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The Hopf bifurcation of van der Pol system with random parameter is studied. Firstly according to the orthogonal polynomial approximation in Hilbert space, the van der Pol system with random parameter can be reduced into the equivalent deterministic system. Then the Hopf bifurcation can be explored by the traditional methods in deterministic bifurcation theory. After the critical point of Hopf bifurcation in stochastic van der Pol system is obtained, the influence of the random parameter on Hopf bifurcation in stochastic van der Pol system is analyzed. At last we verified these results by numerical simulations.
[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 Matre 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 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 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 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 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
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[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 Matre 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 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 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 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 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
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