The cyclicity of hyperbolic hemicycles

We consider families of planar polynomial vector fields of degree $n$ and study the cyclicity of a type of unbounded polycycle~$\Gamma$ called hemicycle. Compactified to the Poincar\'e disc,~$\Gamma$ consists of an affine straight line together with half of the line at infinity and has two singular points, which are hyperbolic saddles located at infinity. We prove four main results. Theorem A deals with the cyclicity of~$\Gamma$ when perturbed without breaking the saddle connections. For the other results we consider the case $n=2$. More concretely they are addressed to the quadratic integrable systems belonging to the class $Q_3^R$ and having two hemicycles, $\Gamma_u$ and $\Gamma_\ell$, surrounding each one a center. Theorem B gives the cyclicity of $\Gamma_u$ and $\Gamma_\ell$ when perturbed inside the whole family of quadratic systems. In Theorem C we study the number of limit cycles bifurcating simultaneously from $\Gamma_u$ and $\Gamma_\ell$ when perturbed as well inside the whole family of quadratic systems. Finally, in Theorem D we show that for three specific cases there exists a simultaneous alien limit cycle bifurcation from $\Gamma_u$ and $\Gamma_\ell$.