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Bifurcations in Nonsmooth Dynamical Systems
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2008 (English)In: SIAM Review, ISSN 0036-1445, E-ISSN 1095-7200, Vol. 50, no 4, 629-701 p.Article, review/survey (Refereed) Published
Abstract [en]

A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to defining a discontinuity-induced bifurcation (DIB) as a nontrivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local are considered, that is, bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows. Three classes of systems are considered, involving either state jumps, jumps in the vector field, or jumps in some derivative of the vector field. A rich array of dynamics are revealed, involving the sudden creation or disappearance of attractors, jumps to chaos, bifurcation diagrams with sharp corners, and cascades of period adding. For each kind of bifurcation identified, where possible, a kind of "normal form" or discontinuity mapping (DM) is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators, impact oscillators, DC-DC converters, and problems in control theory.

Place, publisher, year, edition, pages
2008. Vol. 50, no 4, 629-701 p.
Keyword [en]
nonsmooth, dynamical system, bifurcation, discontinuity, piecewise, equilibria, limit cycles, border-collision bifurcations, piecewise-smooth systems, stick-slip, vibrations, grazing bifurcations, impact oscillators, linear-oscillator, sliding bifurcations, mechanical systems, buck, converter, dry friction
Identifiers
URN: urn:nbn:se:kth:diva-17974DOI: 10.1137/050625060ISI: 000260905600001Scopus ID: 2-s2.0-47749117392OAI: oai:DiVA.org:kth-17974DiVA: diva2:336019
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved

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CiteExportLink to record
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  • apa
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