An example of a wild strange attractor

It is proved that in the space of $C^r$-smooth ($r\geqslant 4$) flows in $\mathbb R^n$ ($n\geqslant 4$) there exist regions filled by systems that each have an attractor (here: a completely stable chain-transitive closed invariant set) containing a non-trivial basic hyperbolic set together with its unstable manifold, which has points of non-transversal intersection with the stable manifold. A construction is given for such a wild attractor containing an equilibrium state of saddle-focus type.