Stochastic Resonance of Fractional-Order System with Multiplicative Noise and Random Delay
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摘要: 时间延迟是动力系统的共同特性，在科学领域中得到广泛应用. 分数阶微积分具有时间记忆性和长程空间相关性，能更好地描述具有记忆、路径依赖性的物理过程，但很少文献研究延迟分数阶系统中的随机共振现象. 为此，研究乘性噪声作用下延迟分数阶系统中的随机共振. 基于线性系统理论，利用拉普拉斯变换和小延迟近似方法，得到了分数阶系统输出幅度增益（output amplitude gain，OAG）表达式. 研究结果表明：OAG是延迟时间的非单调函数；在OAG与乘性噪声的强度和相关率、随机延迟的相关率，以及分数指数和系统驱动信号频率的关系曲线上出现了随机共振现象；当乘性噪声强度较小，以及乘性噪声相关率相对较小或相对较大时，OAG随阻尼系数的增大而减小；而当乘性噪声强度较大，以及乘性噪声相关率取中间值时，OAG随阻尼系数的增大而增大.Abstract: Time delay is a common characteristic of a dynamic system, which has been widely used in the science field. Fractional calculus has the characteristics of time memory and long-range spatial correlation, which can better describe the physical process with memory and path dependence, but rare literature have studied the phenomenon of stochastic resonance (SR) in a time-delayed fractional system. Therefore, the SR behavior for a fractional-order system with multiplicative noise and random delay is investigated. Based on linear system theory, Laplace transform and small delay approximation approach are used to derive the expression of the output amplitude gain (OAG) for the fractional-order system. It is found that the OAG is a non-monotonous function of the delay-time. The SR behavior appears on the relationship curves between the OAG and the strength and correlate rate of the multiplicative noise, between the OAG and the correlate rate of delay noise, and between the OAG and the fractional exponent and the driving frequency of the system. For relatively small multiplicative noise strength, and for the relatively small or relatively large correlation rate of multiplicative noise, the OAG decreases with the increase of the damping coefficient; while for the relatively large multiplicative noise strength, and for the medium correlation rate of multiplicative noise, the OAG increases with the damping coefficient.
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Key words:
- stochastic resonance /
- fractional-order system /
- random delay /
- multiplicative noise
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图 1
$\alpha $ 取不同值时$G$ 与${\tau _0}$ 的关系曲线Figure 1. Amplitude gain
$G$ versus random delay-time${\tau _0}$ for different values of fractional exponent$\alpha $ 
图 2
$\beta $ 取不同值时$G$ 与${\tau _0}$ 的关系曲线Figure 2. Amplitude gain
$G$ versus random delay-time${\tau _0}$ for different values of friction coefficeint$\beta $ 
图 3
$r$ 取不同值时$G$ 与$D$ 的关系曲线Figure 3. Amplitude gain
$G$ versus multiplicative noise strength$D$ for different values of damping coefficient$r$ 
图 4
$r$ 取不同值时$G$ 与${\lambda _1}$ 的关系曲线Figure 4. Amplitude gain G versus correlate rate
${\lambda _1}$ of the multiplicative noise for different values of damping coefficient r
图 5 系统频率
$\omega $ 取不同值时$G$ 与${\lambda _2}$ 的关系曲线Figure 5. Amplitude gain
$G$ versus correlate rate${\lambda _2}$ of random delay noise for different values of system frequency$\omega $ 
图 6
$\beta $ 取不同值时$G$ 与$\alpha $ 的关系曲线Figure 6. Amplitude gain
$G$ versus fractional exponent$\alpha $ for different values of friction coefficeint$\beta $ -
