Regularisation of isolated codimension-2 discontinuity sets

TL;DR

This paper develops a framework for analyzing piecewise-smooth dynamical systems with isolated codimension-2 discontinuity sets, extending traditional methods to higher degeneracy cases like Coulomb friction.

Contribution

It introduces a regularization and blowup approach for codimension-2 discontinuities, generalizing Filippov's framework to more complex systems.

Findings

Provides a general framework for codimension-2 discontinuities

Derives local dynamics for specific classes of problems

Extends Filippov sliding and crossing concepts to higher codimension

Abstract

Work on standard piecewise-smooth (PWS) dynamical systems, with codimension-1 discontinuity sets, relies on the Filippov framework, which does not always readily generalise to systems with higher codimension discontinuities. These higher order degeneracies occur in applications, with spatial Coulomb friction being the prominent example. In this work, we consider PWS systems with isolated codimension-2 discontinuity sets using regularization and blowup to study the dynamics. We present a general framework and give specific results for the local dynamics in a class of problems, and generalising Filippov sliding, crossing and sliding vector fields.