Abstract

We consider analytic planar differential systems with a limit cycle of multiplicity exactly m. We are here considering the classical notion of multiplicity of a limit cycle for a planar differential system. We show that such a limit cycle exists if, and only if, there exists an analytic m-solution f0(x,y)=0 of the differential system such that the limit cycle is contained in its zero--set. The classical way to compute the multiplicity and stability of a limit cycle consists on studying the Poincaré return map. Using our result, we are able to give an alternative way to study the multiplicity and stability of a limit cycle, which can be effectively computed in some cases.
This talk is based on a work still in progress with Armengol Gasull (UAB) and Jaume Gine (UdL).