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一类随机van der Pol系统的Hopf 分岔研究

马少娟

一类随机van der Pol系统的Hopf 分岔研究

马少娟
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  • 研究了一类随机van der Pol 系统的Hopf分岔行为.首先根据Hilbert空间的正交展开理论,含有随机参数的van der Pol系统被约化为等价确定性系统,然后利用确定性分岔理论分析了等价系统的Hopf分岔,得出了随机van der Pol 系统的Hopf 分岔临界点,探究了随机参数对系统Hopf分岔的影响.最后利用数值模拟验证了理论分析结果.
    • 基金项目: 国家自然科学基金(批准号: 10872165,10972181,11002001), 国家民族事务委员会科研基金 (批准号08XBEO)及宁夏回族自治区高校科研基金(批准号: 2008JY007) 资助的课题.
    [1]

    Van der Pol B 1927 Phil. Mag. 7 3

    [2]

    Venkatasubramanian V 1994 IEEE Trans. Circuits System I 41 765

    [3]

    Buonomo A 1998 SIAM J. Appl. Math. 59 156

    [4]

    Qin Q, Gong D, Li R, Wen X 1989 Phys. Lett. A 141 412

    [5]

    Parlitz U, Lauterborn W 1987 Phys. Rev. A 36 1428

    [6]

    Mettin R, Parlitz U, Lauterborn W 1993 International Journal of Bifurcation and Chaos 36 1529

    [7]

    Xu J X, Jiang J 1996 Chaos, Solitons and Fractals 7 3

    [8]

    Liao X, Wong K, Wu Z 2001 Nonlinear Dynamics 26 23

    [9]

    Leung H K 1998 Physica A 254 146

    [10]

    Shinozuka M 1972 Journal of the Engineerring Mechanics Division ASCE 98 1433

    [11]

    Ghamem R, Spans P 1991 Stochastic finite element: a spectral approach. (Berlin: Springer)

    [12]

    Jensen H, Iwan W D 1992 ASCE. Eng. Mech. 118 1012

    [13]

    Xiu D B, Karniadakis G E 2002 Computer Methods in Applied Mechanics and Engineering 191 4927

    [14]

    Le Matre O P, Najm H N, Ghanem R G, Knio O M 2004 Journal of Computational Physics 197 502

    [15]

    Wan X L, Karniadakis G E 2005 Journal of Computational Physics 209 617

    [16]

    Pulch R 2009 Applied Numerical Mathematics 59 2610

    [17]

    Li J 1996 Stochastic Structural System-Analysis and Modeling (Beijing: Science Press)(in Chinese)[李 杰1996 随机结构系统-分析与建模 (北京: 科学出版社)]

    [18]

    Fang T, Leng X L, Song C Q 2003 J. Sound Vib. 226 198

    [19]

    Leng, X L, Wu C L, Ma X P, Meng G, Fang T 2005 Nonlinear Dynamics 42 185

    [20]

    Wu C L, Ma S J, Sun Z K, Fang T 2006 Acta Phys. Sin. 55 6253 (in Chinese)[吴存利、马少娟、孙中奎、方 同 2006 物理学报 55 6253]

    [21]

    Ma S J, Xu W, Li W, Jin Y F 2005 Acta Phys. Sin. 54 3508 (in Chinese)[马少娟、徐 伟、李 伟、 靳艳飞 2005 物理学报 54 3508]

    [22]

    Ma S J, Xu W, Jin Y F, Li W, Fang T 2007 Commumications in Nonlinear Science and Numerical Simulation 12 366

    [23]

    Ma S J, Xu W, Li W, Fang T 2006 Chin. Phys. 15 1231

    [24]

    Ma S J, Xu W, Li W 2006 Acta Phys. Sin. 55 4013 (in Chinese)[马少娟、徐 伟、李 伟 2006 物理学报 55 4013]

    [25]

    Sun X J, Xu W, Ma S J 2006 Acta Phys. Sin. 55 610 (in Chinese) [孙小娟、徐 伟、马少娟 2006 物理学报 55 610]

    [26]

    Xu W, Ma S J, Xie W X 2008 Chin. Phys. B 17 857

    [27]

    Liu B C 2004 Functional analysis (Beijing: Science Press) (in Chinese)[刘炳初 2004 泛函分析 (北京:科学出版社)]

    [28]

    Borwein P, Erdélyi T 1995 Polynomials and Polynomial Inequality(New York: Springer)

    [29]

    Liu S K, Liu S D 1988 Special Function (Beijing: China Meteorological Press) (in Chinese) [刘式适、刘式达 1988 (特殊函数, 北京:气象出版社)]

    [30]

    Kamerich E 1999 A Guide to Maple (New York: Springer).

