Research Workshop on Bifurcations in Oscillators with Elastic and Impact Constraints

In this talk, I document the existence of degenerate discontinuity-induced bifurcation scenarios in the lowcontact-velocity dynamics during tapping-mode atomic-force microscopy. Specifically, numerical analysis of a model of the microscope dynamics shows branch point and isola bifurcations associated with the emergence of two families of saddle-node bifurcation points along a branch of low-amplitude oscillations. The talk argues for the origin of the degenerate bifurcations in the existence of a periodic steady-state trajectory that (i) achieves tangential contact with a discontinuity surface in a piecewise smooth model of the cantilever response and (ii) retracts from the surface under variations in either direction along a line segment in parameter space. Specifically, the discontinuity-mapping technique is here rigorously applied to a general situation of such degenerate contact showing the codimension-two nature of these bifurcations for appropriately chosen parameter values. The discontinuity-mapping-based normal form derived here is a novel extension of that derived in Dankowicz and Nordmark (Physica D, vol. 136, no. 3-4, pp. 280-302, 2000) in the case that (ii) does not hold. In addition, the discussion includes a quantitative reflection on the relative importance of discontinuities in the attractive and repulsive force components in producing the predicted bifurcations.