The period of the limit cycle bifurcating from a persistent polycycle

Abstract

We consider smooth families of planar polynomial vector fields {X mu}mu is an element of Lambda, where Lambda is an open subset of RN, for which there is a hyperbolic polycycle Gamma that is persistent (i.e., such that none of the separatrix connections is broken along the family). It is well known that in this case the cyclicity of Gamma at mu 0 is zero unless its graphic number r(mu 0) is equal to one. It is also well known that if r(mu 0) = 1 (and some generic conditions on the return map are verified), then the cyclicity of Gamma at mu 0 is one, i.e., exactly one limit cycle bifurcates from Gamma. In this paper we prove that this limit cycle approaches Gamma exponentially fast and that its period goes to infinity as 1/|r(mu)-1| when mu -> mu 0. Moreover, we prove that if those generic conditions are not satisfied, although the cyclicity may be exactly 1, the behavior of the period of the limit cycle is not determined.