Dynamical analysis of an SDOF quasi-linear system with jump noises and multitime-delayed feedback forces

In this paper, stochastic dynamics of a single degree-of-freedom quasi-linear system with multitime delays and Poisson white noises are investigated using an approximate procedure based on the stochastic averaging method. The simplified equations, including the averaged stochastic differential equation and the averaged generalized Fokker-Planck-Kolmogorov equation, are obtained to calculate the probability density functions (PDFs) to explore stationary responses. The expression of the Lyapunov exponent is presented to examine the asymptotic stochastic Lyapunov stability. An illustrative example of a quasi-linear oscillator with two Poisson white noises controlled by two time-delayed feedback forces is worked out to demonstrate the validity of the proposed method. The approximate stationary PDFs of stochastic responses and asymptotic stochastic stability are demonstrated numerically and theoretically. The results show that the Gaussian white noise has a stronger influence on the dynamics than the Poisson white noise with a small mean arrival rate. Moreover, the influence of the time delay and noise parameters on stochastic dynamics is investigated. It is found that the PDFs under the Poisson white noise approach those under Gaussian white noise as the mean arrival rate increases. The time delay can induce stochastic P-bifurcation of the system. It is also found that the increase of time delay and the mean arrival rates of the Poisson white noises will broaden the unstable parameter region. The comparison between numerical and theoretical results shows the effectiveness of the proposed method.