In this paper, a study of the characteristics of period-doubling bifurcations in the dynamic behavior of an atomic force microscope probe for off-resonance excitation is presented. Using a three-mode approximation and excitation at two-and-a-half times the fundamental frequency, the relationship between the characteristics of the period-doubling bifurcation and the material properties is studied by using numerical simulations. Simulations are first used to successfully reproduce nonlinear response data collected experimentally by using a commercial atomic force microscope system and then to conduct a parametric study in order to examine the influence of variations in other system parameters on the relationship. These parameters are the excitation magnitude, the damping level, the cantilever stiffness, and the characteristics of the force model. Based upon the results of the parametric study, a new operation mode for obtaining localized material properties through an efficient scanning process is proposed. A preliminary scan simulation demonstrates the successful implementation of the relationship and its potential for providing localized material property information with nanoscale resolution.