TY - JOUR

T1 - Pitchfork and Hopf bifurcation thresholds in stochastic equations with delayed feedback

AU - Gaudreault, Mathieu

AU - Lépine, Françoise

AU - Viñals, Jorge

PY - 2009/12/31

Y1 - 2009/12/31

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevE.80.061920

DO - 10.1103/PhysRevE.80.061920

M3 - Article

C2 - 20365203

AN - SCOPUS:75349097365

SN - 1539-3755

VL - 80

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

IS - 6

M1 - 061920

ER -