Mean-field approximation of time-delayed feedback control of noise-induced oscillations in the Van der Pol system

We study time-delayed feedback control of noise-induced oscillations analytically and numerically for the paradigmatic model of the Van der Pol oscillator under the influence of white noise. We focus on the regime below the Hopf bifurcation where the deterministic system has a stable fixed point and does not exhibit oscillations. Analytical expressions for the power spectral density and the coherence properties of the stochastic delay differential equation in dependence upon noise intensity, delay time, arid feedback strength are derived on the basis of a mean-field approximation, and are in good agreement with our numerical simulations of the full nonlinear model. Our analytical results elucidate how the correlation time of the controlled stochastic oscillations can be maximized as a function of delay time and feedback strength.