Scaling and crossovers in activated escape near a bifurcation point

Near a bifurcation point a system experiences a critical slowdown. 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 $\mathrm{ln}\phantom{\rule{0.2em}{0ex}}W\ensuremath{\propto}{({A}_{c}\ensuremath{-}A)}^{\ensuremath{\xi}}$, where ${A}_{c}$ is the bifurcational value of $A$. With increasing field frequency the critical exponent $\ensuremath{\xi}$ changes from $\ensuremath{\xi}=3∕2$ for stationary systems to a dynamical value $\ensuremath{\xi}=2$ and then again to $\ensuremath{\xi}=3∕2$. 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.