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 R
N , 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