Bifurcation of singularities of fluctuational paths for a noise-driven overdamped two-well system

Noise-induced escape in a 2D generalized Maier-Stein model with two parameters μ and α is investigated in the weak noise limit. With the WKB approximation, the patterns of extreme paths and singularities are displayed. By employing the Freidlin-Wentzell action functional and the asymptotic series, critical parameters α inducing singularity bifurcation are determined analytically for μ=1. The switching line will appear with singularities and is equivalent to the sliding set in the Filippov system. The pseudo-saddle-node bifurcation on the switching line is found. Then, when -1<μ<1, it is found that all bifurcation values α will decrease as μ decreases and the second-order bifurcation values are bigger than all first-order ones. In addition, the variation of the switching line is also analyzed and a new switching line will emerge when the location of the minimum quasi-potential on the boundary changes. At last, when the noise is anisotropic, only the noise intensity ratio will affect the bifurcation value α.