Nonlinear and additive white noise perturbations of linear delay differential equations at the verge of instability: An averaging approach

N. Lingala, N. Sri Namachchivaya

Research output: Contribution to journalArticle

Abstract

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. We deal with linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation (eigenvalues) lie on the imaginary axis of the complex plane, and all other roots have negative real parts. We show that, when the system is perturbed by small noise, under an appropriate change of time scale, the law of the amplitude of projection onto the critical eigenspace is close to the law of a certain one-dimensional stochastic differential equation (SDE) without delay. Further, we show that the projection onto the stable eigenspace is small. These results allow us to give an approximate description of the delay-system using an SDE (without delay) of just one dimension. The proof is based on the martingale problem technique.

Original languageEnglish (US)
Article number1650013
JournalStochastics and Dynamics
Volume16
Issue number4
DOIs
StatePublished - Aug 1 2016

Keywords

  • Delay differential equation
  • averaging
  • martingale problem

ASJC Scopus subject areas

  • Modeling and Simulation

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