

Artikel
Twoparameter discontinuityinduced bifurcations of limit cycles: classification and open problems
Författare: 
Kowalczyk, P., di Bernardo, M., Champneys, A. R. , Hogan, S. J., Homer, M. E., Piiroinen, P T, Kuznetsov, Yu. A., Nordmark, A. 
Dokumenttyp: 
Artikel 
Tillstånd: 
Publicerad 
Tidskrift: 
International Journal of bifurcation and chaos . 
Volym: 
16(3)
601629 
År: 
2006 
AbstractThis paper proposes a strategy for the classification of codimensiontwo discontinuityinduced bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (also known as Cbifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a nongeneric way, such as grazing contact. Several such codimensionone events have recently been identified, causing for example, periodadding or sudden onset of chaos. Here, the focus is on codimensiontwo grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincaré map from a neighborhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimensiontwo 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 codimensiontwo scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

