Codimension 2 bifurcations of double homoclinic loops

In this work, the double-homoclinic-loop bifurcations in four dimensional vector fields are investigated by setting up local coordinates near the double homoclinic loops. We get the existence, uniqueness and incoexistence of the large 1-hom and large 1-per orbit, and their corresponding existence regions are located. Furthermore, the inexistence of the large 2-hom and large 2-per orbit are also demonstrated.