Dynamics of vibrating atomic force microscopy

An atomic force microscope which is operated in the oscillating mode is an example of an impact oscillator. The description of such dynamical systems can be reduced to a mathematical mapping, which displays a square-root sing ularity. A direct consequence of this property is the emergence of an infin ite series of period-adding bifurcations. This extremely characteristic phe nomenon should be observed in atomic force microscopes. We consider an atom ic force microscope in which the tip-substrate forces are modelled by a liq uid-bridge interaction. By integrating the dynamical equations we show that the atomic force microscopy (AFM) dynamical behaviour has the same charact eristic bifurcation scenario as the square-root map. We point to the remark able role of the energy that is dissipated upon impact. We finally suggest ways to improve the operation of AFM.