受Mittag-Leffler噪声激励的广义朗之万方程的随机共振研究

Stochastic resonance is one of preceding issues in nonlinear science. The investigation of stochastic resonance in generalized Langevin equations is very necessary and meaningful. Recently, the most research is focused on the stochastic resonance in Langevin equation with multiplicative dichotomous noise. Some papers studied the stochastic resonance in Langevin equation driven by both multiplicative dichotomous and Mittag-Leffler noise. The stochastic resonance in generalized Langevin equations with multiplicative dichotomous noise and Mittag-Leffler noise requires further research and improvement. The project is expected to study the relationship between generalized Langevin equation and fractional order Langevin equation. Using the Shapiro-Loginov formula, Laplace transform theory, convolution theorem and steady-state response theory, the project will obtain the exact expressions of the first moment and the output amplitude gain of systems. The project is expected to give the numerical simulation for the exact expression of the indicators of stochastic resonance. Then the project will study the generalized stochastic resonance, stochastic multiresonace and stochastic reverse resonance. The investigation of this project will not only enrich the research ideas of theory of stochastic resonance, but also provide a theoretical basis for the random fluctuation, anomous diffusion and potential applications of biological models.

随机共振是非线性科学若干前沿问题之一，研究广义朗之万方程的随机共振是非常有必要和有意义的。目前已有研究主要集中于朗之万方程受乘性二值噪声激励的随机共振，少许文献研究广义朗之万方程同时受乘性二值噪声和Mittag-Leffler噪声激励的随机共振。受乘性二值噪声扰动和Mittag-Leffler噪声激励的广义朗之万方程的随机共振需要进一步研究和完善。本项目拟利用涨落耗散定理揭示广义朗之万方程与分数阶朗之万方程之间的关系。项目拟利用Shapiro-Loginov公式，Laplace变换理论，卷积定理以及稳态响应理论，得出系统的一阶稳态矩以及输出幅度增益的解析表达式。项目拟对衡量随机共振指标的解析表达式进行数值仿真，进而研究广义随机共振现象，随机多共振现象，随机逆共振现象。项目的研究既可拓展随机共振理论的研究思路,又为随机涨落,反常扩散及生物模型潜在的应用提供理论依据。