BIFURCATIONS OF CUSPIDAL LOOPS

A cuspidal loop (X, p, Gamma) for a planar vector field X consists of a homoclinic orbit Gamma through a singular point p, at which X has a nilpotent cusp. This is the simplest non-elementary singular cycle (or graphic) in the sense that its singularities are not elementary (i.e. hyperbolic or semihyperbolic). Cuspidal loops appear persistently in three-parameter families of planar vector fields. The bifurcation diag rams of unfoldings of cuspidal loops are studied here under mild gener icity hypotheses: the singular point p is of Bogdanov-Takens type and the derivative of the first return map along the orbit Gamma is differ ent from 1. An analytic and geometric method based on the blowing up f or unfoldings is proposed here to justify the two essentially differen t models for generic bifurcation diagrams presented in this work. This method can be applied for the study of a large class of complex multi parametric bifurcation problems involving non-elementary singularities , of which the cuspidal loop is the simplest representative. The proof s are complete in a large part of parameter space and can be extended to the complete parameter space module a conjecture on the time functi on of certain quadratic planar vector fields. In one of the cases we c an prove that the generic cuspidal loop bifurcates into four limit cyc les that art: close to it in the Hausdorff sense.