Exposing hidden periodic orbits in scanning force microscopy

The nonlinear interaction between the tip of a scanning probe microscope (SPM) and a sample is manifested in the emergence of bifurcations and unstable branches in the frequency response of a driven cantilever. While extensively investigated theoretically, exploring the unstable branch in an actual SPM experiment is lacking so far, reflecting the broader challenge in studying nanoscopic oscillators under strongly varying external forces. Here we demonstrate experimental tracking of unstable periodic orbits between two saddle-node bifurcation points in the attractive regime, revealing the full set of stationary states. This is achieved by a minimally invasive control scheme based on fast adaptive phase extraction and Fourier discretisation of the tip dynamics. Stabilization of unstable branches of oscillating AFM cantilevers opens avenues for novel experimental modes, potentially enabling ultrasensitive surface detection at considerably large amplitudes with minimal tip-surface interaction, new insights in tip-surface interaction mechanisms, as well as new AFM modes enabling arbitrary setpoint choice while inherently avoiding discontinuities. Raw data sets corresponding to the named figures in the submitted article as well as complete code required to reproduce the control expiriment using an ADwin real time signal processing unit. For details of data-collection and specifics see methods section