TY - JOUR

T1 - Bifurcations in Nonsmooth Dynamical Systems

AU - di Bernardo, M

AU - Budd, C J

AU - Champneys, A R

AU - Kowalczyk, P

AU - Nordmark, A B

AU - Tost, G O

AU - Piiroinen, P T

N1 - ID number: 000260905600001

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=47749117392&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1137/050625060

U2 - 10.1137/050625060

DO - 10.1137/050625060

M3 - Article

SN - 0036-1445

VL - 50

SP - 629

EP - 701

JO - Siam Review

JF - Siam Review

IS - 4

ER -