Stability analysis of a fractional-order Rayleigh system with time-delayed feedback

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Journal of Vibration and Shock ›› 2023, Vol. 42 ›› Issue (2) : 1-6.

CHEN Jufeng1,2，SHEN Yongjun1,3，ZHANG Jing4，LI Xianghong2，WANG Xiaona5

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Abstract

The stability and existence conditions of Hopf bifurcation of a commensurate Rayleigh system with time-delayed feedback are studied. Firstly, the necessary and sufficient conditions for the asymptotic stability of the equilibrium point of fractional-order Rayleigh system with linear velocity feedback are obtained, and it is found that the conditions are not only related to the feedback gain, but also to the fractional order. Secondly, regarding time delay as a bifurcation parameter, the stability of the commensurate fractional-order Rayleigh system with time-delayed feedback is investigated based on the characteristic equation. Under some conditions, the critical value of time delay is calculated. The equilibrium point is stable when the parameter is less than the critical value and will be unstable if the parameter is greater than it. Moreover, the conditions for the occurrence of Hopf bifurcation are obtained. Finally, choosing three typical system parameters, some numerical simulations are carried out to verify the correctness of the obtained theoretical results.

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fractional-order Rayleigh system / time delay / stability / Hopf bifurcation

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CHEN Jufeng1,2，SHEN Yongjun1,3，ZHANG Jing4，LI Xianghong2，WANG Xiaona5. Stability analysis of a fractional-order Rayleigh system with time-delayed feedback[J]. Journal of Vibration and Shock, 2023, 42(2): 1-6

