Dynamical responses to time-dependent control parameters in the presence of noise: a normal form approach

The response of a dynamical system to systematic variations of a control parameter in time in the presence of noise is analyzed by a reduction of the multivariate dynamics to a normal form in the vicinity of bifurcations of the pitchfork and of the limit point type. Mean-field responses, mean values, second- and fourth-order cumulants, probability densities, and entropy-like quantities are evaluated as the system sweeps across the bifurcation point, moving forward toward a multiple state region or moving backward out of this region. Depending on the case stabilization of unstable states, delays and slowing down are found and their signatures on particular observables are identified with an emphasis on the role of noise and on global properties beyond linearized theory.