Lévy noise-induced global bifurcation with the compatible cell mapping method

Stochastic bifurcation of dynamical system induced by Lévy noise is investigated in this paper from the global viewpoint, which is demonstrated based on the global attractor obtained with the compatible cell mapping (CCM) method. Compared with the traditional cell mapping method, the efficiency of CCM method is greatly improved by concentrating the computational domain on the covering set of the global attractor rather than the whole state space. The computation time can be further reduced by a simple parallel procedure with the block matrix strategy. With the proposed method, the evolutionary process of global properties has been presented. It shows that the shape and size of stochastic attractors and stochastic saddles vary along the direction of unstable manifold with the parameters of Lévy noise. Stochastic bifurcation occurs when stochastic attractor collides with stochastic saddle. It can be also found that the global bifurcation under Lévy noise is discontinuous because of the jump property. The results are significantly different from those of Gaussian noise, which reveals a distinct global evolutionary mechanism of stochastic bifurcation. Furthermore, it also indicates that the CCM method with block matrix is an effective tool for global analysis.