Center Conditions and Cyclicity for Generic Planar Polynomial Vector Fields

TL;DR

This paper investigates the center-focus problem for planar polynomial vector fields, providing new bounds on the Bautin ideal and cyclicity using Lyapunov functions, advancing understanding of local phase portrait behavior.

Contribution

It introduces novel bounds on the Bautin ideal and cyclicity for generic polynomial vector fields of any degree, based on Lyapunov function methods.

Findings

Established upper bounds on the Bautin ideal size.

Provided bounds on the cyclicity of polynomial vector fields.

Applied Lyapunov functions to analyze center conditions.

Abstract

We study the center-focus problem for planar polynomial vector fields, which can be viewed as a local version of Hilbert's 16th problem. Based on a Lyapunov function approach, we establish novel results regarding the center-focus conditions. More precisely, under generic conditions, and for any degree of a polynomial vector field, we find an upper bound on the size of the Bautin ideal generated by the Lyapunov constants. This also provides an upper bound on the cyclicity of the systems we consider.