We study stochastic resonance (SR) phenomenon in the fractional oscillator with time delay and damping fluctuation, analyze the impact of time delay and fractional damping as two memory ingredients on SR, and put forward firstly the concept of the robustness of GSR. By moment method, we obtain the analytical expression of the output amplitude gain and find that fluctuations in the output amplitude gain are non-monotonic. Using numerical simulations we verify the accuracy of the analytical results. We find (i) that the length of time delay and system order are parameters related to memory; (ii) that the output amplitude gain could attain a maximum by increasing driving frequency close to system frequency, and small time delay and system order contribute to enhance the resonance intensity; (iii) that the evolution of the output amplitude gain versus the noise intensity exhibits one-peak resonance, and small time delay can enhance the resonance intensity and the robustness of SR regarding to driving frequency and system frequency; (iv) that the evolution of the output amplitude gain versus the noise correlation rate presents one-peak resonance in the presence of small time delay, critical time delay is bigger with the increasing system order when noise intensity is fixed and critical time delay is smaller with the increasing noise intensity when system order is fixed.

我们研究了具有时滞和阻尼波动的分数振荡器的随机共振现象，分析了时滞和分数阻尼作为两种记忆成分对SR的影响，首先提出了GSR鲁棒性的概念。通过矩量法，我们获得了输出幅度增益的解析表达式，并发现输出幅度增益的波动是非单调的。使用数值模拟，我们验证了分析结果的准确性。我们发现（i）时间延迟的长度和系统顺序是与内存相关的参数；（ii）通过增加接近系统频率的驱动频率可以使输出幅度增益达到最大值，并且较小的时延和系统阶数有助于增强谐振强度；（iii）输出幅度增益相对于噪声强度的变化呈现出一个峰值谐振，并且较小的时间延迟可以增强谐振强度和SR相对于驱动频率和系统频率的鲁棒性；（iv）在存在较小的时间延迟的情况下，输出幅度增益与噪声相关速率的变化呈现出一个峰值谐振，当噪声强度固定且临界时间延迟较小时，临界时间延迟随系统阶数的增加而增大当系统顺序固定时，噪声强度会增加。