Extension of catastrophe theory to Dulac unfoldings

We consider a perturbation of a Hamiltonian planar vector field. The bifurcation set of limit cycles is studied. If the vector field is defined in an annulus, limit cycles are in one to one correspondence to the zeros of a polynomial. Catastrophe theory is relevant for the study of the ensuing bifurcations. A similar conclusion is obtained if the Hamiltonian vector field has a center. In these two cases the geometry of the bifurcation set is polynomial. We focus on the case where the Hamiltonian is defined near a homoclinic loop of a hyperbolic saddle. The study now reduces to the zeros of a displacement function that involves perturbations of Dulac unfoldings. The latter expand on a logarithmic scale. In this note, we show that, after a blow up in the parameter space, the geometry of the bifurcation set of limit cycles is again polynomial in an exponentially small domain.