Monodromic singularities without curves of zero angular speed

Abstract

We consider planar analytic vector fields $ \mathcal{X} $ having a monodromic singular point with Poincaré map $ \Pi $. We use the fact that there always exists a real analytic invariant curve $ F = 0 $ of $ \mathcal{X} $ in a neighborhood of that singularity. We find some relations between $ \Pi $ and $ F $ that can be used to determine new conditions that guarantee the analyticity of $ \Pi $ at the singularity. In the special case that $ F $ becomes an inverse integrating factor of $ \mathcal{X} $, we rediscover formulas obtained previously by other methods. Applications to the center-focus problem and also to vector fields with degenerate infinity are given.

Keywords

• center
• periodic orbits
• poincaré map
• invariant curve