Cyclicity of a simple focus via the vanishing multiplicity of inverse integrating factors

Abstract

First we provide new properties about the vanishing multiplicity of the inverse integrating factor of a planar analytic differential system at a focus. After we use this vanishing multiplicity for studying the cyclicity of some simple foci of several classes of planar analytic differential systems.

The first and third authors are partially supported by a MINECO grant number MTM2011-22877 and by a CIRIT grant number 2009SGR 381. The second author is partially supported by a MINECO/FEDER grant number MTM2008-03437, by an AGAUR grant number 2009SGR 410, by ICREA Academia and by FP7-PEOPLE-2012-IRSES-316338 and 318999.