On the stochastic response regimes of a tristable viscoelastic isolation system under delayed feedback control

In this paper, a tristable viscoelastic isolation system with stochastic excitation under both displacement and velocity delayed feedback control is considered. Firstly, the theoretical expressions of the mean first-passage time (MFPT) to measure the activated escape between different potential wells are derived. Induced nonlinear transition dynamics due to the noise and time delays are mainly discussed. It is found that the delay-induced behaviours affect the transitions between the equilibrium points of the system, the corresponding phenomenon of the delay-enhanced stability is observed. In this respect, the existence of the maxima of the MFPT₁ and the MFPT₂ is found in one period. Meanwhile, the MFPT₁ and the MFPT₂ show the monotonic behaviour with the increase of the noise intensity. Additionally, the stationary probability density of the amplitude and the stationary mean amplitude are derived. The influence regimes of the system parameters on both stationary probability density of the amplitude and the stationary mean amplitude are explored. This paper establishes the relationship between system parameters and dynamical properties of the tristable viscoelastic isolation system. This provides a fundamental guidance for the optimization of the viscoelastic isolation by utilizing the technique of delayed feedback control.