SINGULAR PERTURBATIONS ARISING IN HILBERTS 16TH PROBLEM FOR QUADRATICVECTOR-FIELDS

A solution to the second part of Hilbert's 16th problem consists of fi nding out (or proving the existence of) an upper bound to the number o f limit cycles in the family of polynomial planar vector fields. In th is article, we indicate a way to tackle the singular perturbation prob lems that have to be studied in the quadratic case. In particular, for perturbations from the family (lambda x - y)(x + 1) partial derivativ e/partial derivative + (x + lambda y)(x + 1) partial derivative/partia l derivative y, we prove that the cyclicity of certain limit periodic sets is bounded by 1. The proposed method is applicable in any multi-p arameter bifurcation problem and forms an extension to the known techn ique of ''significant degeneration'', i.e. the rescaling of parameters by means of different weights.