Atomic force microscope (AFM) is an important tool for measuring the topographical and other properties of a sample with nanometer resolution. The cantilever probe of the AFM is influenced by nonlinear interaction forces which act between the probe and the sample. For certain non-standard excitation conditions, this can result in bifurcations in the probe's response. This work numerically examines a period-doubling bifurcation observed to occur for interactions with soft materials. The influence of the sample properties and other conditions on the bifurcation is studied and a method is proposed for sample material characterization. The identified values from simulated 1-D and 2-D scans agree well with the true values. The proposed method does not require the use of special probes and it uses the control algorithm for traditional tapping mode AFM with minor modification. This proposed method could also achieve high scan speeds and prevent strong, destructive interaction forces.