董小娟,晏爱君. 双稳态系统中随机共振和相干共振的相关性[J]. 物理学报,2013,62(7): 56-62.DONG Xiaojuan, YAN Aijun. The relationship between stochastic resonance and coherence resonance in a bi-stable system[J]. Acta Physica Sinica, 2013, 62(7): 56-62. 焦尚彬,杨蓉,张青,等. α稳定噪声驱动的非对称双稳随机共振现象[J]. 物理学报,2015,64(2): 49-57.JIAO Shangbin, YANG Rong, ZHANG Qing, et. al. Stochastic resonance of asymmetric bistable system with Alpha stable noise[J]. Acta Physica Sineca, 2015, 64(2): 49-57. GUO Feng, ZHOU Yurong, JIANG Shiqi, et al. Stochastic resonance in a mono-stable system with multiplicative and additive noise[J]. Journal of Physics A: Mathematical and General, 2006, 39: 1386.1-1386.8. doi: 10.1088/0305-4470/39/45/002 LIU Yulei, LIANG Jun, JIAO Shangbin, et al. Stochastic resonance of a tri-stable system with α stable noise[J]. Chinese Journal of Physics, 2017, 55: 355-366. doi: 10.1016/j.cjph.2016.12.010 钟苏川,蔚涛,张路,等. 具有质量及频率涨落的欠阻尼线性谐振子的随机共振[J]. 物理学报,2105,64(2): 28-34.ZHONG Suchuan, YU Tao, ZHANG Lu, et al. Stochastic resonance of an underdamped linear harmonic oscillator with fluctuating mass and fluctuating frequency[J]. Acta Physica Sineca, 2105, 64(2): 28-34. FRANK T D, BEEK P J. Stationary solutions of linear stochastic delay differential equations:applications to biological systems[J]. Physical Review E, 2001, 64: 021917.1-021917.12. MAJER N, SCHOLL E. Resonant control of stochastic spatiotemporal dynamics in a tunnel diode by multiple time-delayed feedback[J]. Physical Review E, 2009, 79: 011109.1-011109.8. ZENG C H, SUN Y L, CHEN G X. The relaxation time of a bistable system with two different kinds of time delays[J]. Modern Physics Letters B, 2009, 23(18): 2281-2292. GUO Feng, ZHOU Yurong, ZHANG Yu. Stochastic resonance in a time-delayed bistable system subject to multiplicative and additive noise[J]. Chinese Physics B, 2010, 19 (7): 90-94. 贺利芳,杨玉蕾,张天骐. 时延反馈EVG系统随机共振特性研究及轴承故障诊断[J]. 仪器仪表学报,2019,40(8): 47-57.HE Lifang, YANG Yulei, ZHANG Tianqi. Stochastic resonance characteristic study and bearing fault diagnosis of time-delayed feedback EVG system[J]. Chinese Journal of Scientific Instrument, 2019, 40(8): 47-57. SHI Peiming, XIA Haifeng, HAN Dongying, et al. Dynamical complexity and stochastic resonance in an asymmetry bistable system with time delay[J]. Chinese Journal of Physics., 2017, 55(1): 133-141. doi: 10.1016/j.cjph.2016.10.013 BAO H B, CAO J D. Delay-distribution-dependent state estimation for discrete-time stochastic neural networks with random delay[J]. Neural Networks, 2011, 24: 19-28. doi: 10.1016/j.neunet.2010.09.010 FINERTY J P. The population ecology of cycles in small mammals[M]. New Haven: Yale University Press, 1980. FLOWEDEW J R. Mammls: their reproductive biology and population ecology[M]. London: Edward Arnold, 1987. KILBAS A A, SARIVASTAVA H M, TRUJILLO J J. Theory and applications of fractional differential equations[M]. New York: Elsevier, 2006. 高仕龙,钟苏川,韦鹏,等. 过阻尼分数阶Langeven方程及其随机共振[J]. 物理学报,2012,61(10): 32-37.GAO Shilong, ZHONG Suchuan, WEI Peng, et al. Overdamped fractional Langevein equation and its stochastic resonance[J]. Acta Physica Sineca, 2012, 61(10): 32-37. MÜLLER S, KÄSTNER M, BRUMMUND J, et al. On the numerical handling of fractional viscoelastic material models in a FE analysis[J]. Computing Mechanics, 2013, 51(6): 999-1012. doi: 10.1007/s00466-012-0783-x 公徐路,许鹏飞. 