Time-delayed feedback control for chaotic systems with coexisting attractors

Erxi Zhu; Erxi Zhu

doi:10.3934/math.2024053

AIMS Mathematics

2024, Volume 9, Issue 1: 1088-1102. doi: 10.3934/math.2024053

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Research article

Time-delayed feedback control for chaotic systems with coexisting attractors

Erxi Zhu ^{1,2
,
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1.
College of Internet of Things Engineering, Jiangsu Vocational College of Information Technology, Wuxi 214153, China
2.
College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

Received: 04 October 2023 Revised: 18 November 2023 Accepted: 27 November 2023 Published: 06 December 2023
MSC : 93B52

This study investigated the Hopf bifurcation of the equilibrium point of chaotic systems with coexisting attractors under the time-delayed feedback control. First, the equilibrium point and Hopf bifurcation of chaotic systems with coexisting attractors were analyzed. Second, the chaotic systems were controlled by time-delayed feedback, the transversality condition of Hopf bifurcation at the equilibrium point was discussed, and the time-delayed value of Hopf bifurcation at the equilibrium point was obtained. Lastly, the correctness of the theoretical analysis was verified by using the numerical results.
- coexisting attractors,
- time-delayed feedback control,
- Hopf bifurcation,
- transverse condition
Citation: Erxi Zhu. Time-delayed feedback control for chaotic systems with coexisting attractors[J]. AIMS Mathematics, 2024, 9(1): 1088-1102. doi: 10.3934/math.2024053

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Abstract

This study investigated the Hopf bifurcation of the equilibrium point of chaotic systems with coexisting attractors under the time-delayed feedback control. First, the equilibrium point and Hopf bifurcation of chaotic systems with coexisting attractors were analyzed. Second, the chaotic systems were controlled by time-delayed feedback, the transversality condition of Hopf bifurcation at the equilibrium point was discussed, and the time-delayed value of Hopf bifurcation at the equilibrium point was obtained. Lastly, the correctness of the theoretical analysis was verified by using the numerical results.

