Analytical determination of the bifurcation thresholds in stochastic differential equations with delayed feedback

Analytical expressions for pitchfork and Hopf bifurcation thresholds are given for a nonlinear stochastic differential delay equation with feedback. Our results assume that the delay time τ is small compared to other characteristic time scales, not a significant limitation close to the bifurcation line. A pitchfork bifurcation line is found, the location of which depends on the conditional average x (t) | x (t-τ) where x (t) is the dynamical variable. This conditional probability incorporates the combined effect of fluctuation correlations and delayed feedback. We also find a Hopf bifurcation line which is obtained by a multiple scale expansion around the oscillatory solution near threshold. We solve the Fokker-Planck equation associated with the slowly varying amplitudes and use it to determine the threshold location. In both cases, the predicted bifurcation lines are in excellent agreement with a direct numerical integration of the governing equations. Contrary to the known case involving no delayed feedback, we show that the stochastic bifurcation lines are shifted relative to the deterministic limit and hence that the interaction between fluctuation correlations and delay affect the stability of the solutions of the model equation studied.