Theoretical analysis for critical fluctuations of relaxation trajectory near a saddle-node bifurcation

TL;DR

This paper provides a theoretical analysis of how relaxation trajectories behave near a saddle-node bifurcation in a stochastic system, revealing divergent statistical properties and a systematic method for analyzing path probabilities.

Contribution

It introduces a systematic formulation based on singular perturbation to analyze path probabilities near bifurcations, explaining critical fluctuations in relaxation trajectories.

Findings

Statistical properties diverge near the bifurcation point in the weak-noise limit.

The final deterministic solution value changes discontinuously at the bifurcation.

Theoretical results align with numerical simulations.

Abstract

A Langevin equation whose deterministic part undergoes a saddle-node bifurcation is investigated theoretically. It is found that statistical properties of relaxation trajectories in this system exhibit divergent behaviors near a saddle-node bifurcation point in the weak-noise limit, while the final value of the deterministic solution changes discontinuously at the point. A systematic formulation for analyzing a path probability measure is constructed on the basis of a singular perturbation method. In this formulation, the critical nature turns out to originate from the neutrality of exiting time from a saddle-point. The theoretical calculation explains results of numerical simulations.