  Many networked systems are governed by non-pairwise interactions between
nodes. The resulting higher-order interaction structure can then be encoded by
means of a hypernetwork. In this paper we consider dynamical systems on
hypernetworks by defining a class of admissible maps for every such
hypernetwork. We explain how to classify robust cluster synchronisation
patterns on hypernetworks by finding balanced partitions, and we generalize the
concept of a graph fibration to the hypernetwork context. We also show that
robust synchronisation patterns are only fully determined by polynomial
admissible maps of high order. This means that, unlike in dyadic networks,
cluster synchronisation on hypernetworks is a higher-order, i.e., nonlinear
effect. We give a formula, in terms of the order of the hypernetwork, for the
degree of the polynomial admissible maps that determine robust synchronisation
patterns. We also demonstrate that this degree is optimal by investigating a
class of examples. We conclude by displaying how this effect may cause
remarkable synchrony breaking bifurcations that occur at high polynomial
degree.
