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This paper aims to investigate the influence of Lévy stable noise on a bistable Duffing-van der Pol oscillator. We obtain the stationary probability density function of amplitude for the Duffing-van der Pol oscillator by use of Monte Carlo method, and analyze the influences of the noise intensity and the stability index on the stationary probability density. Stochastic bifurcations are further discussed though a qualitative change of the stationary probability distribution, which indicates that not only system parameters and noise intensity can be treated as bifurcation parameters, but also the change of the stability index will induce stochastic bifurcations.
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Keywords:
- Lévy stable noise /
- Duffing-van der Pol oscillator /
- stationary probability density function /
- stochastic bifurcations
[1] Ushakov O,Wünsche H,Henneberger F,Khovanov I,Schimansky L,Zaks M A 2005 Phys. Rev. Lett. 95 123903
[2] Mankin R,Laas T,Sauga A,Ainsaar A,Reiter E 2006 Phys. Rev. E 74 021101
[3] Hao B L,Zhang S Y 1983 Acta Phys.Sin.32 198(in Chinese) [郝柏林、张淑誉 1983 32 198]
[4] Zhao Y G 1991 Acta Phys.Sin.40 731(in Chinese)[赵一广 1991 40 731]
[5] Masataka K,Hayato C 2009 Chaos 19 043121
[6] Tang J S,Ouyang K J 2006 Acta Phys.Sin.55 4437(in Chinese) [唐驾时、欧阳克俭 2006 55 4437]
[7] Zhao D,M Zhang Q C 2010 Chin. Phys. B 19 030518
[8] Jiang G R,Xu B G,Yang Q G 2009 Chin. Phys. B 18 5235
[9] Arnold L 1998 Random Dynamical Systems (Springer: New York)
[10] Majumdar S N,Ziff R M 2008 Phys. Rev. Lett. 101 050601
[11] Romanelli A,Siri R,Micenmacher V 2007 Phys. Rev. E 76 037202
[12] Dybiec B,Gudowska E 2007 Phys. Rev. E 75 021109
[13] Ponomarev A V,Denisov S,Hanggi P 2010 Phys. Rev. A 81 043615
[14] Zeng L Z,Xu B H 2010 Physica A 389 5128
[15] Zakharova A 2010 Phys. Rev. E 81 011106
[16] Xu W,He Q,Rong H W,Fang T 2003 Acta Phys.Sin.52 1165(in Chinese)[徐 伟、贺 群、戎海武、方 同 2003 52 1165]
[17] Sun X J,Xu W,Ma S J 2006 Acta Phys.Sin.55 610(in Chinese)[孙晓娟、徐 伟、马少娟 2006 55 610]
[18] Li G J,Xu W,Wang L,Feng J Q 2008 Acta Phys.Sin.57 2107(in Chinese)[李高杰、徐 伟、王 亮、冯进钤 2008 57 2107]
[19] Leccardi M 2005 the Fifth Euromech Nonlinear Dynamics Conference Eindhoven, Netherland August 7—12 2005
[20] Zhu W Q 1996 ASME Appl. Mech. Rev. 49 72
[21] Zeng L Z,Bao R H, Xu B H 2007 J. Phys. A: Math. Theor. 40 7175
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[1] Ushakov O,Wünsche H,Henneberger F,Khovanov I,Schimansky L,Zaks M A 2005 Phys. Rev. Lett. 95 123903
[2] Mankin R,Laas T,Sauga A,Ainsaar A,Reiter E 2006 Phys. Rev. E 74 021101
[3] Hao B L,Zhang S Y 1983 Acta Phys.Sin.32 198(in Chinese) [郝柏林、张淑誉 1983 32 198]
[4] Zhao Y G 1991 Acta Phys.Sin.40 731(in Chinese)[赵一广 1991 40 731]
[5] Masataka K,Hayato C 2009 Chaos 19 043121
[6] Tang J S,Ouyang K J 2006 Acta Phys.Sin.55 4437(in Chinese) [唐驾时、欧阳克俭 2006 55 4437]
[7] Zhao D,M Zhang Q C 2010 Chin. Phys. B 19 030518
[8] Jiang G R,Xu B G,Yang Q G 2009 Chin. Phys. B 18 5235
[9] Arnold L 1998 Random Dynamical Systems (Springer: New York)
[10] Majumdar S N,Ziff R M 2008 Phys. Rev. Lett. 101 050601
[11] Romanelli A,Siri R,Micenmacher V 2007 Phys. Rev. E 76 037202
[12] Dybiec B,Gudowska E 2007 Phys. Rev. E 75 021109
[13] Ponomarev A V,Denisov S,Hanggi P 2010 Phys. Rev. A 81 043615
[14] Zeng L Z,Xu B H 2010 Physica A 389 5128
[15] Zakharova A 2010 Phys. Rev. E 81 011106
[16] Xu W,He Q,Rong H W,Fang T 2003 Acta Phys.Sin.52 1165(in Chinese)[徐 伟、贺 群、戎海武、方 同 2003 52 1165]
[17] Sun X J,Xu W,Ma S J 2006 Acta Phys.Sin.55 610(in Chinese)[孙晓娟、徐 伟、马少娟 2006 55 610]
[18] Li G J,Xu W,Wang L,Feng J Q 2008 Acta Phys.Sin.57 2107(in Chinese)[李高杰、徐 伟、王 亮、冯进钤 2008 57 2107]
[19] Leccardi M 2005 the Fifth Euromech Nonlinear Dynamics Conference Eindhoven, Netherland August 7—12 2005
[20] Zhu W Q 1996 ASME Appl. Mech. Rev. 49 72
[21] Zeng L Z,Bao R H, Xu B H 2007 J. Phys. A: Math. Theor. 40 7175
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