Bifurcations of cuspidal loops

A cuspidal loop for a planar vector field X consists of a homoclinic orbit 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 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 modulo 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 are close to it in the Hausdorff sense.

[1]  P. Bonckaert.  Conjugacy of vector fields respecting additional properties , 1997 .

[2]  Freddy Dumortier,et al.  Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics , 1996 .

[3]  Robert H. Roussarie.  Bifurcations of planar vector fields and Hilbert''''s 16th Problem , 1995 .

[4]  M. Jebrane,et al.  Finite cyclicity of nontrivial double loops , 1994 .

[5]  P. Joyal.  The Generalized Homoclinic Bifurcation , 1994 .

[6]  P. Joyal.  Regular ArticleThe Generalized Homoclinic Bifurcation , 1994 .

[7]  P. Bonckaert.  On the Continuous Dependence of the Smooth Change of Coordinates in Parametrized Normal Form Theorems , 1993 .

[8]  A method of desingularization for analytic two-dimensional vector field families , 1991 .

[9]  Y. Ilyashenko,et al.  Finitely-smooth normal forms of local families of diffeomorphisms and vector fields , 1991 .

[10]  J. Sotomayor,et al.  GENERIC 3-PARAMETER FAMILIES OF PLANAR VECTOR-FIELDS, UNFOLDINGS OF SADDLE, FOCUS AND ELLIPTIC-SINGULARITIES WITH NILPOTENT LINEAR PARTS , 1991 .

[11]  A. Mourtada,et al.  Cyclicite finie des polycycles hyperboliques de champs de vecteurs du plan mise sous forme normale , 1990 .

[12]  J. Ecalle,et al.  Finitude des cycles-limites et accelero-sommation de l’application de retour , 1990 .

[13]  J. Guckenheimer,et al.  On a four parameter family of planar vector fields , 1987 .

[14]  Freddy Dumortier,et al.  Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3 , 1987, Ergodic Theory and Dynamical Systems.

[15]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[16]  Floris Takens,et al.  Singularities of vector fields , 1974 .

[17]  Shlomo Sternberg,et al.  On the Structure of Local Homeomorphisms of Euclidean n-Space, II , 1958 .

[18]  H. Dulac,et al.  Sur les cycles limites , 1923 .