How an anomalous cusp bifurcates in a weak-noise system

The pattern of activated trajectories in a symmetric double well system without detailed balance may contain cusps and other singularities, similar to the caustics of geometrical optics. We derive a scaling law and nonpolynomial "equations of state" that govern the bifurcation of an anomalous cusp (a cusp coinciding with the saddle) into conventional cusps. The bifurcation is reflected in the system quasipotential, much as a phase transition is reflected in the free energy of a thermodynamic system. The anomalous cusp is analogous to a nonclassical critical point. Besides showing how critical phenomena occur in noise-perturbed systems, our results extend classical catastrophe theory.