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

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 are developed and applied to this AFM model. The discontinuity mapping analysis predicts the existence of a branch of period one solutions emanating from the grazing point. In addition, the analysis predicts that 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 cell cycle mitosis at two grazing points corresponding to trajectories that have tangential contact with the discontinuity surface used to trigger a halving of the cell mass. Again, the analysis predicts a branch of period one solutions emanating from the grazing point, and an eigenvalue whose magnitude grows without bounds as the grazing point 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 agree with 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 family of attractors that exist for the AFM model and that warrant further investigation.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2010

Author dc:creator

• Katzenbach, Michael T.

Contributors dc:contributor

• Dankowicz, Harry

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

Statement dc:rights

• © 2010 by MICHAEL KATZENBACH. All rights reserved.

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/16923

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/16923