Noise-induced escape from bifurcating attractors: Symplectic approach in the weak-noise limit

TL;DR

This paper investigates how noise causes escape from metastable states in one-dimensional maps near bifurcations, using a symplectic approach to analytically derive universal escape rate scalings.

Contribution

It introduces a symplectic two-dimensional map framework for analyzing noise-induced escape in bifurcating systems, providing universal scaling laws near bifurcations.

Findings

Identifies heteroclinic orbits critical for escape rate calculation

Derives universal scaling behavior of escape rates at bifurcations

Uses continuous-time approximation for analytic results

Abstract

The effect of noise is studied in one-dimensional maps undergoing transcritical, tangent, and pitchfork bifurcations. The attractors of the noiseless map become metastable states in the presence of noise. In the weak-noise limit, a symplectic two-dimensional map is associated with the original one-dimensional map. The consequences of their noninvertibility on the phase-space structures are discussed. Heteroclinic orbits are identified which play a key role in the determination of the escape rates from the metastable states. Near bifurcations, the critical slowing down justifies the use of a continuous-time approximation replacing maps by flows, which allows the analytic calculation of the escape rates. This method provides the universal scaling behavior of the escape rates at the bifurcations.