Response of non-linear oscillator driven by fractional derivative term under Gaussian white noise

This paper aimed to investigate the stationary response of non-linear system with fractional derivative damping term under Gaussian white noise excitation. The corresponding Fokker–Plank–Kolmogorov (FPK) equation can be deduced by utilizing the stochastic averaging method and Stratonovich–Khasminskii theorem in the first place. And then we can solve the FPK equation to obtain the stationary probability densities (SPDs) of amplitude, which in fact can be used to describe the response of system. Furthermore, the analytical results coincide with the Monte Carlo results. Finally, one found that reducing fractional derivative order is able to enhance the response of system and increasing fractional coefficient can weaken the response of system. So the fractional derivative damping term has a great effect on the response of Duffing-Van der Pol oscillator. In addition, the response can also be influenced by other system parameters.