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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 ofnon-trivial equivariant biharmonic maps from $\mathbf{R}^5$ into $S^5$ asheteroclinic orbits of an associated dynamical system. Moreover, we prove theexistence of such non-trivial equivariant biharmonic maps. Finally, in starkcontrast to the harmonic map analogue, we show the existence of an equivariantbiharmonic map from $B^5(0, 1)$ into $S^5$ that winds around $S^5$ infinitelymany times.

Supercritical equivariant biharmonic maps from $\mathbf{R}^5$ into $S^5$