The period of the limit cycle bifurcating from a persistent polycycle

We consider smooth families of planar polynomial vector fields {Xµ}µ∈Λ, where Λ is an open subset of RN , for which there is a hyperbolic polycycle Γ 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 Γ at µ0 is zero unless its graphic number r(µ0) is equal to one. It is also well known that if r(µ0) = 1 (and some generic conditions on the return map are verified), then the cyclicity of Γ at µ0 is one, i.e., exactly one limit cycle bifurcates from Γ. In this paper we prove that this limit cycle approaches Γ exponentially fast and that its period goes to infinity as 1/|r(µ) − 1| when µ → µ0. 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.