  We study supercritical $O(d)$-equivariant biharmonic maps with a focus on $d
= 5$, where $d$ is the dimension of the domain. We give a characterisation of
non-trivial equivariant biharmonic maps from $\mathbf{R}^5$ into $S^5$ as
heteroclinic orbits of an associated dynamical system. Moreover, we prove the
existence of such non-trivial equivariant biharmonic maps. Finally, in stark
contrast to the harmonic map analogue, we show the existence of an equivariant
biharmonic map from $B^5(0, 1)$ into $S^5$ that winds around $S^5$ infinitely
many times.
