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

Yuri Bakhtin, Zsolt Pajor-Gyulai

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
JournalStochastics and Dynamics
DOIs
StateAccepted/In press - Jan 1 2018

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Jordan Block
Nonlinear dynamical systems
Scaling Limit
White noise
Linearization
Unstable
Exit Problem
Exit Time
Limiting Behavior
Nonlinear Dynamical Systems
Term
Remainder
Critical point
Logarithmic
Perturbation

Keywords

  • Exit problem
  • small noise
  • unstable equilibrium

ASJC Scopus subject areas

  • Modeling and Simulation

Cite this

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AB - 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.

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