Singularities of fluctuational paths for an overdamped two-well system driven by white noise

Noise-induced escape from a metastable state in an overdamped double-well system is investigated. With the WKB approximation employed, the pattern of fluctuational paths is given. By using the action plot, which gives rise to MPEPs, the pattern is divided into various regions where fluctuational paths inside show qualitatively distinguished characteristics. The global quasi-potential is calculated and its value at the saddle on the boundary agrees with the one obtained by action plot. We show a contour and 3D plot of the quasi-potential to illustrate the relative stabilities of metastable states. The bifurcations of singularities as the system parameter varies are analyzed by the second-order term of the WKB expansion in the vicinity of x axis. Its asymptotic behavior determines the positions of cusps and furthermore, its evolution turns out to delineate the twisting structure of the Lagrangian manifold.