The period of the limit cycle bifurcating from a persistent polycycle

Abstract

We consider smooth families of planar polynomial vector fields $\{X_\mu\}_{\mu\in\Lambda}$, where $\Lambda$ is an open subset of $\mathbb{R}^N$, 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 $\mu\to\mu_0.$ 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.