  We prove that in the space of $C^r$ maps $(r=2,\ldots,\infty,\omega)$ of a
smooth manifold of dimension at least 4 there exist open regions where maps
with infinitely many corank-2 homoclinic tangencies of all orders are dense.
The result is applied to show the existence of maps with universal
two-dimensional dynamics, i.e. maps whose iterations approximate the dynamics
of every map of a two-dimensional disk with an arbitrarily good accuracy. We
show that maps with universal two-dimensional dynamics are $C^r$-generic in the
regions under consideration.
