Stochastic P-bifurcation of fractional derivative Van der Pol system excited by Gaussian white noise

Abstract

This paper aimed to investigate the stochastic P-bifurcation of Van der Pol oscillator with a fractional derivative damping term driven by Gaussian white noise excitation. Firstly, based on the method of stochastic averaging method and Stratonovich–Khasminskii theorem, the corresponding Fokker–Plank–Kolmogorov (FPK) equation is deduced. To describe the P-bifurcation of system, the stationary probability densities of amplitude can be obtained by solving the FPK equation. Then, the effects of the fractional order, the fractional coefficient, and the intensity of Gaussian white noise on the fractional systems are discussed in detail. The results show that increasing order α will change obviously the number and the height of peaks under certain parameter conditions. Finally, comparing the analytical and numerical results, a very satisfactory agreement can be found.