Cyclicity of period annuli and principalization of Bautin ideals

Abstract

AbstractLet Π be an open period annulus of a plane analytic vector field X0. We prove that the maximal number of limit cycles which bifurcate from Π under a given multi-parameter analytic deformation Xλ of X0 is the same as in an appropriate one-parameter analytic deformation Xλ(ε), provided that this cyclicity is finite. Along the same lines, we also give a bound for the cyclicity of homoclinic saddle loops.