The influence of perturbations (a small, near-resonant signal and noise) on a driven dissipative dynamical system that is close to undergoing a period-doubling bifurcation is investigated. It is found that the system is very sensitive, and that periodic perturbations change its stability in a well-defined way that is a function of the amplitude and the frequency of the signal. New scaling laws between the amplitude of the signal and the detuning \ensuremath{\delta} are found; these scaling laws apply to a variety of quantities, e.g., to the shift of the bifurcation point. It is also found that the stability and the amplification of the system are not stationary, bu \ensuremath{\delta}. The results in this paper are found from a linear analysis of the dynamics in the Poincar\'e map. It is thereby shown that several effects, which previously were believed to be caused by nonlinearit linear in the lowest order. Numerical and analog simulations of a microwave-driven Josephson junction confirm the theory. Results should be of interest in parametric-amplification studies.