High order homoclinic tangencies of corank 2

We prove that in the space of $C^r$ maps $(r=2,\dots ,\mathrm{\infty },\omega )$ 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.