Invariant slow manifolds of an Atomic Force Microscope system under the effects of Lennard-Jones forces and a slow harmonic base motion

• We model an AFM system subject to Lennard-Jones forces and to a slow harmonic base motion.
• We approximate the invariant slow manifolds of the system and we study their bifurcations.
• It is shown that the tapping mode takes place through a saddle-node bifurcation of the contact and noncontact slow manifolds respectively.
• Chart of behaviors of different modes are determined and the time contact is computed.

We study the nonlinear vibrations of an AFM system, modeled as a linear mass-spring-damper system, under the Lennard-Jones forces and an imposed harmonic base displacement. The frequency of this latter is very low with respect to the natural fundamental frequency of the system. The invariant slow manifolds of the system are approximated and their bifurcations are investigated. It is shown that two dynamic saddle-node bifurcations, during one period of the base oscillation, of the contact and the noncontact invariant slow manifolds are responsible for triggering the tapping mode. It is also shown that these dynamic bifurcations govern the contact time between the probe and the sample during the tapping mode.