The period of the limit cycle bifurcating from a persistent polycycle

We consider smooth families of planar polynomial vector fields $\{X_{\mu }{\}}_{\mu \in \mathrm{\Lambda }}$, where $\mathrm{\Lambda }$ is an open subset of ${\mathbb{R}}^N$, for which there is a hyperbolic polycycle $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ at ${\mu }_0$ is one, i.e., exactly one limit cycle bifurcates from $\mathrm{\Gamma }$. In this paper we prove that this limit cycle approaches $\mathrm{\Gamma }$ exponentially fast and that its period goes to infinity as $1\mathrm{/}\mathrm{\mid }r(\mu )-1\mathrm{\mid }$ when $\mu \to {\mu }_0\mathrm{.}$ 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.