Projective limit cycles

In this article we study projective cycles in $\mathbb P^2_\mathbb R$. Our inspiring example is the Jouanolou foliation of odd degree which has a hyperbolic projective limit cycle. We prove that only odd degree foliations may have projective cycles and that foliations with exactly one real simple singularity have a projective cycle. We also prove that after a perturbation of a generic Hamiltonian foliation with a projective cycle, we have a projective limit cycle if and only if the perturbation is not Hamiltonian.