (2010) 707{715.
[28] Z. Wang, W. Sun, Z. Wei, S. Zhang, Dynamics and delayed feedback control for a 3D jerk system with hidden attractor, Nonlinear Dyn (2015) 1{12.
Journal of Mahani Mathematical Research
Volume 12, Issue 1 - Serial Number 24
January 2023
Pages 183-195

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History

  • Receive Date: 15 March 2022
  • Revise Date: 30 June 2022
  • Accept Date: 14 July 2022

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