Scaling and crossovers in activated escape near a bifurcation point

TL;DR

This paper investigates how escape rates near a saddle-node bifurcation scale with driving amplitude and frequency, revealing three regimes and crossovers supported by analytical, numerical, and simulation results.

Contribution

It uncovers three distinct scaling regimes and crossovers in escape rates near bifurcations under periodic driving, with analytical and numerical validation.

Findings

Escape rate scales as  _c-A)^{} for stationary systems

Critical exponent hanges from 3/2 to 2 and back to 3/2 with increasing frequency

Analytical results agree with asymptotic calculations and simulations.

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.