Stochastic P-bifurcations of a noisy nonlinear system with fractional derivative element

Abstract

This paper investigates the stochastic P-bifurcation (SPB) of a fractionally damped oscillator subjected to additive and multiplicative Gaussian white noise. Variable transformation and the stochastic averaging technique are applied to derive the expression of probability density function (PDF) of the system response. Critical conditions of the stochastic bifurcation induced by system parameters are presented based on the change in the number of extreme points of the probability density function. Numerical results are given to show the effectiveness of the proposed approach. Stochastic P-bifurcations for additive and multiplicative noise are studied in detail according to the critical conditions. Stochastic P-bifurcation (SPB) for additive noise can be investigated by the critical conditions. SPB induced by the fractional order is described in the bifurcation plane, see Fig. a. Figure b shows that the probability density function (PDF) for (0.1, 0.005) ∈ R2 has one peak, while the PDF for (0.8, 0.005) ∈ R2 has two peaks. It implies that the change of the fractional order leads to the variation of the structure of the PDF, which indicates the occurrence of the SPB.