Degenerate discontinuity-induced bifurcations in tapping-mode atomic-force microscopy

作者：Misra Sambit; Dankowicz Harry^*; Paul Mark R

来源：Physica D: Nonlinear Phenomena , 2010, 239(1-2): 33-43.

DOI：10.1016/j.physd.2009.10.001

• Tapping-mode
• Atomic-force microscopy
• Discontinuity mappings
• Codimension-two bifurcations
• Branch point bifurcation
• Isola bifurcation

This paper documents the existence of degenerate bifurcation scenarios in the low-contact-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 paper 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 (2000) [28] in the case that (ii) does not hold. In addition, the paper includes a quantitative reflection on the relative importance of discontinuities in the attractive and repulsive force components in producing the predicted bifurcations.

• 出版日期2010-1