  Non-reciprocal interactions play a crucial role in many social and biological
complex systems. While directionality has been thoroughly accounted for in
networks with pairwise interactions, its effects in systems with higher-order
interactions have not yet been explored as deserved. Here, we introduce the
concept of M-directed hypergraphs, a general class of directed higher-order
structures, which allow to investigate dynamical systems coupled through
directed group interactions. As an application we study the synchronization of
nonlinear oscillators on 1-directed hypergraphs, finding that directed
higher-order interactions can destroy synchronization, but also stabilize
otherwise unstable synchronized states.

Adaptation and Self-Organizing Systems;Dynamical Systems;Chaotic Dynamics;Pattern Formation and Solitons