Taming stochastic bifurcations in fractional-order systems via noise and delayed feedback

The dynamics in fractional-order systems have been widely studied during the past decade due to the potential applications in new materials and anomalous diffusions, but the investigations have been so far restricted to a fractional-order system without time delay(s). In this paper, we report the first study of random responses of fractional-order system coupled with noise and delayed feedback. Stochastic averaging method has been utilized to determine the stationary probability density functions (PDFs) by means of the principle of minimum mean-square error, based on which stochastic bifurcations could be identified through recognizing the shape of the PDFs. It has been found that by changing the fractional order the shape of the PDFs can switch from unimodal distribution to bimodal one, or from bimodal distribution to unimodal one, thus announcing the onset of stochastic bifurcation. Further, we have demonstrated that by merely modulating the time delay, the feedback strengths, or the noise intensity, the shapes of PDFs can transit between a single peak and a double peak. Therefore, it provides an efficient candidate to control, say, induce or suppress, the stochastic bifurcations in fractional-order systems.