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The stochastic response of the smooth and discontinuous (SD) oscillator under additive and multiplicative Poisson white noise excitation is studied by the generalized cell mapping method. Based on the digraph analysis algorithm, the attractors, basins of attraction, basin boundaries, saddles and invariant manifolds of the SD oscillator can be obtained. The transient and stationary responses of the SD oscillator under Poisson white noise excitation are computed based on the matrix analysis algorithm. It can be found that there is a close relationship between the evolution direction of probability density and the unstable manifold. Monte Carlo results are used to verify the efficiency and accuracy of the matrix analysis algorithm.
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Keywords:
- SD oscillator /
- generalized cell mapping method /
- stochastic response /
- Poisson white noise
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[5] Pirrotta A, Santoro R 2011 Probab. Eng. Mech. 26 26
[6] Köyloğlu H U, Nielsen S R K, Çakmak A Ş1995 Struct. Saf. 17 151
[7] Wu Y, Zhu W Q 2008 Phys. Rev. E 77 041911
[8] Er G K, Zhu H T, Iu V P, Kou K P 2009 Nonlinear Dynam. 55 337
[9] Zhu H T, Er G K, Iu V P, Kou K P 2011 J. Sound Vib. 330 2900
[10] Grigoriu M 1995 Probab. Eng. Mech. 10 45
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[14] Hong L, Xu J X 1999 Phys. Lett. A 262 361
[15] He Q, Xu W, Li S, Xiao Y Z 2008 Acta Phys. Sin. 57 743 (in Chinese) [贺群, 徐伟, 李爽, 肖玉柱 2008 57 743]
[16] He Q, Xu W, Li S, Xiao Y Z 2008 Acta Phys. Sin. 57 4021 (in Chinese) [贺群, 徐伟, 李爽, 肖玉柱 2008 57 4021]
[17] Zou H L, Xu J X 2009 Sci. China Tech. Sci. 52 787
[18] Hong L, Xu J X 2002 Acta Phys. Sin. 51 2694 (in Chinese) [洪灵,徐健学 2002 51 2694]
[19] Hong L 2010 Chin. Phys. B 19 030513
[20] Sun J Q, Hsu C S 1988 J. Sound Vib. 124 233
[21] Sun J Q, Hsu C S 1990 J. Appl. Mech. 57 1018
[22] Hsu C S 1995 Int. J. Bifurc. Chaos 5 1085
[23] Yue X L, Xu W, Wang L, Zhou B C 2012 Probab. Eng. Mech. 30 70
[24] Cao Q J, Wiercigroch M, Pavlovskaia E E, Grebogi C, Thompson J M T 2006 Phys. Rev. E 74 046218
[25] Cao Q J, Wiercigroch M, Pavlovskaia E E, Grebogi C, Thompson J M T 2008 Int. J. Non-Linear Mech. 43 462
[26] Tian R L, Cao Q J, Yang S P 2010 Nonlinear Dynam. 59 19
[27] Tian R L, Cao Q J, Li Z X 2010 Chin. Phys. Lett. 27 074701
[28] Cao Q J, Xiong Y P, Wiercigroch M 2011 J. Appl. Anal. Comput. 1 183
[29] Yue X L, Xu W, Wang L 2013 Sci. China-Phys. Mech. Astron. 56 1010
-
[1] Vasta M 1995 Int. J. Non-linear Mech. 30 407
[2] Proppe C 2003 Int. J. Non-linear Mech. 38 557
[3] Wu Y, Zhu W Q 2008 Phys. Lett. A 372 623
[4] Zeng Y, Zhu W Q 2011 J. Appl. Mech. 78 021002
[5] Pirrotta A, Santoro R 2011 Probab. Eng. Mech. 26 26
[6] Köyloğlu H U, Nielsen S R K, Çakmak A Ş1995 Struct. Saf. 17 151
[7] Wu Y, Zhu W Q 2008 Phys. Rev. E 77 041911
[8] Er G K, Zhu H T, Iu V P, Kou K P 2009 Nonlinear Dynam. 55 337
[9] Zhu H T, Er G K, Iu V P, Kou K P 2011 J. Sound Vib. 330 2900
[10] Grigoriu M 1995 Probab. Eng. Mech. 10 45
[11] Hsu C S 1980 J. Appl. Mech. 47 931
[12] Hsu C S 1981 J. Appl. Mech. 48 634
[13] Jiang J, Xu J X 1994 Phys. Lett. A 188 137
[14] Hong L, Xu J X 1999 Phys. Lett. A 262 361
[15] He Q, Xu W, Li S, Xiao Y Z 2008 Acta Phys. Sin. 57 743 (in Chinese) [贺群, 徐伟, 李爽, 肖玉柱 2008 57 743]
[16] He Q, Xu W, Li S, Xiao Y Z 2008 Acta Phys. Sin. 57 4021 (in Chinese) [贺群, 徐伟, 李爽, 肖玉柱 2008 57 4021]
[17] Zou H L, Xu J X 2009 Sci. China Tech. Sci. 52 787
[18] Hong L, Xu J X 2002 Acta Phys. Sin. 51 2694 (in Chinese) [洪灵,徐健学 2002 51 2694]
[19] Hong L 2010 Chin. Phys. B 19 030513
[20] Sun J Q, Hsu C S 1988 J. Sound Vib. 124 233
[21] Sun J Q, Hsu C S 1990 J. Appl. Mech. 57 1018
[22] Hsu C S 1995 Int. J. Bifurc. Chaos 5 1085
[23] Yue X L, Xu W, Wang L, Zhou B C 2012 Probab. Eng. Mech. 30 70
[24] Cao Q J, Wiercigroch M, Pavlovskaia E E, Grebogi C, Thompson J M T 2006 Phys. Rev. E 74 046218
[25] Cao Q J, Wiercigroch M, Pavlovskaia E E, Grebogi C, Thompson J M T 2008 Int. J. Non-Linear Mech. 43 462
[26] Tian R L, Cao Q J, Yang S P 2010 Nonlinear Dynam. 59 19
[27] Tian R L, Cao Q J, Li Z X 2010 Chin. Phys. Lett. 27 074701
[28] Cao Q J, Xiong Y P, Wiercigroch M 2011 J. Appl. Anal. Comput. 1 183
[29] Yue X L, Xu W, Wang L 2013 Sci. China-Phys. Mech. Astron. 56 1010
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