The harmonic balance method is applied to the analysis of period-doubling bifurcations in a general class of nonlinear feedback systems. Compact conditions for the prediction and stability analysis of period-doubling bifurcations are obtained. Specializations of these conditions for systems in which the nonlinear subsystem is static are given. The implications of the results to control design are illustrated through the development of a design procedure for solving a representative control problem. The objectives of the control design can include delay of a given period-doubling bifurcation as well as stability objectives. The results are illustrated with a detailed example.