Singular perturbations arising in Hilbert's 16th problem for quadratic vector fields

Abstract

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