Controlling Bifurcations in Fractional-Delay Systems with Colored Noise

Comparing with the traditional integer-order model, fractional-order systems have shown enormous advantages in the analysis of new materials and anomalous diffusion dynamics mechanism in the past decades, but the research has been confined to fractional-order systems without delay. In this paper, we study the fractional-delay system in the presence of both the colored noise and delayed feedback. The stationary density functions (PDFs) are derived analytically by means of the stochastic averaging method combined with the principle of minimum mean-square error, by which the stochastic bifurcation behaviors have been well identified and studied. It can be found that the fractional-orders have influences on the bifurcation behaviors of the fractional-order system, but the bifurcation point of stationary PDF for amplitude differs from the bifurcation point of joint PDF. By merely changing the colored noise intensity or correlation time the shape of the PDFs can switch between unimodal distribution and bimodal one, thus announcing the occurrence of stochastic bifurcation. Further, we have demonstrated that modulating the time delay or delayed feedback may control bifurcation behaviors. The perfect agreement between the theoretical solution and the numerical solution obtained by the predictor-corrector algorithm confirms the correctness of the conclusion. In addition, fractional-order dominates the bifurcation control in the fractional-delay system, which causes the sensitive dependence of other bifurcation parameters on fractional-order.