Bifurcation of degenerate homoclinics

We analyze the continuation and bifurcation of homoclinic orbits near a given degenerate homoclinic orbit. We show that the existence of such degenerate homoclinic orbit is a codimension three phenomenon, and that generically the set of parametervalues at which a nearby homoclinic exists forms a codimension one surface which shows a singularity of Whitney umbrella type at the critical parametervalue. The line of self-intersecting points of such surface corresponds to systems which have two nearby homoclinics.