The Painlevé paradox and geometric singular perturbation theory

当一支粉笔在黑板上拖动时，粉笔会颤抖，有时还会发出高亢的尖叫声，这是一种常见且通常不愉快的经历。这种行为与Painlevé悖论（Painlevé 1905）有关。从物理上讲，接触点的摩擦力矩足够高，可以克服刚性表面的阻力，这意味着粉笔应该进入黑板。由于这不可能发生，粉笔跳跃。最近发现，悖论可以发生在机器人操作器，在那里它影响可控性，加上一些优秀的实验证据（赵等. 2008），激起了强烈的现代兴趣，这个老问题。这个项目旨在处理一些悬而未决的问题有关的潘勒维悖论。对于在粗糙表面上滑动的细长杆，刚体方程中的不确定性或不一致性表示建模失败。必须放松假定的刚性。Hogan & Kristiansen（2016）已经表明，当接触点有一定的顺应性时，就会出现物理上看到的行为（例如粉笔瞬间跳离棋盘）。这种顺应性（或正则化）是非常小的，由此产生的方程导致了一个慢-快系统，其中有丰富的现有数学理论。然而，为了捕捉刚体的分段光滑（PWS）极限，我们需要几何奇异摄动理论，在这方面已经有了很多进展。最近开发的“blowup方法”（Krupa & Szmolyan 2001）能够在可修改为动力系统理论中经典约化方法的框架中识别与正则化相关的尺度。该项目旨在解决的一个突出问题是Dupont和Yamajako（1997）提出的两个粗糙表面之间的杆。其目的是建立在Hogan & Kristiansen（2016）的框架上，其中放松了刚体动力学的基本建模假设，并通过正则化将PWS系统替换为平滑系统。然后将blowup方法应用于问题的分析。

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.