Discontinuity-induced bifurcations in models of mechanical contact, capillary adhesion, and cell division: A common framework

Abstract

This paper collects four distinct instances of grazing contact of a periodic trajectory in a hybrid dynamical system under a common abstract framework and establishes selected general properties of the associated near-grazing dynamics. In particular, it is shown that for critical choices of parameter values, commonly used physical models of rigid or compliant mechanical contact, capillary adhesion, and cell division satisfy the conditions required by the general framework. The paper relies on the well-known discontinuity-mapping formalism. In contrast to previous treatments, the proposed abstract framework more clearly establishes the origin of the large state-space stretching in the initial (and possibly only) step of the construction of a discontinuity mapping. It further highlights the nonuniqueness in the formulation of the discontinuity mapping and its connection to the choice of a locally smooth map with which the discontinuity mapping is composed to describe the near-grazing dynamics. The analysis is illustrated with examples from tapping-mode atomic force microscopy in the presence of thin fluid layers on the sample and the probe tip and from the study of protein activity during a eukaryotic cell cycle.