# Pitchfork and Hopf bifurcation thresholds in stochastic equations with delayed feedback

## TL;DR

This paper investigates the effects of stochastic noise and delayed feedback on bifurcation thresholds in a model differential equation, revealing how noise shifts the bifurcation point and characterizing the transition using the stationary distribution.

## Contribution

It introduces a novel analysis of bifurcation thresholds in stochastic delay differential equations, combining analytic and numerical methods to understand noise-induced shifts.

## Key findings

- Bifurcation threshold shifts linearly with noise intensity.
- Delay influences the bifurcation structure and thresholds.
- Analytic results agree with numerical solutions for small delays.

## 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.