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Generalized Stochastic Resonance for a Fractional Noisy Oscillator with Random Mass and Random Damping

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In this paper, we consider the random dichotomous fluctuations on both mass and damping in a fractional oscillator, which is subjected to an additive fractional Gaussian noise and driven by a periodic force. In order to investigate the generalized stochastic resonance (GSR) phenomena, we acquire the exact expression of the first-order moment of system’s steady response by applying generalized fractional Shapiro–Loginov formula and Laplace transform. Meanwhile, we discuss the evolutions of the output amplitude amplification (OAA) with driving frequency, noise parameters, fractional order, and damping strength. It is observed that the non-monotonic resonance behaviors of one-peak GSR, double-peak GSR and triple-peak GSR existing in this fractional system. Moreover, the interplay of mass fluctuation, damping fluctuation, and memory effect can generate a rich variety of non-equilibrium cooperation phenomena, especially the stochastic multi-resonance (SMR) behaviors. It is worth emphasizing that the triple-peak GSR was not observed in previously proposed fractional oscillator subjected to dichotomous noise. Finally, the numerical simulations are also carried out based on predictor-corrector approach to verify the effectiveness of analytic result.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11701086), the Natural Science Foundation of Fujian Province (No. 2017J01550), the China Postdoctoral Science Foundation funded project (No. 2016M602047), and the Basic and Cutting-edge Research Program of Chongqing (No. cstc2017jcyjAX0412). Also, special thanks should go to Prof. Giuseppe Gaeta at Department of Mathematics, the University of Milan.

Funding

This study was funded by the National Natural Science Foundation of China (No. 11701086), the Natural Science Foundation of Fujian Province (No. 2017J01550), the China Postdoctoral Science Foundation funded project (No. 2016M602047), and the Basic and Cutting-edge Research Program of Chongqing (No. cstc2017jcyjAX0412).

Author information

Authors and Affiliations

  1. College of Computer and Information Science, Fujian Agriculture and Forestry University, Fuzhou, 350002, China

    Xipei Huang & Lifeng Lin

  2. College of Crop Science, Fujian Agriculture and Forestry University, Fuzhou, 350002, China

    Lifeng Lin

  3. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Huiqi Wang

  4. Chongqing Key Laboratory of Analytic Mathematics and Applications, Chongqing, 401331, China

    Huiqi Wang

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  1. Xipei Huang
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Appendix: The coefficients in Eq. (18)

Appendix: The coefficients in Eq. (18)

The coefficients in Eq. (18) are given by

$$\begin{aligned} \mu _1&=(f_5f_{17}-f_6f_{18})-f_{13}(f_7f_9-f_8f_{10})+f_{14}(f_7f_{10}+f_8f_9),\\ \mu _2&=(f_5 f_{18}+f_6f_{17})-f_{13}(f_7f_{10}+f_8f_9)-f_{14}(f_7f_9-f_8f_{10}),\\ \mu _3&=f_{17}f_{19}-f_{18}f_{20}+f_{21}+f_{23},\\ \mu _4&=f_{17}f_{20}+f_{18}f_{19}+f_{22}+f_{24}, \end{aligned}$$

in which, all the related parameters can be calculated by the following expressions:

