"We study the existence of of transverse homoclinic points in two settings of area preserving maps The first one corresponds to the so-called generalized standard maps having a weak saddle and the second one deals with the hyperbolic islands appearing near elliptic points. We prove an asymptotic formula measuring the splitting of separatrices which turns out to be exponentially small in some singular parameter. Our methods are constructive and suitable to be implemented by computer assisted proofs.

This a join work with I. de Blasi, D. Gil, M. Guardia, P. Martin and T. Seara"