Robust cycles unfolding from conservative bifocal homoclinic orbits

We prove that suspended robust heterodimensional cycles and suspended robust homoclinic tangencies can be found arbitrarily close to any non-degenerate bifocal homoclinic orbit of a Hamiltonian vector field. In order to achieve this result, we show that any diffeomorphism with a saddle-node periodic point, which has both quasi-transversal and tangential strong homoclinic intersections, can be approximated by diffeomorphisms with both, robust homoclinic tangencies and a robust heterodimensional cycles. As explained in the paper, Hamiltonian vector fields with non-degenerate bifocal homoclinic orbits are exhibited in the limit family of any generic unfolding of a four-dimensional nilpotent singularity of codimension four. Supported by the achieved results, we conjecture that suspended robust cycles can be generically unfolded from such singularities.