# Predictability of escape for a stochastic saddle-node bifurcation: when rare events are typical

## TL;DR

This paper explores how stochastic fluctuations influence the predictability of escape events near a saddle-node bifurcation, revealing a transition from rare to typical escapes governed by a control parameter.

## Contribution

It introduces a unified framework describing the interplay between deterministic bifurcations and stochastic noise, identifying regimes where escapes are rare or typical, with distinct trajectories.

## Key findings

- Small noise regime: escapes are rare, described by Freidlin-Wentzell theory.
- Large noise regime: escapes become typical, with peaked escape time distribution.
- Different escape trajectories: algebraic divergence in rare events, exponential in typical events.

## Abstract

Transitions between multiple stable states of nonlinear systems are ubiquitous in physics, chemistry, and beyond. Two types of behaviors are usually seen as mutually exclusive: unpredictable noise-induced transitions and predictable bifurcations of the underlying vector field. Here, we report a new situation, corresponding to a fluctuating system approaching a bifurcation, where both effects collaborate. We show that the problem can be reduced to a single control parameter governing the competition between deterministic and stochastic effects. Two asymptotic regimes are identified: when the control parameter is small (e.g. small noise), deviations from the deterministic case are well described by the Freidlin-Wentzell theory. In particular, escapes over the potential barrier are very rare events. When the parameter is large (e.g. large noise), such events become typical. Unlike pure noise-induced transitions, the distribution of the escape time is peaked around a value which is asymptotically predicted by an adiabatic approximation. We show that the two regimes are characterized by qualitatively different reacting trajectories, with algebraic and exponential divergence, respectively.