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The resonant behavior of a fractional linear oscillator subjected to both parametric excitation of colored noise and external excitation of periodically modulated noise is considered. Using Laplace transformation technique and Shapiro-Loginov formula, exact expressions of the first moment for the system response and its long-time amplitude are presented. The influence of the system parameters on the long-time behavior of the system response is discussed, such as fractional order, friction coefficient, driving frequency, noise intensity and relevant rate. It is found that the long-time amplitude of the fractional oscillator behaves non-monotonical, that is, there exist stochastic resonances in a wide sense. Moreover, generalized stochastic resonance with two peaks can be found subject to some appropriate parameters.
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Keywords:
- fractional linear oscillator /
- periodically modulated noise /
- stochastic resonance /
- multi-peak generalized stochastic resonance
[1] Benzi R, Sutera A, Vulpiana A 1981 J. Phys. A 14 L453
[2] Gammaitoni L, Hänggi P, Jung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223
[3] McNamara B, Wiesenfeld K, Roy R 1988 Phys. Rev. Lett. 60 2626
[4] Gang H, Nicolis G, Nicolis C 1990 Phys. Rev. A 42 2030
[5] Hu G 1994 Stochastic Forces and Nonlinear Systems (Shanghai: Shanghai Science and Technology Education Press) (in Chinese) [胡岗 1994 随机力与非线性系统 (上海: 上海科技教育出版社)]
[6] Gitterman M 2003 Phys. Rev. E 67 057103
[7] Jia Y, Yu S N, Li J R 2000 Phys. Rev. E 62 1869
[8] Berdichevsky V, Gitterman M 1996 Europhys. Lett. 36 161
[9] Luo X, Zhu S 2003 Phys. Rev. E 67 021104
[10] Berdichevsky V, Gitterman M 1999 Phys. Rev. E 60 1494
[11] Jin Y F, Hu H Y 2009 Acta Phys. Sin. 58 2895 (in Chinese) [靳艳飞, 胡海岩 2009 58 2895]
[12] Ning L J, Xu W 2009 Acta Phys. Sin. 58 2889 (in Chinese) [宁丽娟, 徐伟 2009 58 2889]
[13] Qian M, Wang Y, Zhang X J 2003 Chin. Phys. Lett. 20 810
[14] Liu F, Anh V, Turner I, Zhuang P 2003 J. Appl. Math. Comput. 13 233
[15] Huang F, Liu F 2005 Anziam J. 46 317
[16] Bao J D 2009 Stochastic Simulation Method of Classic and Quantum Dissipative Sysmtem (Beijing: Science Press) p160 (in Chinese) [包景东 2009 经典和量子耗散系统的随机模拟方法 (北京: 科学出版社) 第84页]
[17] Zhou Y Q 2006 Stochastic Process Theory (2 Edn.) (Beijing: Publishing House of Electronics Industry) p94 (in Chinese) [周荫清 2006 随机过程理论 (第2版) (北京: 电子工业出版社) 第94页]
[18] Shapiro V E, Loginov V M 1978 Physica A 91 563
[19] Kempfle S, Schäfer I, Beyer H 2002 Nonlinear Dynam. 29 99
[20] Laas K, Mankin R, Reiter E 2011 Int. J. Math. Mod. Meth. Appl. S 5 280
[21] Soika E, Mankin R, Ainsaar A 2010 Phys. Rev. E 81 011141
[22] Kubo R, Toda M, Hashitsume N 1985 Statistical Physics II (Berlin: Springer)
[23] Sauga A, Mankin R, Ainsaar A 2010 WSEAS Transactions on Systems 18 21
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[1] Benzi R, Sutera A, Vulpiana A 1981 J. Phys. A 14 L453
[2] Gammaitoni L, Hänggi P, Jung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223
[3] McNamara B, Wiesenfeld K, Roy R 1988 Phys. Rev. Lett. 60 2626
[4] Gang H, Nicolis G, Nicolis C 1990 Phys. Rev. A 42 2030
[5] Hu G 1994 Stochastic Forces and Nonlinear Systems (Shanghai: Shanghai Science and Technology Education Press) (in Chinese) [胡岗 1994 随机力与非线性系统 (上海: 上海科技教育出版社)]
[6] Gitterman M 2003 Phys. Rev. E 67 057103
[7] Jia Y, Yu S N, Li J R 2000 Phys. Rev. E 62 1869
[8] Berdichevsky V, Gitterman M 1996 Europhys. Lett. 36 161
[9] Luo X, Zhu S 2003 Phys. Rev. E 67 021104
[10] Berdichevsky V, Gitterman M 1999 Phys. Rev. E 60 1494
[11] Jin Y F, Hu H Y 2009 Acta Phys. Sin. 58 2895 (in Chinese) [靳艳飞, 胡海岩 2009 58 2895]
[12] Ning L J, Xu W 2009 Acta Phys. Sin. 58 2889 (in Chinese) [宁丽娟, 徐伟 2009 58 2889]
[13] Qian M, Wang Y, Zhang X J 2003 Chin. Phys. Lett. 20 810
[14] Liu F, Anh V, Turner I, Zhuang P 2003 J. Appl. Math. Comput. 13 233
[15] Huang F, Liu F 2005 Anziam J. 46 317
[16] Bao J D 2009 Stochastic Simulation Method of Classic and Quantum Dissipative Sysmtem (Beijing: Science Press) p160 (in Chinese) [包景东 2009 经典和量子耗散系统的随机模拟方法 (北京: 科学出版社) 第84页]
[17] Zhou Y Q 2006 Stochastic Process Theory (2 Edn.) (Beijing: Publishing House of Electronics Industry) p94 (in Chinese) [周荫清 2006 随机过程理论 (第2版) (北京: 电子工业出版社) 第94页]
[18] Shapiro V E, Loginov V M 1978 Physica A 91 563
[19] Kempfle S, Schäfer I, Beyer H 2002 Nonlinear Dynam. 29 99
[20] Laas K, Mankin R, Reiter E 2011 Int. J. Math. Mod. Meth. Appl. S 5 280
[21] Soika E, Mankin R, Ainsaar A 2010 Phys. Rev. E 81 011141
[22] Kubo R, Toda M, Hashitsume N 1985 Statistical Physics II (Berlin: Springer)
[23] Sauga A, Mankin R, Ainsaar A 2010 WSEAS Transactions on Systems 18 21
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