Stochastic resonance in a fractal dimensional bistable system

A fractal dimensional bistable system driven by multiplicative and additive noises and a periodic signal is investigated. We have derived analytically the fractal Fokker–Planck equation of the system, and obtained exact expression of its signal-to-noise ratio (SNR). Numerical results indicate that: (1) The curve of the SNR as a function of multiplicative noise intensity D or additive noise intensity Q exhibits a peak in the fractal dimensional system, i.e., a stochastic resonance phenomenon; (2) For the smaller values of D, the SNR first decreases then increases with increment of dimensionality α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. At the integer dimension of α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1$$\end{document}, response of the system to the weak periodic signal displays a minimum. Yet the SNR increases monotonically for the greater values of D. Our further investigation shows that the height of the potential barrier depends on the dimensionality, and influences on the SNR of the system.