  This paper is a theoretical and a numerical investigation of the stability of
a tip-cantilever system used in noncontact atomic force microscopy (NC-AFM)
when it oscillates close to a surface. No additional dissipative force is
considered. The theoretical approach is based on a variational method
exploiting a coarse grained operation that gives the temporal dependance of the
non-linear coupled equations of motion in amplitude and phase of the
oscillator. Stability criterions for the resonance peak are deduced and predict
a stable behavior of the oscillator in the vicinity of the resonance. The
numerical approach is based on the results obtained with a virtual NC-AFM
developed by our group. The effect of the size of the stable domain in phase is
investigated. These results are in particularly good agreement with the
theoretical predictions. Also they show the influence of the phae shifter in
the feedback loop and the way it can affect the damping signal.
