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一个三时滞生物捕食被捕食系统分岔的混合控制

张丽萍 王惠南 徐敏

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一个三时滞生物捕食被捕食系统分岔的混合控制

张丽萍, 王惠南, 徐敏

Hybrid control of bifurcation in a predator-prey system with three delays

Zhang Li-Ping, Xu Min, Wang Hui-Nan
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  • 针对多时滞捕食被捕食系统的Hopf分岔控制问题,提出一种基于状态反馈和参数调节的混合控制方法,这种混合控制方法可以延迟有害Hopf分岔的发生或使Hopf分岔消失.分析了该控制系统的稳定性和Hopf分岔的存在性,并通过规范型理论和中心流形定理,给出了分岔周期解的稳定性和分岔方向的计算公式.最后通过数值模拟验证了该理论的正确性.
    The problem of Hopf bifurcation control for a predator-prey system with three delays is considered. A new hybrid strategy is proposed to control the Hopf bifurcation, in which the state feedback and parameter perturbation are used to delay the onset of an inherent bifurcation or make the bifurcation disappear. The stability and the existence of bifurcation are researched. In particular, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are derived by using the normal form theory and center manifold theorem. Finally, numerical simulation results confirm that the new hybrid controller is efficient in controlling Hopf bifurcation.
    • 基金项目: 南京航空航天大学基本科研业务费专项科研项目(批准号:NS2010112)资助的课题.
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    May R M 1973 Ecology 54 315

    [2]

    Sarker R, Petrovskii S, Biswas M, Gupta A, Chattopadhyay J 2006 Ecol Model 193 589

    [3]

    Luo X S, Chen G R, Wang B H 2003 Chaos Solit. Fract. 18 775

    [4]

    Liu Z R, Chung K W 2005 Int. J. Bifurcation Chaos 15 3895

    [5]

    Sun B D 2005 Principle of Automatic Control (Beking:China Machine Press) (in Chinese)[孙炳达 2005 自动控制原理 (北京: 机械工业出版社)]

    [6]

    Faria T 2001 Math. Anal. Appl. 254 433

    [7]

    Hassard B D, Azarinoff N D K, Wan Y H 1981 Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge)

    [8]

    Freedman H I, Rao V S H 1986 SIAM J. Appl. Math. 46 552

    [9]

    Hale J K 1977 Theory of Functional Differential Equations (NewYork: Springer)

    [10]

    He X 1996 Math. Anal. Appl. 198 355

    [11]

    Mao Z S, Zhao H Y 2007 Phys. Lett. A 364 38

    [12]

    Zhao H Y, Chen L, Mao Z S 2009 Nonlinear Analysis: Real World Applications 9 663

    [13]

    Song Y L, Wei J J 2005 Math. Anal. Appl. 301 1

    [14]

    Yan X P, Li W T 2006 Appl. Math. Comput. 177 427

    [15]

    Mao Z S, Zhao H Y, Wang X F 2007 Physica D 234 11

    [16]

    Yan X P, Zhang C H 2008 Nonlinear Anal. 9 114

    [17]

    Ruan S G, Wei J J 2003 Dynamics Continuous, Discrete Impulsive Systems Ser. A: Math. Anal. 10 863

  • [1]

    May R M 1973 Ecology 54 315

    [2]

    Sarker R, Petrovskii S, Biswas M, Gupta A, Chattopadhyay J 2006 Ecol Model 193 589

    [3]

    Luo X S, Chen G R, Wang B H 2003 Chaos Solit. Fract. 18 775

    [4]

    Liu Z R, Chung K W 2005 Int. J. Bifurcation Chaos 15 3895

    [5]

    Sun B D 2005 Principle of Automatic Control (Beking:China Machine Press) (in Chinese)[孙炳达 2005 自动控制原理 (北京: 机械工业出版社)]

    [6]

    Faria T 2001 Math. Anal. Appl. 254 433

    [7]

    Hassard B D, Azarinoff N D K, Wan Y H 1981 Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge)

    [8]

    Freedman H I, Rao V S H 1986 SIAM J. Appl. Math. 46 552

    [9]

    Hale J K 1977 Theory of Functional Differential Equations (NewYork: Springer)

    [10]

    He X 1996 Math. Anal. Appl. 198 355

    [11]

    Mao Z S, Zhao H Y 2007 Phys. Lett. A 364 38

    [12]

    Zhao H Y, Chen L, Mao Z S 2009 Nonlinear Analysis: Real World Applications 9 663

    [13]

    Song Y L, Wei J J 2005 Math. Anal. Appl. 301 1

    [14]

    Yan X P, Li W T 2006 Appl. Math. Comput. 177 427

    [15]

    Mao Z S, Zhao H Y, Wang X F 2007 Physica D 234 11

    [16]

    Yan X P, Zhang C H 2008 Nonlinear Anal. 9 114

    [17]

    Ruan S G, Wei J J 2003 Dynamics Continuous, Discrete Impulsive Systems Ser. A: Math. Anal. 10 863

计量
  • 文章访问数:  5032
  • PDF下载量:  904
  • 被引次数: 0
出版历程
  • 收稿日期:  2008-08-06
  • 修回日期:  2010-06-24
  • 刊出日期:  2011-01-15

一个三时滞生物捕食被捕食系统分岔的混合控制

  • 1. (1)南京航空航天大学理学院,南京 210016;南京航空航天大学自动化学院,南京 210016; (2)南京航空航天大学信息科学与技术学院,南京 210016; (3)南京航空航天大学自动化学院,南京 210016
    基金项目: 

    南京航空航天大学基本科研业务费专项科研项目(批准号:NS2010112)资助的课题.

摘要: 针对多时滞捕食被捕食系统的Hopf分岔控制问题,提出一种基于状态反馈和参数调节的混合控制方法,这种混合控制方法可以延迟有害Hopf分岔的发生或使Hopf分岔消失.分析了该控制系统的稳定性和Hopf分岔的存在性,并通过规范型理论和中心流形定理,给出了分岔周期解的稳定性和分岔方向的计算公式.最后通过数值模拟验证了该理论的正确性.

English Abstract

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