含时滞反馈与涨落质量的记忆阻尼系统的随机共振[J]. 力学学报,2018,50(4): 880-889. doi: 10.6052/0459-1879-18-051GONG Xulu, XU Pengfei. Stochastic resoancne of a memorial-damped system with time delay feedback and fluctuating mass[J]. Chinses Journal of Theoretical and Applied Mechanics, 2018, 50(4): 880-889. doi: 10.6052/0459-1879-18-051 XU Yong, LI Yongge, LIU Di, et al. Responses of Duffing oscillator with fractional damping and random phase[J]. Nonlinear Dynamics, 2013, 74: 745-753. doi: 10.1007/s11071-013-1002-9 SHEN Yongjun, YANG Shaopu, XING Haijun, et al. Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives[J]. International Journal of Non-Linear Mechanics, 2012, 47: 975-983. doi: 10.1016/j.ijnonlinmec.2012.06.012 LEUNG A Y T, GUO Zhongjin, YANG H X. Fractional derivative and time delay damper characteristics in Duffing-van der Pol oscillators[J]. Communicaton on Nonlinear Science and Numerical Simulation, 2013, 18: 2900-2915. doi: 10.1016/j.cnsns.2013.02.013 ZHU Jianqu, JIN Weidong, GUO Feng. Stochastic resonance for a linear oscillator with two kinds of fractional derivatives and random frequency[J]. Journal of Korean Physical Society, 2017, 70(8): 745-750. doi: 10.3938/jkps.70.745 ERKKI S, ROMI M, AIN A. Resonant behavior of a fractional oscillator with fluctuating frequency[J]. Physical Review E, 2010, 81(1): 011141.1-011141.11. LIN Lifeng, CHEN Cong, WANG Huiqi. Trichotomous noise induced stochastic resonance in a fractional oscillator with random damping and random frequency[J]. Journal of Statistical Mechanics:Theory and Experement, 2016, 2: 023201.1-023201.21. GUO Feng, ZHU Chengyin, CHENG Xiaofeng, et al. Stochastic resonance in a fractional harmonic oscillator subject to random mass and signal-modulated noise[J]. Physica A:Statistical Mechanics and Its Application, 2016, 459: 86-91. doi: 10.1016/j.physa.2016.04.011 DU Luchun, MEI Dongcheng. Effects of time delay on stochastic resonance of a periodically driven linear system with multiplicative and periodically modulated additive white noises[J]. Chinese Physics B, 2009, 18(3): 946-951. doi: 10.1088/1674-1056/18/3/018 GAO Shilong. Generalized stochastic resonance in a linear fractional system with a random delay[J]. Journal of Statistical Mechanics:Theory and Experiment, 2012: P12011.1-P12011.16. VAN KAMPEN N G. Stochastic processes in physics and chemistry[M]. Amsterdam: [s.n.], 1992. FULINSKI A. Non-Markovian noise[J]. Physical Review E, 1994, 50: 2668-2681. doi: 10.1103/PhysRevE.50.2668 GUILLOUZIC S, HEUREUX I L, LONGTIN A. Small delay approximation of stochastic delay differential equations[J]. Physical Review E, 1999, 59: 3970-3982. doi: 10.1103/PhysRevE.59.3970 SHAPIRO V E, LOGINOV V M. Formulae of differentiation and their use for solving stochastic equations[J]. Physica A, 1978, 91: 563-574. doi: 10.1016/0378-4371(78)90198-X SELLERIO A L, MARI D, GREMAUD G. Fractional Brownian motion and anomalous diffusion in vibrated granular materials[J]. Journal of Statistical Mechanics:Theory and Experiment, 2012: P01002.1-P01002.18. -
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