Stochastic bifurcation in Duffing–Van der Pol oscillators

Global analysis of bifurcation in the Duffing–van der Pol oscillators under both additive and multiplicative random excitations is explored in detail by the generalized cell mapping method using digraph. System parameters are chosen in the range that two co-exist attractors and a saddles. As an alternative definition, stochastic bifurcation may be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value. It is found that under certain conditions stochastic bifurcation always occurs when a stochastic attractor collides with a stochastic saddle. Our study reveals that the generalized cell mapping method with digraph is also a powerful tool for global analysis of stochastic bifurcation. By this global analysis the mechanism of development, occurrence, and evolution of stochastic bifurcation can be explored clearly and vividly.