Stochastic bifurcation of generalized Van der Pol systems with fractional order delay under colored noise excitation

The stochastic bifurcation problem of a generalized Van der Pol system with fractional order delay excited by colored noise is studied. Firstly, the fractional derivative term is transformed into a linear combination of restoring force and damping force, using the principle of minimum mean square error, by calculating the error with the original system, the original fractional order system can be transformed into integer order system. Secondly, the generalized Van der Pol transformation is introduced, based on the stochastic averaging method for system of Itô stochastic differential equations and Fokker-Planck-Kolmogorov(FPK) equations. Finally, through the steady-state probability density function diagram of the system, it is verified that the change of fractional derivative order pand nonlinear damping coefficient μcan induce stochastic P-bifurcation of the system.