Dynamics near discontinuity events in systems with hysteresis and systems with finite state resets

Atomic Force Microscopy (AFM);Grazing;Atomic Force;Discontinuity;Capillary

Katzenbach, Michael T. ; Dankowicz ; Harry

关键词: Atomic Force Microscopy (AFM);    Grazing;    Atomic Force;    Discontinuity;    Capillary;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/16923/1_Katzenbach_Michael.pdf?sequence=2&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

In this thesis, we study the dynamics near grazing for a model of an atomic force microscope in tapping mode and a model of cell cycle mitosis. In particular, period one behavior is studied near grazing points corresponding to tangential contact with the discontinuity surface used to initiate capillary interactions in a tapping mode AFM model. Two different discontinuity mapping analysis methods aredeveloped and applied to this AFM model. The discontinuity mappinganalysis predicts the existence of a branch of period one solutionsemanating from the grazing point. In addition, the analysis predictsthat one eigenvalue of a suitable Poincaré map approaches minus infinity as the varied parameter approaches its grazing value. The second, more general, method is then applied to a model of cellcycle mitosis at two grazing points corresponding to trajectoriesthat have tangential contact with the discontinuity surface used totrigger a halving of the cell mass. Again, the analysis predicts abranch of period one solutions emanating from the grazing point, and an eigenvalue whose magnitude grows without bounds as the grazingpoint is approached in parameter space.In the case of both the AFM model and the cell cycle model, predictions produced using the discontinuity mapping analysis are shown to agreewith results produced using numerical continuation. In addition, we remark on the limitations of the discontinuity mapping analysis and provide suggestions for future work. Specifically, we identify a familyof attractors that exist for the AFM model and that warrant further investigation.