Pitchfork and Hopf bifurcation threshold in stochastic equations with delayed feedback

TL;DR

This paper investigates how stochastic delayed feedback influences bifurcation behavior in nonlinear Langevin equations, revealing shifted thresholds and new distribution characteristics through numerical and analytical methods.

Contribution

It provides the first detailed numerical and analytical analysis of bifurcation thresholds in stochastic equations with delayed feedback, highlighting the effects of noise and delay.

Findings

Bifurcation diagram obtained numerically showing direct and oscillatory bifurcations.

Bifurcation threshold is shifted by noise, scaling linearly with noise intensity.

Analytic expression for stationary distribution p(x) matches numerical results.

Abstract

The bifurcation diagram of a model nonlinear Langevin equation with delayed feedback is obtained numerically. We observe both direct and oscillatory bifurcations in different ranges of model parameters. Below threshold, the stationary distribution function p(x) is a delta function at the trivial state x=0. Above threshold, p(x) ~ x^alpha at small x, with alpha = -1 at threshold, and monotonously increasing with the value of the control parameter above threshold. Unlike the case without delayed feedback, the bifurcation threshold is shifted by fluctuations by an amount that scales linearly with the noise intensity. With numerical information about time delayed correlations, we derive an analytic expression for p(x) which is in good agreement with the numerical results.