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 $ -
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