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

Sambit Misra, Harry Dankowicz, Mark R. Paul

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)33-43
Number of pages11
JournalPhysica D: Nonlinear Phenomena
Issue number1-2
StatePublished - Jan 2010


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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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