Noise-induced enhancement of stability and resonance in a tri-stable system with time-delayed feedback

In this paper, the mean first-passage times and the stochastic resonance in an asymmetric tri-stable system are studied by considering time-delayed feedback, which is driven by a periodic signal and correlated noises. Both the theoretical expressions for small time delay and the numerical results for large time delay are provided to explore the noise-induced enhancement in stability and resonance. The obtained results suggest that the correlation between the additive and multiplicative noises plays a constructive role on system dynamical behaviors, such as the appearing of noise enhanced stability and stochastic resonance, which can be optimized by choosing proper time-delayed feedback gain for fixed time delay. When the feedback gain is fixed, the time delay can greatly accelerate the crossing between left well and middle well, while cannot effectively enhance the crossing velocity between right well and middle well. Moreover, for small time delay, theoretical analysis predicts that stochastic resonance effect and control performance can be substantially improved by choosing appropriate combination of time delay and feedback gain. For large time delay, the phenomenon of stochastic resonance is enhanced by selecting small feedback gain and large asymmetry.