Bifurcations of cuspidal loops

Abstract

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 diagrams of unfoldings of cuspidal loops are studied here under mild genericity hypotheses: the singular point p is of Bogdanov-Takens type and the derivative of the first return map along the orbit Gamma is different from 1. An analytic and geometric method based on the blowing up for unfoldings is proposed here to justify the two essentially different models for generic bifurcation diagrams presented in this work. This method can be applied for the study of a large class of complex multiparametric bifurcation problems involving non-elementary singularities, of which the cuspidal loop is the simplest representative. The proofs are complete in a large part of parameter space and can be extended to the complete parameter space module a conjecture on the time function of certain quadratic planar vector fields. In one of the cases we can prove that the generic cuspidal loop bifurcates into four limit cycles that art: close to it in the Hausdorff sense.