Abstract: The stochastic stability and stochastic bifurcation behavior  of Van der pol vibration damping system constructed based on fractional-order viscoelastic material was studied under external broadband noise excitation. Considering the influence of constraints condition, a non-smooth Zhuravlev transformation was introduced to transform the collision system into a collision-free dynamic system. A set of quasi-periodic functions was used to replace the fractional-order differential element approximately, the stochastic average method was used to obtain the Ito stochastic differential equation of the system. The stochastic stability of the system was classified and discussed based on the maximum Lyapunov exponent method and singular boundary theory. The stochastic bifurcation behavior of the system under the linear Ito equation was analyzed by using the pseudo Halmiton system stochastic average method, and the critical condition for D-bifurcation was obtained. Furthermore, the stationary probability density function related to the amplitude of the system was obtained. Using the steady-state probability density curves drawn by MATLAB to visually display the changes of steady state that occurred in the system. The results show that the system can generate P-bifurcation behavior when the fractional-order and noise intensity change within a certain threshold.

Key words: stochastic P-bifurcation, stochastic average method, collision system, non-smooth Zhuravlev transformation, broadband noise