Pitchfork and Hopf bifurcation thresholds in stochastic equations with delayed feedback

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) ~ x^alpha at small x (with alpha = -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 tau -> 0 that compare quite favorably with the numerical solutions even for tau = 1.

References in corpus (3)

• Noise-based switches and amplifiers for gene expression
• Adiabatic reduction near a bifurcation in stochastically modulated systems
• Onset of oscillatory instabilities under stochastic modulation

Cited by in corpus (4)

• Time-delayed feedback control of coherence resonance near subcritical Hopf bifurcation: theory versus experiment
• Activation process in excitable systems with multiple noise sources: One and two interacting units
• Activation process in excitable systems with multiple noise sources: Large number of units
• Persistence and failure of mean-field approximations adapted to a class of systems of delay-coupled excitable units