Hilbert′s 16th Problem for Quadratic Vector Fields

We discuss here a systematic approach towards a positive answer to Hilbert′s 16th problem for quadratic systems, namely the existence of a uniform bound for the number of limit cycles of a quadratic system. The method is the following: describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity. In this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features. We also indicate how to find or where to find concrete examples of these limit periodic sets.