The two-fold singularity of nonsmooth flows: Leading order dynamics in n-dimensions

Alessandro Colombo, Mike R. Jeffrey*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)

11 Citations (Scopus)
255 Downloads (Pure)

Abstract

A discontinuity in a system of ordinary differential equations can create allow that slides along the discontinuity locus. Prior to sliding, the flow may have collapsed onto the discontinuity, making the reverse flow non-unique, as happens when dry-friction causes objects to stick. Alternatively, a flow may slide along the discontinuity before escaping it at some indeterminable time, implying non-uniqueness in forward time. At a two-fold singularity these two behaviours are brought together, so that a single point may have multiple possible futures as well as histories. Two-folds are a generic consequence of discontinuities in three or more dimensions, and play an important role in both local and global dynamics. Despite this, until now nothing was known about two-fold singularities in systems of more than 3 dimensions. Here, the normal form of the two-fold is extended to higher dimensions, where we show that much of its lower dimensional dynamics survives. (C) 2013 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)1-10
Number of pages10
JournalPhysica D: Nonlinear Phenomena
Volume263
Early online date6 Aug 2013
DOIs
Publication statusPublished - 15 Nov 2013

Keywords

  • Filippov
  • Singularity
  • Two-fold
  • Non-determinism
  • Nonsmooth
  • Discontinuous
  • DISCONTINUOUS VECTOR-FIELDS
  • PIECEWISE-SMOOTH FLOWS
  • BIFURCATIONS
  • SYSTEMS
  • STABILITY

Projects

Cite this