Influence of perturbations on period-doubling bifurcation

The influence of noise and resonant perturbation on a dynamical system in the vicinity of a period-doubling bifurcation is investigated. It is found that the qualitative dynamics can be revealed by simple considerations of the Poincaré map. These considerations lead to a shift of the bifurcation point which is proportional to the square of the amplitude of the perturbation. The results of this investigation are in agreement with numerical calculations for the microwave-driven Josephson junction.