The stochastic resonance (SR) phenomenon for a fractional underdamped bistable system with multiplicative and additive noise driven by one high-frequency and one low-frequency periodic signal is investigated. The Fokker–Plank equation of the system for the low-frequency motion and the transition rate out of the system stable states are obtained. The signal-to-noise ratio (SNR) is derived based on two-state theory. It is shown that bona fide SR and bona fide vibration resonance (VR) phenomenon occur when the SNR varies with the frequency of the low-frequency signal and with that of the high-frequency signal, respectively. Single resonance peak appears when the SNR changes with the increase of the fractional exponent. Damping-induced SR takes place on the SNR curves versus the system damping coefficient. The SNR obtains one maximum value at certain levels of the multiplicative and additive noise.

Graphical abstract

中文翻译：

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由一种高频和一种低频周期信号驱动的具有乘性和加性噪声的分数欠阻尼双稳态系统的谐振行为

研究了由一种高频和一种低频周期信号驱动的具有乘性和加性噪声的分数欠阻尼双稳态系统的随机共振（SR）现象。得到了系统低频运动的Fokker-Plank方程和系统稳定状态的转变率。信噪比 (SNR) 是根据二态理论推导出来的。结果表明，当信噪比分别随低频信号频率和高频信号频率变化时，会出现真正的SR和真正的振动共振（VR）现象。当信噪比随着分数指数的增加而变化时，出现单共振峰。阻尼引起的 SR 发生在 SNR 与系统阻尼系数的曲线上。SNR 在乘性和加性噪声的特定水平处获得一个最大值。

图形概要