The stochastic P-bifurcation analysis of the impact system via the most probable response

The most probable response, which acts as a deterministic geometric tool for the response analysis of stochastic systems, offers an attractive alternative to traditional methods for analyzing the P-bifurcation of the stochastic impact system. Specifically, the stochastic impact system perturbed by multiplicative Gaussian white noises is considered to research the P-bifurcations under the most probable response angle. Firstly, the non-smooth coordinate transformation of state variables is applied to convert the impact system into an equivalent system without the velocity jump. Then, the stochastic averaging method of energy envelope is exploited to the transformed system and the most probable response is obtained by the combination of the Fokker-Planck equation and the extreme value theory. Finally, based on the most probable response, the bifurcation behavior of the stochastic impact system is investigated qualitatively from a new perspective. It is found that the stochastic P-bifurcation can be induced or suppressed by modulating the noise intensity D₂ or the restitution coefficient r in the stochastic impact system. However, there is no influence of the noise intensity D₁ on the most probable response of the stochastic impact system. Therefore, the noise intensity D₁ will not trigger the P-bifurcation of the stochastic impact system. Meanwhile, the validity of the proposed procedure is verified by numerical simulation.