Higher order interactions lead to "reluctant" synchrony breaking

To model dynamical systems on networks with higher order (non-pairwise) interactions, we recently introduced a new class of ODEs on hypernetworks. Here we consider one-parameter synchrony breaking bifurcations in such ODEs. We call a synchrony breaking steady state branch ``reluctant'' if it is tangent to a synchrony space, but does not lie inside it. We prove that reluctant synchrony breaking is ubiquitous in hypernetwork systems, by constructing a large class of examples that support it. We also give an explicit formula for the order of tangency to the synchrony space of a reluctant steady state branch.