{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T18:03:12Z","timestamp":1773252192190,"version":"3.50.1"},"reference-count":30,"publisher":"World Scientific Pub Co Pte Ltd","issue":"05","funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/J001317\/2"],"award-info":[{"award-number":["EP\/J001317\/2"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2016,5]]},"abstract":"<jats:p>When a flow suffers a discontinuity in its vector field at some switching surface, the flow can cross through or slide along the surface. Sliding along the switching surface can be understood as the flow along an invariant manifold inside a switching layer. It turns out that the usual method for finding sliding modes \u2014 the Filippov convex combination or Utkin equivalent control \u2014 results in a degeneracy in the switching layer whenever the flow is tangent to the switching surface from both sides. We derive the general result and analyze the simplest case here, where the flow curves parabolically on either side of the switching surface (the so-called fold\u2013fold or two-fold singularities). The result is a set of zeros of the fast switching flow inside the layer, which is structurally unstable to perturbation by terms nonlinear in the switching parameter, terms such as [Formula: see text] [where the superscript does mean \u201csquared\u201d]. We provide structurally stable forms, and show that in this form the layer system is equivalent to a generic singularity of a two timescale system. Finally we show that the same degeneracy arises when a discontinuity is smoothed using standard regularization methods.<\/jats:p>","DOI":"10.1142\/s0218127416500875","type":"journal-article","created":{"date-parts":[[2016,5,31]],"date-time":"2016-05-31T06:25:31Z","timestamp":1464675931000},"page":"1650087","source":"Crossref","is-referenced-by-count":16,"title":["Hidden Degeneracies in Piecewise Smooth Dynamical Systems"],"prefix":"10.1142","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3325-7211","authenticated-orcid":false,"given":"Mike R.","family":"Jeffrey","sequence":"first","affiliation":[{"name":"Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UB, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2016,5,30]]},"reference":[{"key":"S0218127416500875BIB001","doi-asserted-by":"publisher","DOI":"10.1209\/0295-5075\/89\/17003"},{"key":"S0218127416500875BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2013.07.015"},{"key":"S0218127416500875BIB003","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/23\/3\/017"},{"key":"S0218127416500875BIB004","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/24\/5\/014"},{"key":"S0218127416500875BIB006","volume-title":"Piecewise-Smooth Dynamical Systems: Theory and Applications","author":"di Bernardo M.","year":"2008"},{"key":"S0218127416500875BIB007","doi-asserted-by":"crossref","DOI":"10.1201\/9781498701822","volume-title":"Sliding Mode Control","author":"Edwards C.","year":"1998"},{"key":"S0218127416500875BIB008","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(79)90152-9"},{"key":"S0218127416500875BIB009","doi-asserted-by":"publisher","DOI":"10.1137\/120869134"},{"key":"S0218127416500875BIB010","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-7793-9"},{"key":"S0218127416500875BIB011","doi-asserted-by":"publisher","DOI":"10.1137\/08073113X"},{"key":"S0218127416500875BIB012","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2014.02.003"},{"key":"S0218127416500875BIB013","first-page":"1650068-1","volume":"26","author":"Jeffrey M. 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