Scaling limit for escapes from unstable equilibria in the vanishing noise limit: nontrivial Jordan block case

TL;DR

This paper analyzes the behavior of a nonlinear dynamical system near an unstable equilibrium with a Jordan block structure under small noise, focusing on exit times and locations, and deriving detailed asymptotic expansions.

Contribution

It provides a detailed asymptotic analysis of exit times and locations for systems with Jordan block linearization under vanishing noise, including new expansion terms.

Findings

Exit occurs near two deterministic points associated with eigendirections.

Leading correction to exit location is deterministic and logarithmic in noise magnitude.

Random remainder satisfies a specific scaling limit.

Abstract

We consider white noise perturbations of a nonlinear dynamical system in the neighborhood of an unstable critical point with linearization given by a Jordan block of full dimension. For the associated exit problem, we study the joint limiting behavior of the exit location and exit time, in the vanishing noise limit. The exit typically happens near one of two special deterministic points associated with the eigendirection, and we obtain several more terms in the expansion for the exit point. The leading correction term is deterministic and logarithmic in the noise magnitude, while the random remainder satisfies a scaling limit.