A universal method to approach the Poincar\'e center problem

TL;DR

This paper introduces a universal approach to the Poincaré center problem by proving the existence of Laurent inverse integrating factors for analytic centers and providing a method to identify parameter conditions for centers in polynomial vector fields.

Contribution

It establishes a general theoretical framework linking Laurent inverse integrating factors to the Poincaré center problem and offers a procedure to determine parameter constraints for centers.

Findings

Every analytic center admits a Laurent inverse integrating factor in weighted polar coordinates.

If no local curves of zero angular speed exist, the Poincaré map is analytic.

The method can identify parameter conditions characterizing centers in polynomial vector fields.

Abstract

We address the classical (degenerate or non-degenerate) center problem posed by Poincar\'e in the 19th century for monodromic singularities of analytic families of planar vector fields $\mathcal{X}$. We prove that every analytic center admits a Laurent inverse integrating factor $V$ in weighted polar coordinates. Moreover, we show that when $\mathcal{X}$ has no local curves of zero angular speed, the Poincar\'e map is analytic, and if, in addition, $V$ has an essential singularity, then the singularity of $\mathcal{X}$ is a center. Based on this result, we derive a theoretical procedure to determine parameter constraints within the family that characterize any center of a polynomial vector field. Applications to nontrivial families that have resisted other methods are also provided.