The permanent magnet synchronous generator (PMSG) for wind turbine system operating under inevitable stochastic disturbance from wind power is a nonlinear stochastic dynamical system. With the random interaction and nonlinearity, the intense nonlinear stochastic oscillation is likely to happen in such a system, causing the system to be unstable or even collapse. However, the PMSG is usually considered as a deterministic system when analyzing its nonlinear dynamic behaviors in the past researches. Such a simplification can lead to wrong predictions for the system stability and reliability. This paper aims to discuss the effect of the stochastic disturbance on the nonlinear dynamic behavior of the PMSG. Based on the derived PMSG model considering the stochastic disturbance from the input mechanical torque, the evolution of the system global structure with the stochastic intensity is investigated using the generalized cell mapping digraph method. Meanwhile, the occurrence process and development process of the stochastic bifurcation are illustrated. Based on this global analysis, the intrinsic mechanism for the effect of the stochastic disturbance on the operating performances of the PMSG is revealed. Finally, the numerical simulations based on the Euler-Maruyama algorithm are carried out to validate the results of the theoretical analysis. The results present that as the intensity of the stochastic disturbance increases, two kinds of stochastic bifurcations can be observed in the PMSG system according to the definition of a sudden change in characteristic of the stochastic attractor. One is the stochastic interior crisis that occurs when a stochastic attractor collides with a stochastic saddle in its attraction basin interior, leading to the abrupt increase of the attractor and the disappearance of the saddle. This kind of bifurcation results in the intense stochastic oscillation and instability of the PMSG system. Another stochastic bifurcation is the stochastic boundary crisis which occurs when a stochastic attractor collides with the boundary of its attraction basin and results in the disappearance of the attractor. This sudden change of the number of stochastic attractors induces the stable solution set to vanish and thus the PMSG system to collapse. In a word, even the stochastic disturbance with small intensity may lead to the complete destruction of the stable structure of the PMSG, inducing the system to suffer a strong disordered oscillation or the operation to collapse. The results of this paper can provide significant theoretic reference for both practically operating and designing the PMSG for wind turbine systems.