Pitchfork and Hopf bifurcation thresholds in stochastic equations with delayed feedback

Mathieu Gaudreault, Françoise Lépine, Jorge Viñals

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

The bifurcation diagram of a model stochastic differential equation with delayed feedback is presented. We are motivated by recent research on stochastic effects in models of transcriptional gene regulation. We start from the normal form for a pitchfork bifurcation, and add multiplicative or parametric noise and linear delayed feedback. The latter is sufficient to originate a Hopf bifurcation in that region of parameters in which there is a sufficiently strong negative feedback. We find a sharp bifurcation in parameter space, and define the threshold as the point in which the stationary distribution function p (x) changes from a delta function at the trivial state x=0 to p (x) ∼ xα at small x (with α=-1 exactly at threshold). We find that the bifurcation threshold is shifted by fluctuations relative to the deterministic limit by an amount that scales linearly with the noise intensity. Analytic calculations of the bifurcation threshold are also presented in the limit of small delay τ→0 that compare quite favorably with the numerical solutions even for moderate values of τ.

Original languageEnglish (US)
Article number061920
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume80
Issue number6
DOIs
StatePublished - Dec 31 2009

Fingerprint

Pitchfork Bifurcation
Delayed Feedback
Hopf Bifurcation
Stochastic Equations
thresholds
Bifurcation
negative feedback
Transcriptional Regulation
Negative Feedback
Gene Regulation
gene expression
Delta Function
noise intensity
delta function
Bifurcation Diagram
Stationary Distribution
Normal Form
Parameter Space
Multiplicative
Trivial

Cite this

Pitchfork and Hopf bifurcation thresholds in stochastic equations with delayed feedback. / Gaudreault, Mathieu; Lépine, Françoise; Viñals, Jorge.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 80, No. 6, 061920, 31.12.2009.

Research output: Contribution to journalArticle

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