$$\begin{aligned}&f_1=N+\gamma \Omega ^\alpha \cos \left( \frac{\pi }{2}\alpha \right) , \qquad \qquad \qquad \qquad \qquad \!\!\!\!\! f_2=\gamma \Omega ^\alpha \sin \left( \frac{\pi }{2}\alpha \right) ,\\&f_3=\gamma C_2^\alpha \cos \left( \alpha \theta _2\right) ,\qquad \qquad \qquad \qquad \qquad \qquad \!\! f_4=\gamma C_2^\alpha \sin (\alpha \theta _2),\\&f_5=\lambda _1^2 + N +\gamma C_1^\alpha \cos \left( \alpha \theta _1\right) ,\qquad \qquad \qquad \qquad \!\!\!\! f_6=2\lambda _1\Omega +\gamma C_1^\alpha \sin (\alpha \theta _1),\\&f_7=\gamma C_3^\alpha \cos (\alpha \theta _3),\qquad \qquad \qquad \qquad \qquad \quad \quad \quad \!\! \!\!\!\! f_8=\gamma C_3^\alpha \sin (\alpha \theta _3),\\&f_9=\lambda _2^2 +N+\gamma C_2^\alpha \cos (\alpha \theta _2),\qquad \qquad \qquad \qquad \!\!\! f_{10}=2\lambda _2\Omega +\gamma C_2^\alpha \sin (\alpha \theta _2),\\&f_{11}=\left( \lambda _1+\lambda _2 \right) ^2-\Omega ^2,\qquad \qquad \qquad \qquad \qquad \quad \!\!\! f_{12}=2(\lambda _1+\lambda _2)\Omega ,\\&f_{13}=\gamma \sigma _2^2C_1^\alpha \cos (\alpha \theta _1),\qquad \qquad \qquad \qquad \qquad \quad \!\! f_{14}=\gamma \sigma _2^2C_1^\alpha \sin (\alpha \theta _1),\\&f_{15}=(\lambda _1+\lambda _2)^2 +N+\gamma C_3^\alpha \cos (\alpha \theta _3),\qquad \qquad \! f_{16}=2(\lambda _1+\lambda _2)\Omega + \gamma C_3^\alpha \sin (\alpha \theta _3),\\&f_{17}=f_{9}f_{15}-f_{10}f_{16}-f_{11}\sigma _1^2(\lambda _2^2-\Omega ^2),\qquad \qquad \!\!\! f_{18}=f_{10}f_{15}+f_{9}f_{16}-\sigma _1^2\left[ f_{12}(\lambda _2^2-\Omega ^2)+2f_{11}\lambda _2\right] ,\\&f_{19}=f_5(N+f_1)-f_2f_6+\sigma _1^2\Omega ^2(\lambda _1^2-\Omega ^2),\quad \quad f_{20}=f_2f_5+f_6(N+f_1)+2\lambda _1\sigma _1^2\Omega ^3,\\ \end{aligned}$$
$$\begin{aligned} f_{21}&= (f_1f_{10}f_{14}+f_2f_9f_{14}+f_2 f_{10}f_{13}-f_1f_9f_{13})f_7+(f_1f_9f_{14}+f_1f_{10} f_{13}\\&+f_2f_9f_{13}-f_2f_{10}f_{14})f_8\\&-\sigma _1^2\sigma _2^2\left[ (\lambda _1^2-\Omega ^2)(\lambda _2^2-\Omega ^2)f_1-2\lambda _2(\lambda _1^2-\Omega ^2) \Omega f_2- 4\lambda _1\lambda _2\Omega ^2 f_1\right. \\&\left. -2\lambda _1(\lambda _2^2-\Omega ^2)\Omega f_2\right] f_7\\&+\sigma _1^2\sigma _2^2\left[ 2\lambda _2\Omega (\lambda _1^2-\Omega ^2) f_1+(\lambda _1^2-\Omega ^2) (\lambda _2^2-\Omega ^2)f_2 +2\lambda _1\Omega (\lambda _2^2-\Omega ^2) f_1\right. \\&\left. - 4\lambda _1\lambda _2\Omega ^2 f_2\right] f_8,\\ f_{22}&= (f_2 f_{10} f_{14}-f_1 f_9f_{14}-f_1f_{10}f_{13}-f_2 f_9f_{13})f_7+(f_1f_{10}f_{14}+f_2f_9f_{14}\\&+f_2f_{10}f_{13}-f_1f9 f_{13})f_8\\&-\sigma _1^2\sigma _2^2\left[ 2\lambda _2\Omega (\lambda _1^2-\Omega ^2) f_1+(\lambda _1^2-\Omega ^2) (\lambda _2^2-\Omega ^2) f_2 +2\lambda _1\Omega (\lambda _2^2-\Omega ^2)f_1\right. \\&\left. - 4\lambda _1\lambda _2\Omega ^2 f_2\right] f_7\\&-\sigma _1^2\sigma _2^2\left[ (\lambda _1^2-\Omega ^2)(\lambda _2^2-\Omega ^2)f_1-2\lambda _2(\lambda _1^2-\Omega ^2) \Omega f_2- 4\lambda _1\lambda _2 \Omega ^2 f_1 \right. \\&\left. - 2\lambda _1(\lambda _2^2-\Omega ^2)\Omega f_2\right] f_8,\\&f_{23}= \sigma _1^2\Omega ^2\left[ (f_{11}f_{13}-f_{12}f_{14})f_3\right. \\&\left. -(f_{11}f_{14}+f_{12}f_{13})f_4\right] \\&-\sigma _2^2f_3\left[ (f_5f_{15}-f_6f_{16})f_1-(f_5f_{16}+f_6f_{15})f_2-(f_1f_{13}-f_2f_{14})f_7\right. \\&\left. +(f_1f_{14}+f_2f_{13})f_8\right] \\&-\sigma _2^2f_4\left[ (f_5f_{15}-f_6f_{16})f_2-(f_5f_{16}+f_6f_{15})f_1+(f_1f_{14}+f_2f_{13})f_7\right. \\&\left. +(f_1f_{13}-f_2f_{14})f_8\right] \\ f_{24}&= \sigma _1^2\Omega ^2\left[ (f_{11}f_{14}+f_{12}f_{13})f_3\right. \\&\left. +(f_{11}f_{13}-f_{12}f_{14})f_4\right] \\&-\sigma _2^2f_3\left[ (f_5f_{15}-f_6f_{16})f_2+(f_5f_{16}+f_6f_{15})f_1-(f_1f_{14}+f_2f_{13})f_7\right. \\&\left. -(f_1f_{13}-f_2f_{14})f_8\right] \\&-\sigma _2^2f_4\left[ (f_5f_{15}-f_6f_{16})f_1-(f_5f_{16}+f_6f_{15})f_2-(f_1f_{13}-f_2f_{14})f_7\right. \\&\left. +(f_1f_{14}+f_2f_{13})f_8\right] , \end{aligned}$$

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Huang, X., Lin, L. & Wang, H. Generalized Stochastic Resonance for a Fractional Noisy Oscillator with Random Mass and Random Damping. J Stat Phys 178, 1201–1216 (2020). https://doi.org/10.1007/s10955-020-02494-3

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