一类具时滞的超混沌系统的稳定性及控制分析
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摘要
研究一类新的超混沌系统的动力学性质,通过时滞反馈控制方法实现对该系统混沌控制的目的。分析具时滞的超混沌系统的平衡点的稳定性和Hopf分支的存在性,利用多时间尺度方法推导出具时滞超混沌系统Hopf分支的规范型,对极坐标下的规范型给出判断Hopf分支方向及分支周期解稳定性的判别准则,从理论上实现将混沌系统控制成为稳定状态。数值仿真结果验证了理论分析的正确性。
We study the dynamic behaviors of a new hyperchaos system. The aim of chaos control can be achieved by using delayed feedback control. We analyze the stability of the equilibrium and the existence of the Hopf bifurcation of the new system with delayed feedback. Furthermore, we deduce the normal form of Hopf bifurcation by using the multiple time scales method and investigate the criterion associated with the direction of Hopf bifurcation and the stability of periodic solution with the normal forms of polar coordinate. The hyperchaos system is controlled to be stable state in theory. Numerical examples are given to verify the theoretical results.
We study the dynamic behaviors of a new hyperchaos system. The aim of chaos control can be achieved by using delayed feedback control. We analyze the stability of the equilibrium and the existence of the Hopf bifurcation of the new system with delayed feedback. Furthermore, we deduce the normal form of Hopf bifurcation by using the multiple time scales method and investigate the criterion associated with the direction of Hopf bifurcation and the stability of periodic solution with the normal forms of polar coordinate. The hyperchaos system is controlled to be stable state in theory. Numerical examples are given to verify the theoretical results.
引文
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[13]程贝贝,胡志兴,廖福成.具有Beddington-DeAngelis发生率和免疫损害项的带时滞的病毒感染模型的稳定性分析 [J].黑龙江大学自然科学学报,2016,33 (3):281-290.
[14]VERDUGO A.Mathematical analysis of a biochemical oscillator with delay [J].Journal of Computational & Applied Mathematics,2016,291:66-75.
[15]DING Y T.Dynamic analysis of nonlinear variable frequency water supply system with time delay [J].Nonlinear Dynamics,2017,1:1-14.
[16]孙方方,雷银彬.一种新超混沌系统的反馈控制方法研究 [J].西南民族大学学报 (自然科学版),2015,6:27-31.
[17]DEJESUS E X,KAUFMAN C.Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations [J].Physics Letters A,1987,35(12):5288-5290.
[18]NAYFEH H A.Introduction to Perturbation Techniques [M].Hoboken:Wiley,1981.
[2]秦显荣.超混沌系统的混沌控制研究 [D].大庆:东北石油大学,2014.
[3]SUN K,LIU X,ZHU C,SPROTT J C.Hyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system [J].Nonlinear Dynamics,2012,69(3):1383-1391.
[4]AGHABABA M P,KHANMOHAMMADI S,ALIZADEH G.Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique [J].Applied Mathematical Modelling,2011,35(6):3080-3091.
[5]YAU H T,CHEN C L.Chaos control of Lorenz systems using adaptive controller with input saturation [J].Chaos Solitons & Fractals,2007,34(34):1567-1574.
[6]MATOUK A E.Stability conditions,hyperchaos and control in a novel fractional order hyperchaotic system [J].Physics Letters A,2009,373(25):2166-2173.
[7]DING Y T,JIANG W H,WANG H B.Delayed feedback control and bifurcation analysis of Rossler chaotic system [J].Nonlinear Dynamics,2010,61(4):707-715.
[8]李卫东,王秀岩.混沌控制综述 [J].自动化技术与应用,2009,28(1):1-5.
[9]YU W G.Stabilization of three-dimensional chaotic systems via single state feedback controller [J].Physics Letters A,2010,374(13-14):1488-1492.
[10]ZHANG J X,TANG W S.Control and synchronization for a class of new chaotic systems via linear feedback [J].Nonlinear Dynamics,2009,58(4):675-686.
[11]ZHANG X B,ZHU H L,YAO H X.Analysis of a new three-dimensional chaotic system [J].Nonlinear Dynamics,2012,67(1):335-343.
[12]孙丞,孙鹤旭,刁心薇.一类非齐次高阶非线性系统的连续反馈控制设计 [J].自动化学报,2014,40(1):149-155.
[13]程贝贝,胡志兴,廖福成.具有Beddington-DeAngelis发生率和免疫损害项的带时滞的病毒感染模型的稳定性分析 [J].黑龙江大学自然科学学报,2016,33 (3):281-290.
[14]VERDUGO A.Mathematical analysis of a biochemical oscillator with delay [J].Journal of Computational & Applied Mathematics,2016,291:66-75.
[15]DING Y T.Dynamic analysis of nonlinear variable frequency water supply system with time delay [J].Nonlinear Dynamics,2017,1:1-14.
[16]孙方方,雷银彬.一种新超混沌系统的反馈控制方法研究 [J].西南民族大学学报 (自然科学版),2015,6:27-31.
[17]DEJESUS E X,KAUFMAN C.Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations [J].Physics Letters A,1987,35(12):5288-5290.
[18]NAYFEH H A.Introduction to Perturbation Techniques [M].Hoboken:Wiley,1981.