Cyclicity of period annuli and principalization of Bautin ideals

TL;DR

This paper proves that the maximum number of limit cycles bifurcating from a period annulus remains the same under multi-parameter and one-parameter deformations, and provides bounds for homoclinic saddle loops.

Contribution

It establishes the equivalence of cyclicity in multi-parameter and one-parameter deformations for analytic vector fields with finite cyclicity.

Findings

Maximum cyclicity is preserved under multi-parameter deformations.

Provides bounds for cyclicity of homoclinic saddle loops.

Shows cyclicity equivalence under different deformation parameters.

Abstract

We prove that the maximal number of limit cycles which bifurcate from an open period annulus under a given multi-parameter analytic deformation of a given analytic vector field is the same as in an appropriate one-parameter analytic deformation of the field, provided that this cyclicity is finite. Along the same lines we give also a bound of the cyclicity of homoclinic saddle loops.