The near‐resonant, nonlinear dynamic response of microcantilevers in atomic force microscopy is investigated through numerical continuation techniques and simulations of discretized models of the microcantilever interacting with a surface through a Lennard‐Jones potential. The tapping‐mode responses of two representative systems, namely a soft silicon probe‐silicon sample system and a stiff silicon probe‐polystyrene sample system, are studied. Van der Waals interactions are shown to lead to a softening nonlinearity of the periodic solution response, while the short‐range repulsive interactions lead to an overall hardening nonlinear response. Depending on the tip‐sample properties, the dynamics of the microcantilevers occur either in asymmetric single‐well potential regions or in asymmetric double‐well potential regions. In both cases, forced periodic motions of the probe tip destabilize through a sequence of period‐doubling bifurcations, while, in the latter, the tip can also escape the potential well to execute complex and unpredictable cross‐well dynamics. The results predict a broad range of nonlinear dynamic phenomena, many of which have been observed in the literature on experimental atomic force microscopy.