| The bifurcation diagram of a model nonlinear Langevin equation with delay is obtained. A finite delay implies non-Markovian behavior, and is the subject of current interest as such equations model delayed feedback loops in many settings, including the study of cell regulation. We show that the bifurcation remains sharp, both in the ranges of direct and oscillatory instabilities. Below threshold, the stationary distribution function is a delta function at the trivial state x = 0 despite the delay. At threshold, the stationary distribution function becomes a power law p(x) ∼ xα with −1 < α < 0, where α = −1 at threshold and monotonously increases with increasing value of the control parameter. Unlike the case without delay, the bifurcation threshold is shifted by fluctuations, and the shift scales linearly with the noise intensity D.. |