    [31]

    Guckenheimer J, Holmes P J 1983 Nonlinear oscillators, Dynamical system and bifurcation of vector fields (New York: Spring-Verlag)

    [32]

    Hassard B, Kazarinoff N, Wan Y 1981 Theory and application of Hopf bifurcation (Cambridge: Cambridge University Press)

    [33]

    Shen J, Jing Z J 1993 Acta Mathematics Application Sinica 11 79

  • [1]

    Van der Pol B 1927 Phil. Mag. 7 3

    [2]

    Venkatasubramanian V 1994 IEEE Trans. Circuits System I 41 765

    [3]

    Buonomo A 1998 SIAM J. Appl. Math. 59 156

    [4]

    Qin Q, Gong D, Li R, Wen X 1989 Phys. Lett. A 141 412

    [5]

    Parlitz U, Lauterborn W 1987 Phys. Rev. A 36 1428

    [6]

    Mettin R, Parlitz U, Lauterborn W 1993 International Journal of Bifurcation and Chaos 36 1529

    [7]

    Xu J X, Jiang J 1996 Chaos, Solitons and Fractals 7 3

    [8]

    Liao X, Wong K, Wu Z 2001 Nonlinear Dynamics 26 23

    [9]

    Leung H K 1998 Physica A 254 146

    [10]

    Shinozuka M 1972 Journal of the Engineerring Mechanics Division ASCE 98 1433

    [11]

    Ghamem R, Spans P 1991 Stochastic finite element: a spectral approach. (Berlin: Springer)

    [12]

    Jensen H, Iwan W D 1992 ASCE. Eng. Mech. 118 1012

    [13]

    Xiu D B, Karniadakis G E 2002 Computer Methods in Applied Mechanics and Engineering 191 4927

    [14]

    Le Matre O P, Najm H N, Ghanem R G, Knio O M 2004 Journal of Computational Physics 197 502

    [15]

    Wan X L, Karniadakis G E 2005 Journal of Computational Physics 209 617

    [16]

    Pulch R 2009 Applied Numerical Mathematics 59 2610

    [17]

    Li J 1996 Stochastic Structural System-Analysis and Modeling (Beijing: Science Press)(in Chinese)[李 杰1996 随机结构系统-分析与建模 (北京: 科学出版社)]

    [18]

    Fang T, Leng X L, Song C Q 2003 J. Sound Vib. 226 198

    [19]

    Leng, X L, Wu C L, Ma X P, Meng G, Fang T 2005 Nonlinear Dynamics 42 185

    [20]

    Wu C L, Ma S J, Sun Z K, Fang T 2006 Acta Phys. Sin. 55 6253 (in Chinese)[吴存利、马少娟、孙中奎、方 同 2006 物理学报 55 6253]

    [21]

    Ma S J, Xu W, Li W, Jin Y F 2005 Acta Phys. Sin. 54 3508 (in Chinese)[马少娟、徐 伟、李 伟、 靳艳飞 2005 物理学报 54 3508]

    [22]

    Ma S J, Xu W, Jin Y F, Li W, Fang T 2007 Commumications in Nonlinear Science and Numerical Simulation 12 366

    [23]

    Ma S J, Xu W, Li W, Fang T 2006 Chin. Phys. 15 1231

    [24]

    Ma S J, Xu W, Li W 2006 Acta Phys. Sin. 55 4013 (in Chinese)[马少娟、徐 伟、李 伟 2006 物理学报 55 4013]

    [25]

    Sun X J, Xu W, Ma S J 2006 Acta Phys. Sin. 55 610 (in Chinese) [孙小娟、徐 伟、马少娟 2006 物理学报 55 610]

    [26]

    Xu W, Ma S J, Xie W X 2008 Chin. Phys. B 17 857

    [27]

    Liu B C 2004 Functional analysis (Beijing: Science Press) (in Chinese)[刘炳初 2004 泛函分析 (北京:科学出版社)]

    [28]

    Borwein P, Erdélyi T 1995 Polynomials and Polynomial Inequality(New York: Springer)

    [29]

    Liu S K, Liu S D 1988 Special Function (Beijing: China Meteorological Press) (in Chinese) [刘式适、刘式达 1988 (特殊函数, 北京:气象出版社)]

    [30]

    Kamerich E 1999 A Guide to Maple (New York: Springer).

    [31]

    Guckenheimer J, Holmes P J 1983 Nonlinear oscillators, Dynamical system and bifurcation of vector fields (New York: Spring-Verlag)

    [32]

    Hassard B, Kazarinoff N, Wan Y 1981 Theory and application of Hopf bifurcation (Cambridge: Cambridge University Press)

    [33]

    Shen J, Jing Z J 1993 Acta Mathematics Application Sinica 11 79

  • 引用本文:
    Citation:
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出版历程
  • 收稿日期:  2009-12-18
  • 修回日期:  2010-04-20
  • 刊出日期:  2011-01-15

一类随机van der Pol系统的Hopf 分岔研究

  • 1. 北方民族大学信息与计算科学学院,银川 750021
    基金项目: 

    国家自然科学基金(批准号: 10872165,10972181,11002001), 国家民族事务委员会科研基金 (批准号08XBEO)及宁夏回族自治区高校科研基金(批准号: 2008JY007) 资助的课题.

摘要: 研究了一类随机van der Pol 系统的Hopf分岔行为.首先根据Hilbert空间的正交展开理论,含有随机参数的van der Pol系统被约化为等价确定性系统,然后利用确定性分岔理论分析了等价系统的Hopf分岔,得出了随机van der Pol 系统的Hopf 分岔临界点,探究了随机参数对系统Hopf分岔的影响.最后利用数值模拟验证了理论分析结果.

English Abstract

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