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TitleMean, covariance, and effective dimension of stochastic distributed delay dynamics.
Publication TypeJournal Article
Year of Publication2017
AuthorsRené, Alexandre, and Andre Longtin
JournalChaos
Volume27
Issue11
Pagination114322
Date Published2017 Nov
ISSN1089-7682
Abstract

Dynamical models are often required to incorporate both delays and noise. However, the inherently infinite-dimensional nature of delay equations makes formal solutions to stochastic delay differential equations (SDDEs) challenging. Here, we present an approach, similar in spirit to the analysis of functional differential equations, but based on finite-dimensional matrix operators. This results in a method for obtaining both transient and stationary solutions that is directly amenable to computation, and applicable to first order differential systems with either discrete or distributed delays. With fewer assumptions on the system's parameters than other current solution methods and no need to be near a bifurcation, we decompose the solution to a linear SDDE with arbitrary distributed delays into natural modes, in effect the eigenfunctions of the differential operator, and show that relatively few modes can suffice to approximate the probability density of solutions. Thus, we are led to conclude that noise makes these SDDEs effectively low dimensional, which opens the possibility of practical definitions of probability densities over their solution space.

DOI10.1063/1.5007866
Alternate JournalChaos
PubMed ID29195307