Discontinuity-induced bifurcations in systems with hysteretic force interactions

This paper presents the application of the discontinuity-mapping technique to the analysis of discontinuity-induced bifurcations of periodic trajectories in an example hybrid dynamical system in which changes in the vector field associated with the crossing of a discontinuity-surface depend on the direction of crossing. The analysis is motivated by a hysteretic model of the capillary force interactions between an atomic-force-microscope cantilever probe tip and a nanoscale sample surface in the presence of a thin liquid film on the tip and the surface and operating in intermittent-contact mode. The analysis predicts the sudden termination of branches of periodic system responses at parameter values corresponding to grazing contact with the onset of the hysteretic force interactions. It further establishes the increase beyond all bounds of the magnitude of one of the eigenvalues of the linearization of a suitably defined Poincaré mapping, indicating the destabilizing influence of near-grazing contact.