Synchronization and Pinning Control on Circulating Directed Hypergraphs

A pressing open problem in the study of interconnected dynamical systems is how to gauge the impact higher-order interactions have on the onset of collective behaviors. In this paper, we describe multi-body interactions among systems using the formalism of directed hypergraphs. Introducing the novel concept of circulating hypergraphs, we formulate the synchronization and pinning control problems in hypernetwork dynamical systems. Extending the notion of balanced digraphs to the case of directed hypergraphs, we provide for the first time global conditions for both the spontaneous emergence of synchronization, as well as for controlling the network to a desired trajectory set by the pinner, in the presence of nonlinear, time-varying coupling protocols. The effectiveness of the proposed approach is illustrated by select testbed examples.