Bifurcations in Nonsmooth Dynamical Systems

M di Bernardo, C J Budd, A R Champneys, P Kowalczyk, A B Nordmark, G O Tost, P T Piiroinen

Research output: Contribution to journalArticle

198 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)629-701
Number of pages73
JournalSiam Review
Volume50
Issue number4
DOIs
Publication statusPublished - 2008

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Bifurcation (mathematics)
Dynamical systems
Jump
Bifurcation
Dynamical system
Discontinuity
DC-DC converters
Control theory
Chaos theory
Vector Field
Friction
Derivatives
DC-DC Converter
Limit Set
Bifurcation Diagram
Control Theory
Interaction
Limit Cycle
Normal Form
Codimension

Cite this

di Bernardo, M., Budd, C. J., Champneys, A. R., Kowalczyk, P., Nordmark, A. B., Tost, G. O., & Piiroinen, P. T. (2008). Bifurcations in Nonsmooth Dynamical Systems. Siam Review, 50(4), 629-701. https://doi.org/10.1137/050625060

Bifurcations in Nonsmooth Dynamical Systems. / di Bernardo, M; Budd, C J; Champneys, A R; Kowalczyk, P; Nordmark, A B; Tost, G O; Piiroinen, P T.

In: Siam Review, Vol. 50, No. 4, 2008, p. 629-701.

Research output: Contribution to journalArticle

di Bernardo, M, Budd, CJ, Champneys, AR, Kowalczyk, P, Nordmark, AB, Tost, GO & Piiroinen, PT 2008, 'Bifurcations in Nonsmooth Dynamical Systems', Siam Review, vol. 50, no. 4, pp. 629-701. https://doi.org/10.1137/050625060
di Bernardo M, Budd CJ, Champneys AR, Kowalczyk P, Nordmark AB, Tost GO et al. Bifurcations in Nonsmooth Dynamical Systems. Siam Review. 2008;50(4):629-701. https://doi.org/10.1137/050625060
di Bernardo, M ; Budd, C J ; Champneys, A R ; Kowalczyk, P ; Nordmark, A B ; Tost, G O ; Piiroinen, P T. / Bifurcations in Nonsmooth Dynamical Systems. In: Siam Review. 2008 ; Vol. 50, No. 4. pp. 629-701.
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