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This article aims at studying the stochastic P-bifurcation of tri-stable van der Pol-Duffing oscillator subjected to multiplicative colored noise. First, the stationary probability density of amplitude is derived by using the stochastic averaging method. Then the critical parameter conditions of stochastic P-bifurcation are obtained based on the singularity theory. And the different types of stationary probability densities of amplitude are also obtained, which are in good agreement with the results from Monte-Carlo numerical simulation. Based on these results, the effects of the noise correlation time, noise intensity and linear damping coefficient on the P-bifurcation and the stable response behavior of the system are studied.
[1] Zhu W Q 1998 Random Vibration (Beijing: Science Press) p465 (in Chinese) [朱位秋1998 随机振动(北京: 科学出版社)第465页]
[2] Chen L C, Zhu W Q 2010 Chin. J. Appl. Mech. 3 517 (in Chinese) [陈林聪, 朱位秋 2010 应用力学学报 3 517]
[3] Chen L C, Zhu W Q 2011 Int. J. Non-Linear Mech. 46 1324
[4] Rong H W, Wang X D, Xu W, Meng G, Fang T 2005 Acta Phys. Sin. 54 2557 (in Chinese) [戎海武, 王向东, 徐伟, 孟光, 方同 2005 54 2557]
[5] Rong H W, Wang X D, Meng G, Xu W, Fang T 2006 Chin. J. Appl. Mech. 27 1373 (in Chinese) [戎海武, 王向东, 孟光, 徐伟, 方同 2006 应用数学和力学 27 1373]
[6] Zakharova A, Vadivasova T, Anishchenko V, Koseska A, Kurths J 2010 Phys. Rev. E 81 011106
[7] Xu Y, Gu R C, Zhang H Q, Xu W, Duan J Q 2011 Phys. Rev. E 83 056215
[8] Gu R C, Xu Y, Hao M L 2011 Acta Phys. Sin. 60 060513 (in Chinese) [顾仁财, 许勇, 郝孟丽 2011 60 060513]
[9] Wu Z Q, Hao Y 2013 Sci. Sin.: Physica, Mechanica & Astronomica 43 524 (in Chinese) [吴志强, 郝颖 2013 中国科学:中国科学: 物理学 力学 天文学 43 524]
[10] Hao Y, Wu Z Q 2013 Chin. J. Theor. Appl. Mech. 43 257 (in Chinese) [郝颖, 吴志强 2013 力学学报 43 257]
[11] Christiansen L E, Lehn-Schioler T, Mosekilde E, Gránásy P, Matsushita H 2002 Math. Comput. Simulat. 58 385
[12] Dimitriadis G, Li J 2009 AIAA J. 47 2577
[13] Wu Z Q 2012 Proceedings of the 23rd International Congress of Theoretical and Applied Mechanics Beijing August 19-24, 2012
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[1] Zhu W Q 1998 Random Vibration (Beijing: Science Press) p465 (in Chinese) [朱位秋1998 随机振动(北京: 科学出版社)第465页]
[2] Chen L C, Zhu W Q 2010 Chin. J. Appl. Mech. 3 517 (in Chinese) [陈林聪, 朱位秋 2010 应用力学学报 3 517]
[3] Chen L C, Zhu W Q 2011 Int. J. Non-Linear Mech. 46 1324
[4] Rong H W, Wang X D, Xu W, Meng G, Fang T 2005 Acta Phys. Sin. 54 2557 (in Chinese) [戎海武, 王向东, 徐伟, 孟光, 方同 2005 54 2557]
[5] Rong H W, Wang X D, Meng G, Xu W, Fang T 2006 Chin. J. Appl. Mech. 27 1373 (in Chinese) [戎海武, 王向东, 孟光, 徐伟, 方同 2006 应用数学和力学 27 1373]
[6] Zakharova A, Vadivasova T, Anishchenko V, Koseska A, Kurths J 2010 Phys. Rev. E 81 011106
[7] Xu Y, Gu R C, Zhang H Q, Xu W, Duan J Q 2011 Phys. Rev. E 83 056215
[8] Gu R C, Xu Y, Hao M L 2011 Acta Phys. Sin. 60 060513 (in Chinese) [顾仁财, 许勇, 郝孟丽 2011 60 060513]
[9] Wu Z Q, Hao Y 2013 Sci. Sin.: Physica, Mechanica & Astronomica 43 524 (in Chinese) [吴志强, 郝颖 2013 中国科学:中国科学: 物理学 力学 天文学 43 524]
[10] Hao Y, Wu Z Q 2013 Chin. J. Theor. Appl. Mech. 43 257 (in Chinese) [郝颖, 吴志强 2013 力学学报 43 257]
[11] Christiansen L E, Lehn-Schioler T, Mosekilde E, Gránásy P, Matsushita H 2002 Math. Comput. Simulat. 58 385
[12] Dimitriadis G, Li J 2009 AIAA J. 47 2577
[13] Wu Z Q 2012 Proceedings of the 23rd International Congress of Theoretical and Applied Mechanics Beijing August 19-24, 2012
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