New solutions of the Poincaré Center Problem in degree 3

Abstract

Let $\omega$ be a plane autonomous system and C its configuration of algebraic integral curves. If the singularities of C are quasi-homogeneous we we present new criteria that guarantee Darboux integrability. We use this to construct previously unknown components of the center variety in degree 3.

Includes 5 figures; updated references; revised section on the inverse problem; the role of singularities is emphasized throughout the paper