The Painlevé paradox and geometric singular perturbation theory

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When a piece of chalk is dragged across a blackboard, it is a matter of common, and usually unpleasant, experience that the chalk can judder and sometimes emit a high-pitched squeal. Such behaviour is related to the Painlevé paradox (Painlevé 1905). Physically, the frictional torque at the point of contact is high enough to overcome the resistance of the rigid surface, implying that the chalk should enter the blackboard. Since this cannot happen, the chalk jumps.The recent discovery that the paradox can occur in robotic manipulators, where it effects controllability, together with some excellent experimental evidence (Zhao et al. 2008), have provoked strong modern interest in this old problem.This project aims to deal with some outstanding issues relating to the Painlevé paradox. For a slender rod slipping on a rough surface, indeterminacy or inconsistency in the rigid body equations represent failures in modelling. The assumed rigidity must be relaxed. It has been shown by Hogan & Kristiansen (2016) that behaviour like that seen physically (e.g. instantaneous jumping of the chalk away from the board) arises when there is some compliance at the point of contact. This compliance (or regularization) is extremely small, and the resulting equations lead to a slow-fast system for which there is a wealth of existing mathematical theory. However, to capture the piecewise-smooth (PWS) limit of the rigid body, we need geometric singular perturbation theory, in which there have been many advances. The recently developed "blowup method" (Krupa & Szmolyan 2001) enables the identification of scales associated with the regularization, in a framework amendable to classical reduction methods in dynamical system theory. One outstanding problem that this project will aim to resolve was posed by Dupont & Yamajako (1997) of a rod between two rough surfaces. The aim is to build upon the framework in Hogan & Kristiansen (2016), where the underlying modelling assumptions of rigid body dynamics are relaxed and the PWS system is replaced by a smooth one through regularization. Then blowup will be used in the analysis of the problem.