The Painlevé paradox and geometric singular perturbation theory

当粉笔在黑板上拖拽时，粉笔会抖动，有时会发出尖锐的尖叫声，这是一种常见的、通常令人不快的经历。这种行为与painlev<e:1>悖论（painlev<e:1> 1905）有关。从物理上讲，接触点处的摩擦力矩足够高，足以克服刚性表面的阻力，这意味着粉笔应该进入黑板。既然这种情况不可能发生，粉笔就会跳起来。最近发现，悖论可能发生在机器人操纵器中，它会影响可控性，再加上一些优秀的实验证据（Zhao et al. 2008），引起了现代人对这个老问题的强烈兴趣。这个项目旨在处理一些与painlevevle悖论相关的突出问题。对于在粗糙表面上滑动的细长杆，刚体方程的不确定性或不一致性表示建模失败。假定的刚性必须放松。Hogan和Kristiansen（2016）表明，当接触点有一定的依从性时，就会出现像身体上看到的行为（例如粉笔从板子上瞬间跳开）。这种遵从性（或正则化）是非常小的，所得到的方程导致了一个有丰富的现有数学理论的慢-快系统。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，其中有许多进展。最近发展的“放大法”（Krupa & Szmolyan 2001）能够识别与正则化相关的尺度，其框架可修改为动力系统理论中的经典约简方法。该项目旨在解决的一个突出问题是杜邦和Yamajako（1997）提出的两个粗糙表面之间的杆。目的是建立在Hogan和Kristiansen（2016）的框架之上，其中放松了刚体动力学的潜在建模假设，并通过正则化将PWS系统替换为平滑系统。然后将放大法用于问题的分析。

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.