Two-parameter discontinuity-induced bifurcations of limit cycles: Classification and open problems
2006 (English)In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, ISSN 0218-1274, Vol. 16, no 3, 601-629 p.Article in journal (Refereed) Published
This paper proposes a strategy for the classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (also known as C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a nongeneric way, such as grazing contact. Several such codimension-one events have recently been identified, causing for example, period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincare map from a neighborhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the grazing cycle is itself degenerate (e.g. nonhyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that with discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.
Place, publisher, year, edition, pages
2006. Vol. 16, no 3, 601-629 p.
nonsmooth systems, grazing contact, bifurcations, border-collision bifurcations, smooth dynamical-systems, grazing bifurcations, impact oscillators, sliding bifurcations, buck converter, local analysis, dry friction, chaos, dimension
IdentifiersURN: urn:nbn:se:kth:diva-15712ISI: 000237900100006ScopusID: 2-s2.0-29144508275OAI: oai:DiVA.org:kth-15712DiVA: diva2:333754
QC 201005252010-08-052010-08-05Bibliographically approved