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.