Scaling in activated escape of underdamped systems

Noise-induced escape from potentials is ubiquitous in many areas of physics. Here, noise-induced escape from a metastable state of a dynamical system is studied close to a saddle-node bifurcation point, but in a region where the system remains underdamped. We find the activation energy of escape scales as a power of the distance to the bifurcation point. Moreover, we find two types of scaling and the corresponding critical exponents.