Scaling and crossovers in activated escape near a bifurcation point

Abstract

Near a bifurcation point a system experiences critical slowing down. This leads to scaling behavior of fluctuations. We find that a periodically driven system may display three scaling regimes and scaling crossovers near a saddle-node bifurcation where a metastable state disappears. The rate of activated escape $W$ scales with the driving field amplitude $A$ as $\ln W \propto (A_c-A)^{\xi}$, where $A_c$ is the bifurcational value of $A$. With increasing field frequency the critical exponent $\xi$ changes from $\xi = 3/2$ for stationary systems to a dynamical value $\xi=2$ and then again to $\xi=3/2$. The analytical results are in agreement with the results of asymptotic calculations in the scaling region. Numerical calculations and simulations for a model system support the theory.