  We study the effect of structured higher-order interactions on the collective
behavior of coupled phase oscillators. By combining a hypergraph generative
model with dimensionality reduction techniques, we obtain a reduced system of
differential equations for the system's order parameters. We illustrate our
framework with the example of a hypergraph with hyperedges of sizes 2 (links)
and 3 (triangles). For this case, we obtain a set of 2 coupled nonlinear
algebraic equations for the order parameters. For strong values of coupling via
triangles, the system exhibits bistability and explosive synchronization
transitions. We find conditions that lead to bistability in terms of hypergraph
properties and validate our predictions with numerical simulations. Our results
provide a general framework to study synchronization of phase oscillators in
hypergraphs, and they can be extended to hypergraphs with hyperedges of
arbitrary sizes, dynamic-structural correlations, and other features.
