Stochastic P-Bifurcation Analysis of Fractional Smooth and Discontinuous Oscillator with an Extended Fast Method

In this paper, stochastic bifurcations of a fractional-order smooth and discontinuous (SD) oscillator composed of different viscoelastic materials are studied. As a widely applicable algorithm for various fractional-orders cases, an extended fast algorithm is introduced to obtain the statistics of the response, where the fractional derivative is separated into a history part and a local part with a predetermined memory length. The local part is approximated by a highly accurate algorithm while the history part is computed by an efficient convolution algorithm. Through this accurate and fast method, effects of the system parameters on the dynamic behaviors, such as the fractional order, smoothness parameter, and frequency of harmonic force, are thus successfully investigated. Abundant stochastic P-bifurcation phenomena are discussed in detail. Further, it is found that only when the damping material shows nearly elastic behaviors, the probability density functions of the system exhibit the crater shape. Experiments show that the fast algorithm is accurate for different fractional orders.