References

[1] Van der Pol B, Van der mark J. Frequency Demultiplication[J]. Nature, 1927, 120(3019):363.
[2] Kaplan B Z, Gabay I, Sarafian G, et al. Biological applications of the "Filtered" Van der Pol oscillator[J]. Journal of the Franklin Institute, 2008, 345(3):226-232.
[3] Siewe M S, Cao H , Miguel A.F. Sanjuán. On the occurrence of chaos in a parametrically driven extended Rayleigh oscillator with three-well potential [J]. Chaos Solitons & Fractals, 2009, 41(2):772-782.
[4] Trovato A, Kumar A, Erlicher S. Stability analysis of entrained solutions of the non-autonomous modified hybrid van der Pol/Rayleigh oscillator: theory and application to pedestrian modeling [J]. Annals of Solid & Structural Mechanics, 2014, 6(1-2):1-16.
[5] Sinelshchikov D I. Linearizability conditions for the Rayleigh-like oscillators[J]. Physics Letters A, 2020, 384(26):126655.
[6] Van Horssen W T. An asymptotic theory for a class of initial-boundary value problems for weakly nonlinear wave equations[J]. Siam Journal on Applied Mathematics, 1988, 48(6):1227-1243.
[7] Hamdi M, Belhaq M. Self-excited vibration control for axially fast excited beam by a time delay state feedback[J]. Chaos Solitons & Fractals, 2009, 41(2):521-532.
[8] U H Hegazy. Dynamics and control of self-sustained electromechanical seismographs with time varying stiffness. Mecanica 44, 355-368[J]. Meccanica, 2009, 44(4):355-368.
[9] Kitiokwuimy C A K, Woafo P. Dynamics, chaos and synchronization of self-sustained electromechanical systems with clamped-free flexible arm[J]. Nonlinear Dynamics, 2008, 53(3):201-213.
[10] Kwuimy C A K, Nana B, Woafo P. Experimental bifurcations and chaos in a modified self-sustained macro electromechanical system[J]. Journal of Sound and Vibration, 2010, 329(15):3137-3148.
[11] Podlubny I. Fractional differential equations[M]. San Diego. Academic press, 1999.
[12] Petras I. Fractional-order nonlinear systems: modeling, analysis and simulation[M]. Beijing: Higher Education Press, 2011.
[13] Oldham K, Spanier J. The fractional calculus theory and applications of differentiation and integration to arbitrary order[M]. Elsevier, 1974.
[14] Koeller R C. Applications of fractional calculus to the theory of viscoelasticity[J]. Transactions of the Asme Journal of Applied Mechanics, 1984, 51(2):299-307.
[15] Hilfer R. Application of fractional calculus in physics[M]. Singapore: World Scientific, 2000.
[16] Wang Z H, Hu H Y. Stability switches of time-delayed dynamic system with unknown parameters[J]. Journal of Sound and Vibration, 2000, 233 (2): 215-233 .
[17] Li C P, Peng G J. Chaos in Chen's system with a fractional order[J]. Chaos, Solitons &.Fractals, 2004, 22 (2): 443-450.
[18] 申永军, 杨绍普, 邢海军. 含分数阶微分的线性单自由度振子的动力学分析[J]. 物理学报, 2012, 61(11):110505.
Shen Yongjun, Yang Shaopu, Xing Haijun. Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative[J]. Acta Physica Sinica, 2012, 61(11):110505.
[19] Du M L, Wang Z H, Hu H Y. Measuring memory with the order of fractional derivative [J]. Scientific Reports, 2013, 3.
[20] Yang Y G, Xu W, Gu X D, et al. Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise[J]. Chaos, Solitons & Fractals, 2015, 77: 190-204
[21] Xu Y, Li Y G, Liu D. A method to stochastic dynamical systems with strong nonlinearity and fractional damping[J]. Nonlinear Dynamics. 2016, 83 (4): 2311-2321.
[22] Chen L C, Zhu W Q. Stochastic response of fractional-order van der Pol oscillator[J]. Theoretical & Applied Mechanics Letters, 2014, 4 (1): 013010.
[23] Chen J F, Li X H, Tang J H, et al. Primary resonance of van der Pol oscillator under fractional-order delayed feedback and forced excitation[J]. Shock and Vibration, 2017, 5975329(1-9).
[24] 王晓娜, 申永军, 张娜,等. 含分数阶微分项的Van Del Pol振子的动力学分[J].振动与冲击, 2020, 39(20):91-96.
Wang Xiaona, Shen Yongjun, Zhang Na, et al. Dynamical analysis of Van Del Pol oscillator with fractional-order derivative [J].Journal of Vibration and Shock, 2020, 39(20):91-96.
[25] Tavazoei M S, Haeri M, Attari M, et al. More detail on analysis of fractional-order van der Pol oscillator[J], Journal of Vibration and Control, 2009, 15(6): 803-819.
[26] Shen Y J, Wei P, Yang S P. Primary resonance of fractional-order van der Pol oscillator[J]. Nonlinear Dynamics, 2014, 77: 1629-1642.
[27] Zhang Y L, Luo M K. Fractional Rayleigh-Duffing-like system and its synchronization[J]. Nonlinear Dynamics, 2012, 70(2): 1173-1183.
[28] Zhang Y L, Li C Q. Fractional modified Duffing–Rayleigh system and its synchronization[J]. Nonlinear Dynamics, 2017, 88(4):3023-3041.
[29] Zhang R R, Xu W, Yang G D, et al. Response of a Duffing-Rayleigh system with a fractional derivative under Gaussian white noise excitation[J]. Chinese Physics B, 2015, 24(2):020204.
[30] Xiao M, Jiang G P, Zheng W X, et al. Bifurcation control of a fractional-order van der Pol oscillator based on the state feedback[J]. Asian Journal of Control, 2015, 17(5): 1756-1766.
[31] 王在华, 胡海岩. 时滞动力系统的稳定性与分岔:从理论走向应用[J].力学进展, 2013, 43(1): 3-20.
Wang Zaihua, Hu Haiyan. Stability and bifurcation of time-delayed dynamical systems: From theory to application [J]. Advances in Mechanics, 2013, 43(1): 3-20.
[32] 蔡国平, 陈龙祥. 时滞反馈控制及其实验[M]. 北京:科学出版社, 2017.
Cai Guoping, Chen Longxiang. Delayed feedback control and its experiments [M]. Beijing: Science Press, 2017.
[33] Deng W H, Li C P, Lü J H. Stability analysis of linear fractional differential system with multiple time delays[J], Nonlinear Dynamics. 2007, 48 (4): 409-416.
[34] Li X L, Wei J J. On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays[J], Chaos, Solitons and Fractals, 2005, 26 (2) : 519-526.