On the cyclicity of weight-homogeneous centers

TL;DR

This paper establishes an upper bound on the number of limit cycles bifurcating from the period annulus of weight-homogeneous centers under polynomial perturbations, with applications to systems with nilpotent centers.

Contribution

It provides a new upper bound for limit cycles in weight-homogeneous centers and applies it to systems lacking a meromorphic first integral.

Findings

Derived an upper bound for bifurcating limit cycles.

Applied the bound to systems with nilpotent centers.

Enhanced understanding of cyclicity in polynomial systems.

Abstract

Let W be a weight-homogeneous planar polynomial differential system with a center. We find an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation. We apply this result to a particular family of planar polynomial systems having a nilpotent center without meromorphic first integral.