References

[1]	K. Cheng, Z. Lu, Y. Zhen, Multi-level multi-fidelity sparse polynomial chaos expansion based on Gaussian process regression, Comput. Meth. Appl. Mech. Eng., 349 (2019), 360–377. https://doi.org/10.1016/j.cma.2019.02.021 doi: 10.1016/j.cma.2019.02.021
[2]	G. C. Wu, D. Baleanu, Discrete chaos in fractional delayed logistic maps, Nonlinear Dyn., 80 (2014), 1697–1703. https://doi.org/10.1007/s11071-014-1250-3 doi: 10.1007/s11071-014-1250-3
[3]	L. Grigoryeva, A. Hart, J. P. Ortega, Chaos on compact manifolds: differentiable synchronizations beyond Takens, Phys. Rev. E, 103 (2020), 062204. https://doi.org/10.1103/PhysRevE.103.062204 doi: 10.1103/PhysRevE.103.062204
[4]	A. Jahangiri, N. K. A. Attari, A. Nikkhoo, Z. Waezi, Nonlinear dynamic response of an Euler-Bernoulli beam under a moving mass-spring with large oscillations, Arch. Appl. Mech., 90 (2020), 1135–1156. https://doi.org/10.1007/s00419-020-01656-9 doi: 10.1007/s00419-020-01656-9
[5]	F. Fotiadis, K. G. Vamvoudakis, Detection of actuator faults for continuous-time systems with intermittent state feedback, Syst. Control Lett., 152 (2021), 104938. https://doi.org/10.1016/j.sysconle.2021.104938 doi: 10.1016/j.sysconle.2021.104938
[6]	T. Xu, J. Xu, X. Zhang, Inertia-free computation efficient immersion and invariance adaptive tracking control for Euler-Lagrange mechanical systems with parametric uncertainties, Adv. Space Res., 66 (2020), 1902–1910. https://doi.org/10.1016/j.asr.2020.07.004 doi: 10.1016/j.asr.2020.07.004
[7]	M. P. Aghababa, No-chatter variable structure control for fractional nonlinear complex systems, Nonlinear Dyn., 73 (2013), 2329–2342. https://doi.org/10.1007/s11071-013-0944-2 doi: 10.1007/s11071-013-0944-2
[8]	X. Song, L. Chen, K. Wang, D. He, Robust time-delay feedback control of vehicular CACC systems with uncertain dynamics, Sensors, 20 (2020), 1775. https://doi.org/10.3390/s20061775 doi: 10.3390/s20061775
[9]	M. Farazmand, Mitigation of tipping point transitions by time-delay feedback control, Chaos, 301 (2019), 013149. https://doi.org/10.1063/1.5137825 doi: 10.1063/1.5137825
[10]	Y. Ding, L. Zheng, R. Yang, Time-delayed feedback control of improved friction-induced model: application to moving belt of particle supply device, Nonlinear Dyn., 100 (2020), 423–434. https://doi.org/10.1007/s11071-020-05523-8 doi: 10.1007/s11071-020-05523-8
[11]	G. Yang, Hopf birurcation of Lorenz-like system about parameter $h$, Mod. Appl. Sci., 4 (2009), 91–95. https://doi.org/10.5539/mas.v4n1p91 doi: 10.5539/mas.v4n1p91
[12]	Y. Li, H. P. Ju, C. Hua, G. Liu, Distributed adaptive output feedback containment control for time-delay nonlinear multiagent systems, Automatica, 127 (2021), 109545. https://doi.org/10.1016/j.automatica.2021.109545 doi: 10.1016/j.automatica.2021.109545
[13]	J. Kengne, Z. T. Njitacke, H. B. Fotsin, Dynamical analysis of a simple autonomous jerk system with multiple attractors, Nonlinear Dyn., 83 (2016), 751–765. https://doi.org/10.1007/s11071-015-2364-y doi: 10.1007/s11071-015-2364-y
[14]	V. T. Pham, C. Volos, S. Jafari, T. Kapitaniak, Coexistence of hidden chaotic attractors in a novel no-equilibrium system, Nonlinear Dyn., 87 (2017), 2001–2010. https://doi.org/10.1007/s11071-016-3170-x doi: 10.1007/s11071-016-3170-x
[15]	B. C. Bao, H. Bao, N. Wang, M. Chen, Q. Wu, Hidden extreme multistability in memristive hyperchaotic system, Chaos Soliton. Fract., 94 (2017), 102–111. https://doi.org/10.1016/j.chaos.2016.11.016 doi: 10.1016/j.chaos.2016.11.016
[16]	B. Bao, W. Ning, X. Quan, H. Wu, Y. Hu, A simple third-order memristive band pass filter chaotic circuit, IEEE Trans. Circuits Syst. II, 64 (2017), 977–981. https://doi.org/10.1109/TCSII.2016.2641008 doi: 10.1109/TCSII.2016.2641008
[17]	C. Li, W. J. C. Thio, J. C. Sprott, H. H. C. Iu, Y. Xu, Constructing infinitely many attractors in a programmable chaotic circuit, IEEE Access, 2018 (2018), 29003–29012. https://doi.org/10.1109/ACCESS.2018.2824984 doi: 10.1109/ACCESS.2018.2824984
[18]	Q. Lai, Z. Wan, P. D. K. Kuate, H. Fotsin, Coexisting attractors, circuit implementation and synchronization control of a new chaotic system evolved from the simplest memristor chaotic circuit, Commun. Nonlinear Sci. Numer. Simul., 89 (2020), 105341. https://doi.org/10.1016/j.cnsns.2020.105341 doi: 10.1016/j.cnsns.2020.105341
[19]	B. Muthuswamy, L. O. Chua, Simplest chaotic circuit, Int. J. Bifurcation Chaos, 20 (2010), 1567–1580. https://doi.org/10.1142/S0218127410027076
[20]	X. Zhang, Y. Lin, Global stabilization of high-order nonlinear time-delay systems by state feedback, Syst. Control Lett., 65 (2014), 89–95. https://doi.org/10.1016/j.sysconle.2013.12.015 doi: 10.1016/j.sysconle.2013.12.015
[21]	C. C. Hua, X. P. Guan, Smooth dynamic output feedback control for multiple time-delay systems with nonlinear uncertainties, Automatica, 28 (2016), 1–8. https://doi.org/10.1016/j.automatica.2016.01.007 doi: 10.1016/j.automatica.2016.01.007
[22]	E. Fridman, U. Shaked, A descriptor system approach to H$\infty$ control oflinear time-delay systems, IEEE Trans. Autom. Control, 47 (2002), 253–270. https://doi.org/10.1109/9.983353 doi: 10.1109/9.983353

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